Reconfiguration of Basis Pairs in Regular Matroids

In recent years, combinatorial reconfiguration problems have attracted great attention due to their connection to various topics such as optimization, counting, enumeration, or sampling. One of the most intriguing open questions concerns the exchange distance of two matroid basis sequences, a problem that appears in several areas of computer science and mathematics. In 1980, White proposed a conjecture for the characterization of two basis sequences being reachable from each other by symmetric exchanges, which received a significant interest also in algebra due to its connection to toric ideals and Gr'obner bases. In this work, we verify White’s conjecture for basis sequences of length two in regular matroids, a problem that was formulated as a separate question by Farber, Richter, and Shank and Andres, Hochst'attler, and Merkel. Most of previous work on White’s conjecture has not considered the question from an algorithmic perspective. We study the problem from an optimization point of view: our proof implies a polynomial algorithm for determining a sequence of symmetric exchanges that transforms a basis pair into another, thus providing the first polynomial upper bound on the exchange distance of basis pairs in regular matroids. As a byproduct, we verify a conjecture of Gabow from 1976 on the serial symmetric exchange property of matroids for the regular case.


Introduction
The basis exchange axiom of matroids implies that for any pair X, Y of bases, there exists a sequence of exchanges that transforms X into Y .White [56] studied the analogous problem for basis sequences instead of single bases.Let X = (X 1 , . . ., X k ) be a sequence of -not necessarily disjoint -bases of a matroid, and let e ∈ X i − X j and f ∈ X j − X i with 1 ≤ i < j ≤ k be such that both X i − e + f and X j + e − f are bases.Then, the sequence X ′ = (X 1 , . . ., X i−1 , X i − e + f, X i+1 , . . ., X j−1 , X j + e − f, X j+1 , . . ., X k ) is obtained from X by a symmetric exchange.Two sequences X and Y are called equivalent if Y can be obtained from X by a composition of symmetric exchanges.The question naturally arises: what is the characterization of two basis sequences being equivalent?
There is an easy necessary condition for the equivalence of two sequences X and Y: since a symmetric exchange does not change the number of bases in the sequence that contain a given element, the union of the members of X must coincide with the union of the members of Y as multisets.Motivated by this observation, X and Y are called compatible if |{i | e ∈ X i , i ∈ {1, . . ., k}}| = |{i | e ∈ Y i , i ∈ {1, . . ., k}}| for every e ∈ E, where E denotes the ground set of the matroid.White [56] conjectured that compatibility is not only necessary but also sufficient for two sequences to be equivalent.
Conjecture 1 (White).Two basis sequences X and Y of the same length are equivalent if and only if they are compatible.
Conjecture 1 received a significant interest also in algebra due to its connection to toric ideals and Gröbner bases, see [11] and [38,Chapter 13] for further details.However, despite all the efforts, White's conjecture remains open even for sequences of length two.In this special setting, Farber, Richter, and Shank [21] verified the statement for graphic and cographic matroids, and noted that their proof does not seem to generalize for regular matroids.Andres, Hochstättler and Merkel [4] formulated White's conjecture for regular matroids as a separate question, and noted that Seymour's decomposition theorem [46] might help to find a proof for it.
White's conjecture has no implications on the minimum number of exchanges needed to transform two equivalent sequences into each other, called their exchange distance.For basis pairs, Gabow [25] formulated the following problem, later stated as a conjecture by Wiedemann [57] and by Cordovil and Moreira [14], and posed as an open problem in Oxley's book [41,Conjecture 15.9.11].
Conjecture 2 (Gabow).Let X 1 and X 2 be disjoint bases of a rank-r matroid M .Then, the exchange distance of (X 1 , X 2 ) and (X 2 , X 1 ) is r.
Note that Conjecture 2 would imply Conjecture 1 for sequences of the form (X 1 , X 2 ) and (X 2 , X 1 ).Since the rank of the matroid is a trivial lower bound on the minimum number of exchanges needed to transform (X 1 , X 2 ) into (X 2 , X 1 ), the essence of Gabow's conjecture is that rank many steps might always suffice.This also implies that the conjecture can be rephrased as a generalization of the symmetric exchange axiom as follows: If X 1 and X 2 are bases of the same matroid, then there are orderings X 1 = (x 1  1 , . . ., x 1 r ) and X 2 = (x 2  1 , . . ., x 2 r ) such that {x 1 1 , . . ., x 1 i , x 2 i+1 , . . ., x 2 r } and {x 2 1 , . . ., x 2 i , x 1 i+1 , . . ., x 1 r } are bases for i = 0, . . ., r.This property is often referred to as serial symmetric exchange property.
The focus of this paper is on regular matroids, a fundamental class that generalizes graphic and cographic matroids.Our main tool is Seymour's celebrated decomposition theorem, which gives a method for decomposing any regular matroid into matroids which are either graphic, cographic, or isomorphic to a simple 10-element matroid.Regular matroids play a crucial role in both matroid theory and optimization, since those are exactly the matroids that can be represented over R by totally unimodular matrices [41,Theorem 6.6.3].This connection has far reaching implications, e.g. the fastest known algorithm for testing total unimodularity of a matrix is based on the ability to find such a decomposition if one exists [49].
Interest in exchange properties of matroids originally arose in part from the fact that they serve as an abstraction of pivot algorithms of linear algebra [27,56].In the past decades, however, problems appeared in many different areas of computer science and mathematics that are actually based on exchange properties of matroid bases, though these problems have never been explicitly linked together in previous work.Implicitly, one of the goals of the paper is to draw attention to these connections.

The Role of Equivalent Sequences
Though finding a sequence of symmetric exchanges between basis sequences may seem to be a structural question purely on matroids, it has been identified as the key ingredient in a range of problems.In what follows, we give an overview of main applications where the reconfiguration of basis sequences shows up.

Sampling Common Bases
Mihail and Vazirani conjectured that the basis exchange graph of any matroid has edge expansion at least one, see [22].The motivation behind the conjecture was to solve the problem of approximately sampling from bases of a matroid.After the appearance of the conjecture, a long line of work concentrated on designing approximation algorithms to count the number of bases of a matroid, and efficient sampling algorithms were developed for various special classes.Most of these results relied on the Markov Chain Monte Carlo technique: for any matroid, the basis exchange property defines a natural random walk which mixes to the uniform distribution over all bases, also known as "down-up" random walk in the context of high-dimensional expanders.In a breakthrough result, Anari et al. [3] verified the conjecture of Mihail and Vazirani, and thus gave an efficient approximate sampling algorithm for all matroids.
Sampling common bases of two matroids is also of interest.In [2], Anari, Gharan and Vinzant gave a deterministic polynomial time 2 O(r) -approximation algorithm for the number of common bases of any two matroids of rank r.Unlike in the case of a single matroid, the intersection of two matroids does not satisfy the exchange property.Even worse, there are examples showing that the symmetric difference of a common basis with any other common basis might be large, hence there is no hope for defining a simple down-up-type random walk in general.
Partitions of the ground set of a matroid M into two disjoint bases can be identified with common bases of M and its dual M * .From this perspective, White's conjecture states that there is a sequence of exchanges between any pair of common basis of M and M * .Thus verifying Conjecture 1 would open up the possibility for a natural down-up random walk for matroid intersection in the special case when the two matroids are dual to each other.

Equitability of Matroids
There are several further problems that aim at a better understanding of the structure of bases.The probably most appealing one is the Equitability Conjecture for matroids that provides a relaxation of both Conjecture 1 and Conjecture 2. A matroid whose ground set E partitions into disjoint bases is called equitable if for any set Z ⊆ E, there exists a partition into disjoint bases The Equitability Conjecture states that every matroid whose ground set partitions into disjoint bases is equitable.
The existence of such a partition would follow from both Conjecture 1 and Conjecture 2. To see this, first observe that it suffices to consider the case k = 2, since for general k the statement then follows by repeated application of the problem restricted to X i ∪ X j for 1 ≤ i < j ≤ k; see [30, Discussion page] for details.Then, for any partition of the ground set into two bases X 1 and X 2 , both Conjecture 1 and Conjecture 2 imply the existence of a sequence of symmetric exchanges that transforms (X 1 , X 2 ) into (X 2 , X 1 ).One of the basis pairs of the sequence thus obtained must satisfy Apart from the matroid classes for which Conjecture 1 or 2 was settled, the Equitability Conjecture was verified for base orderable matroids [23] only.It is worth mentioning that equitable partitions are closely related to fair representations, introduced by Aharoni, Berger, Kotlar, and Ziv [1].For further details, we refer the interested reader to [9,23,30].

Fair Allocations
Fair allocation of indivisible goods has received a significant interest in the past decade.In such problems, the goal is to find an allocation of a set E of m indivisible items among n agents so that each agent finds the allocation fair.The degree of equality can be measured in various ways, and envy-freeness, introduced by Foley [24] and Varian [53], is one of the most natural fairness concepts.An allocation is considered to be envy-free (EF) if each agent finds the value of her bundle at least as much as that of any other agent.
Though envy-freeness imposes a very natural criterion for the fairness of an allocation, such a solution may not exist.As a workaround, several relaxations have been proposed, including envy-freeness up to one good (EF1).Biswas and Barman [10] and Dror, Feldman, and Segal-Halevi [18] studied the existence of EF1 allocations under matroid constraints.If the agents have different matroid constraints, then such an allocation might not exist even for two agents with identical valuations.Therefore, it is natural to consider identical matroid constraints.Given a matroid M on the set of items, an allocation is called feasible if the bundle of each agent forms an independent set of M .Deciding the existence of a feasible EF1 allocation as well as finding one algorithmically are interesting open problems that have only been solved for very restricted cases: for partition matroids [10], for base orderable matroids with identical valuations [10], for base orderable matroids with two agents [18], and for base orderable matroids with three agents and binary valuations, i.e. when each item has value zero or one for every agent [18].
The problem remains open even when the agents share the same binary valuation.In such a case, there exists a Z ⊆ E such that the value of any subset X of items is |X ∩ Z| for every agent, and a feasible EF1 allocation corresponds to a partition of the ground set into n independent sets X 1 , . . ., X n such that ⌊|Z|/n⌋ ≤ |X i ∩ Z| ≤ ⌈|Z|/n⌉ for every 1 ≤ i ≤ n.The existence of such an allocation would follow from the Equitability Conjecture, and hence from both Conjecture 1 and Conjecture 2. The only difference compared to the setting of the Equitability Conjecture is that here we seek for a partition into independent sets instead of bases.However, by possibly taking the direct sum of M with a free matroid and then truncating it, the sets X i can be assumed to form bases of the matroid.

Carathéodory Rank of Matroid Base Polytopes
A polyhedron P ⊆ R n has the integer decomposition property if for every positive integer k, every integer vector x in kP can be written as x = t i=1 λ i x i , where each λ i is a positive integer together satisfying t i=1 λ i = k, and each x i is an integer vector in P .This notion was introduced by Baum and Trotter [7], and plays an important role in the theory and application of integer programming [43,Section 22.10] as well as in the study of toric varieties [13], [15,Chapter 2]).For a polyhedron P with the integer decomposition property, the smallest number cr(P ) such that we can take t ≤ cr(P ) for every k and every x ∈ kP is called the Carathéodory rank of P .In [16], Cunningham asked whether a sum of bases in M can always be written as a sum using at most n bases, where n is the cardinality of the ground set, which was answered in the affirmative by Gijswijt and Regts [26].That is, cr(P ) ≤ n holds where P is the convex hull of the incidence vectors of bases of M .
For a polytope P ⊆ R n with vertices v 1 , . . ., v p , a triangulation T of P is a collection of simplices on the vertices of P such that (i) if T ∈ T then all faces of T are in T , (ii) if T 1 , T 2 ∈ T then T 1 ∩ T 2 is a face of both T 1 and T 2 , and (iii) T ∈T conv(T ) = P .A triangulation T is unimodular if the volume of every highest dimensional simplex of T are the same.As a geometric variant of Conjecture 1, Haws [28] conjectured that every matroid base polytope has a unimodular triangulation.One motivation behind the conjecture was that the existence of such a triangulation implies a bound of n on the Carathéodory rank of a connected matroid base polytope, a result that was proved only later in [26].Recently, Backman and Liu [6] verified Haws' conjecture by showing that every matroid base polytope admits a regular unimodular triangulation.

Toric Ideals
Describing minimal generating set of a toric ideal is a well-studied and difficult problem.
Then, the quadratic binomial corresponding to the symmetric exchange is y X1 y X2 − y Y1 y Y2 .It is not difficult to see that such binomials belong to the ideal I M , and White [56] conjectured that they in fact generate I M .
More precisely, White stated three conjectures of growing difficulty.Using the notation of [56], two sequences X , Y of bases of equal length are in relation ∼ 1 if Y can be obtained from X by a composition of symmetric exchanges, in relation ∼ 2 if Y can be obtained from X by a composition of symmetric exchanges and permutations of the order of the bases, and in relation ∼ 3 if Y can be obtained from X by a composition of symmetric exchanges of subsets.Let T E(i) denote the class of matroids for which every two compatible sequences X , Y are in relation X ∼ i Y.In algebraic terms, a matroid belongs to T E(3) if and only if its is generated by quadratic binomials, and it belongs to T E(2) if and only if its toric ideal is generated by quadratic binomials corresponding to symmetric exchanges.Property T E(1) is a counterpart of T E(2) for the noncommutative polynomial ring S M .The three conjectures of White state that T E(i) is the class of all matroids for i = 1, 2, 3.In particular, Conjecture 1 corresponds to the choice i = 1.

Reconfiguration Problems
In combinatorial reconfiguration problems, the goal is to study the solution space of an underlying combinatorial optimization problem.The solution space can be represented by a graph, where vertices correspond to feasible solutions and there is an edge between two vertices if the corresponding solutions can be obtained from each other by an elementary step, defined specifically for the given problem.Reconfiguration problems concern the reachability of a solution from another in this graph, and if such a path exists, then finding a shortest one between them.In recent years, such problems have attracted great attention due to their connection to various topics such as optimization, counting, enumeration, and sampling.For further details, we refer the interested reader to [40,52].
The vertices of the exchange graph of a matroid correspond to basis sequences of a given length, two vertices being connected by an edge if the corresponding basis sequences can be obtained from each other by a single symmetric exchange.In this context, Conjecture 1 aims at characterizing reachability in the exchange graph, and states that the connected components are exactly the equivalence classes of compatibility.
An analogous problem can be formulated for the intersection of two matroids, i.e. given two common bases of two matroids, decide if one can be obtained from the other by always changing a single element while maintaining independence in both matroids.Such a sequence of exchanges is known to exist in special cases, e.g. for arborescences, or more generally, for k-arborescences [31].Recently, the problem was shown to be oracle hard by Kobayashi, Mahara, and Schwarcz [31].For sequences of length two, Conjecture 1 is the special case of the common basis reconfiguration problem when the two matroids are dual to each other.

Our Results
Motivated by the significance of equivalent basis sequences in various applications and by the fact that it was formulated as an interesting open problem in [4,21], we study the exchange distance of basis pairs in regular matroids.First, we give a polynomial upper bound on the exchange distance of compatible basis pairs, which proves Conjecture 1 for sequences of length two in regular matroids.Our proof is algorithmic, which allows us to determine a sequence of symmetric exchanges that transforms a given pair of bases into another in polynomial time.As usual in matroid algorithms, we assume that the matroid is given by an independence oracle and the running time is measured by the number of oracle calls and other conventional elementary steps.For the sake of simplicity, by "polynomial number" of oracle calls we mean "polynomial in the number of elements of the ground set".Theorem 1.1.Let X = (X 1 , X 2 ) and Y = (Y 1 , Y 2 ) be compatible basis pairs of a regular matroid M of rank r ≥ 2.Then, there exists a sequence of symmetric exchanges that transforms X into Y, has length at most 2 • r 2 , and uses each element at most 4 • (r − 1) times.Furthermore, such a sequence can be determined using a polynomial number of oracle calls.
A fine grained analysis of the algorithm shows that the number of steps can be bounded better when the basis pairs are inverses of each other.Our second result is an improved upper bound on the exchange distance of such pairs, which proves Conjecture 2 for regular matroids.Theorem 1.2.Let X 1 , X 2 be disjoint bases of a regular matroid M of rank r.Then, there exists a sequence of symmetric exchanges that transforms (X 1 , X 2 ) into (X 2 , X 1 ) and has length r.Furthermore, such a sequence can be determined using a polynomial number of oracle calls.
Our results give the first polynomial bound on the exchange distance of basis pairs and are the first to settle the conjectures of White and Gabow in regular matroids.We hope that our paper will help proving White's conjecture in regular matroids for sequences of arbitrary length, as well as obtaining better bounds for the exchange distance of basis pairs.

Overview of Techniques
We give a high-level overview of the proofs of Theorem 1.1 and Theorem 1.2.

Connectivity and Cogirth
The first step in proving our main results is to deduce structural properties of regular matroids that allow us to reduce the size of the problem.First, we show that the basis pairs can be assumed to consist of disjoint bases, for if not, then the problem size can be decreased by contracting We then consider tight sets, where in a matroid M over a ground set E a subset Z ⊆ E is tight if |Z| = 2 • r M (Z).We show that if the basis pairs cover a tight set, then the problem can be divided into smaller subproblems on the restriction M |Z and contraction M/Z.As a corollary, we get that it is enough to consider 2-connected matroids.Solving the problem for the 2-sum of regular matroids is based on a similar idea, but merging the solutions for the subproblems is significantly more involved.We explain how to schedule the exchanges on the two sides of the 2-sum in such a way that the basis pairs fit together at each step, meaning that they form a basis pair of the original matroid.This observation eventually reduces the problem to the case of 3-connected matroids.While the above simplifications are well-understood, our main contribution is to show that small cocircuits can also be excluded.More precisely, we prove that if the basis pairs cover a cocircuit of size at most three, then the size of the problem can be decreased by contracting and deleting certain elements of the cocircuit.
For almost all of these operations, we prove a stronger statement that allows certain elements not to be involved in the exchange sequence.This observation will play a crucial role in the proof by providing control over the choice of symmetric exchanges to be used.Besides reducing the problem to 3-connected matroids of cogirth at least four, all the reduction steps can be performed using a polynomial number of oracle calls which is essential for achieving an efficient algorithm.
Graphic Matroids Though it is not stated explicitly, the algorithms of [21] and [11] that prove White's conjecture for graphic matroids give a sequence of exchanges of length at most (n − 1) 2 .Using the fact that the union of two forests always contains a vertex of degree two or at least four vertices of degree three, we give a formal proof of this bound even under certain restrictions on the set of exchanges that can be used.To the best of our knowledge, this is the first analysis of the algorithm that proves a polynomial running time and hence might be of independent combinatorial interest.
More precisely, let both note that these edges do not have to change positions between the two bases.Building on the reductions along tight sets and cocircuits of size three, we show that there exists a sequence of exchanges of length at most (|V | − 1) 2 that transforms X into Y and uses none of the edges in F .This result will be used in the proof of Theorem 1.1 as follows: when the matroid is the 3-sum of a regular and a graphic matroid along a cycle F of length three, then one can solve the two subproblems corresponding to the two sides of the 3-sum while restricting the usage of the elements of F , which in turn allows for an efficient merging of the sequences.
Refined Decomposition Theorem Seymour's decomposition theorem provides a way of writing any regular matroid as the 1-, 2-and 3-sums of so-called basic matroids that are graphic, cographic, or isomorphic to R 10 .Andres, Hochstättler and Merkel [4] already noted that such a decomposition might be helpful in proving White's conjecture for regular matroids.Unfortunately, without any further information on the structure of the decomposition, solving the problem for the basic matroids does not suffice, since it is not clear how to merge these solutions together.
To overcome these difficulties, we use a recent result by Aprile and Fiorini [5] that gives a refinement of Seymour's theorem.Roughly speaking, they showed that any 3-connected regular matroid distinct from R 10 admits a decomposition in which the 3-sums are not using nontrivial cuts of the graphs that correspond to cographic basic matroids.McGuiness [35] gave a characterization of the dual of a 3-sum using the socalled ∆ -Y exchange operation.We observe that applying a ∆ -Y exchange corresponding to a trivial cut of the underlying graph transforms a cographic matroid into another.With the help of these results, we can identify a graphic basic matroid that is "sitting at the end of the decomposition".Moreover, by relying on the reduction steps for tight sets and triads mentioned earlier, we show that the graph in question can be assumed to be 4-regular.It is truly amazing that all these results come together so nicely, thus narrowing the problem down to a case that we can then tackle.
An Inductive Approach Based on our previous observations, we write up the matroid as the 3-sum of a regular matroid and the graphic matroid of a 4-regular graph.The basis pairs of the original instance can be naturally restricted to the two sides of the 3-sum, thus resulting in smaller instances for which one can find desired exchange sequences separately.If not chosen carefully, merging these two sequences to get a solution for the original instance would require too many steps or may even be impossible.On the graphic part, however, we use the strengthening of the statement for graphic matroids which exclude certain elements to take part in the exchanges.This allows us to merge the solutions for the two sides of the 3-sum efficiently.

Related Work
When restricted to sequences of length two, White's conjecture was verified for graphic and cographic matroids by Farber, Richter, and Shank [21], for transversal matroids by Farber [20], and for split matroids by Bérczi and Schwarcz [9].For sequences of arbitrary length, Blasiak [11] confirmed the conjecture for graphic matroids.It is not difficult to check that the conjecture holds for a matroid M if and only if it holds for its dual M * , therefore Blasiak's result settles the cographic case as well.Further results include lattice path matroids by Schweig [44], sparse paving matroids by Bonin [12], strongly base orderable matroids by Lasoń and Micha lek [34], and frame matroids satisfying a linearity condition by McGuinness [36].
For the case of basis pairs, a common generalization of the conjectures of White and Gabow was proposed by Hamidoune [14] stating that the exchange distance of compatible basis pairs is at most the rank of the matroid.A strengthening was proposed by Bérczi, Mátravölgyi, and Schwarcz [8] who considered a weighted variant of Hamidoune's conjecture and verified it for strongly base orderable matroids, split matroids, spikes, and graphic matroids of wheel graphs.
An ordered pair (X 1 , X 2 ) of bases of a matroid satisfies the unique exchange property if there exists an element e ∈ X 1 for which there is a unique element f ∈ X 2 that can be symmetrically exchanged with e.The unique exchange graph can be defined for sequences of bases in a straightforward manner, and White [56] conjectured that for regular matroids, the connected components of this graph are exactly the equivalence classes of compatibility.The motivation behind this conjecture comes from the study of the bracket ring of a matroid, see [54,55] for details.McGuinness [35] verified that any pair of bases in a regular matroid has the unique exchange property, implying the the unique exchange graph has no isolated vertices.However, Andres, Hochstättler and Merkel [4] disproved the conjecture, and proposed a relaxation instead in which the element with a unique symmetrically exchangeable pair can be chosen from both X 1 and X 2 .
Paper Organization The rest of the paper is organized as follows.In Section 2, we recall basic definitions, notation, and some results on the decomposition of regular matroids that we will use in our proofs.In Section 3, we show how Conjecture 1 for sequences of length two and Conjecture 2 can be reduced to 3connected matroids not containing cocircuits of size at most three.Then, in Section 4, we explain how a quadratic bound on the number of exchanges can be derived for graphs using the aforementioned reductions, and prove strengthenings of White's and Gabow's conjectures for graphic matroids.The rest of the paper is devoted to proving Theorem 1.1 and Theorem 1.2.Our proofs rely on the regular matroid decomposition theorem of Seymour.Nevertheless, solving the problems for each matroid in the decomposition in parallel and then simply merging the solutions does not work.The key ingredient that leads to Theorem 1.1 and Theorem 1.2 is a careful combination of the solutions of these subproblems that results in a sequence of exchanges whose length is polynomially bounded.For ease of reading, we encourage first-time readers to skip the technical parts of Section 3.

Preliminaries
Basic Notation and Definitions Given a ground set E, the difference of X, Y ⊆ E is denoted by X − Y .If Y consists of a single element y, then X −{y} and X ∪{y} are abbreviated as X −y and X +y, respectively.The symmetric difference of X and Y is defined as Graphs Throughout the paper, we consider loopless graphs that might contain parallel edges.For a graph G = (V, E), the set of edges incident to a vertex v ∈ V is denoted by δ G (v) and the degree of We dismiss the subscript if the graph is clear from the context.For a subset F ⊆ E, we denote the set of vertices of the edges in F by V (F ).For X ⊆ V , we denote by F [X] the set of edges in F induced by X.The graph obtained by deleting F and X is denoted by for some w ∈ V and nontrivial otherwise.A graph is called bispanning if its edge set can be decomposed into two spanning trees.By a classical result of Tutte [51] and Nash-Williams [39], a graph Matroids For basic definitions on matroids, we refer the reader to [41].A matroid M = (E, I) is defined by its ground set E and its family of independent sets I ⊆ 2 E that satisfies the independence axioms: (I1) Members of I are called independent, while sets not in I are called dependent.The rank r M (X) of a set X is the maximum size of an independent set in X.The maximal independent subsets of E are called bases and their family is usually denoted by B. If the matroid is given by its family of bases instead of independent sets, then we write M = (E, B).The dual of M is the matroid M * = (E, I * ) where For technical reasons, we allow the ground set of the matroid to be the empty set, in which case the matroid is simply the empty matroid M = (∅, {∅}).Let X = (X 1 , . . ., X k ) and Y = (Y 1 , . . ., Y k ) be sequences of bases of M .A sequence of symmetric exchanges that transforms X into Y is called an X -Y exchange sequence.The width of an exchange sequence is the maximum number of occurrences of any element in it.If the symmetric exchanges do not involve the elements in F ⊆ E then the exchange sequence is called F -avoiding.The exchange distance of X and Y is the minimum length of an X -Y exchange sequence if one exists and +∞ otherwise.
A circuit is an inclusionwise minimal dependent set, while a loop is a circuit consisting of a single element.A cocircuit is an inclusionwise minimal set that intersects every basis, or equivalently, a circuit of the dual matroid.A set is said to be coindependent if it contains no cocircuit of the matroid, or equivalently, it is independent of the dual matroid.Two elements e, f ∈ E are parallel if they form a circuit of size two.A circuit of size three is called a triangle, while a corcircuit of size three is called a triad.A cycle of a matroid is a (possibly empty) subset of its ground set which can be partitioned into circuits.For a matroid M , we denote its families of independent sets, bases and circuits by I(M ), B(M ) and C(M ), respectively.Unlike in graphs, the intersection of a circuit and a cocircuit of a matroid might have odd size.Nevertheless, the intersection never consists of a single element, see e. Let M = (E, I) be a matroid and E ′ , E ′′ ⊆ E. The restriction to E ′ and the deletion of A matroid N that can be obtained from M by a sequence of restrictions and contractions is called a minor of M .The union or sum of two matroids M 1 = (E, I 1 ) and M 2 = (E, I 2 ) over the same ground set is the matroid M Σ = (E, I Σ ) where I Σ = {I ⊆ E | I = I 1 ∪ I 2 for some I 1 ∈ I 1 , I 2 ∈ I 2 }.We use M 1 + M 2 for denoting the sum of M 1 and M 2 .Edmonds and Fulkerson [19] showed that the rank function of the sum of two matroids is A matroid is representable over some field F if if there exists a family of vectors from a vector space over F whose linear independence relation is the same as the independence relation of the matroid.The matroid is binary if it is representable over GF (2), and is regular if it can be represented over any field.The following lemma gives a characterization of binary matroids in terms of circuits, see e.g.[ We will further rely on the following observation.
Proof.Assume first that at least two of the three sets form bases of M .We may assume that F + t 1 and F + t 2 are bases.Then, there exists a circuit C ⊆ F + t 1 + t 2 and necessarily t 1 , t 2 ∈ C. The set C△T is a cycle such that t 3 ∈ C△T ⊆ F + t 3 , hence F + t 3 is not a basis of M .This shows that at most two of the sets F + t 1 , F + t 2 and F + t 3 are bases.
Suppose now that at most one of the three sets forms a basis.We may assume that F + t 1 and F + t 2 are not bases.If F is not independent, then F + t 3 is clearly not a basis.Otherwise, let C 1 and C 2 be circuits such that is not a basis.This concludes the proof of the lemma.
The matroid R 10 is a binary matroid that can be represented as the ten vectors in the five-dimensional vector space over GF (2) that have exactly three nonzero entries, see Figure 1a.The Fano matroid F 7 is obtained from the Fano plane by calling a set independent if it contains at most two points or it has three points which are not lines of the plane, see Figure 1c.In other words, F 7 is the matroid with ground set E = {a, b, c, d, e, f, g} whose bases are all subsets size of 3 except {a, b, d}, {b, c, e}, {a, c, f }, {a, e, g}, {c, d, g}, {b, f, g} and {d, e, f }.

Decomposition of Regular Matroids
Let M 1 and M 2 be binary matroids on ground sets Then, we denote by M 1 △M 2 the binary matroid on ground set E = E 1 △E 2 with cycles being the sets of the form (a) Representation of the binary matroid R10 over GF (2).When and its family of bases is (see [5, Section 2.1] together with Lemma 2.3) Seymour's fundamental decomposition theorem [46] gives a constructive characterization of regular matroids.
Theorem 2.4 (Seymour's decomposition theorem).A matroid is regular if and only if it is obtained by means of 1-, 2-and 3-sums, starting from graphic and cographic matroids and copies of a certain 10-elements matroid R 10 .
A binary matroid is said to be connected or 2-connected if it is not a 1-sum, and 3-connected if it is not a 1-sum or a 2-sum of two matroids.While studying the extension complexity of the independence polytope of regular matroids, Aprile and Fiorini [5] recently gave a refinement of Seymour's result in the 3-connected case.
Theorem 2.5 (Aprile and Fiorini).Let M be a 3-connected regular matroid distinct from R 10 .There exists a tree T such that each node v ∈ V (T ) is labeled with a graphic or cographic matroid M v , each edge uv ∈ E(T ) has a corresponding 3-sum M u ⊕ 3 M v , and M is the matroid obtained by performing all the 3-sum operations corresponding to the edges of T in arbitrary order.Moreover, if v ∈ V (T ) is such that M v is the cographic matroid of a nonplanar graph G v , then no nontrivial cut of G v is involved in any of the 3-sums.
A tree T satisfying the conditions of Theorem 2.5 is called a decomposition tree of M , and the matroids corresponding to the nodes of T are referred to as basic matroids.It is worth mentioning that an analogous result was proved by Dinitz and Kortsarz in [17], but their decomposition tree may involve 1-and 2-sums, and also 3-sums along nontrivial cuts.
To describe the dual of a 3-sum, we need the notion of ∆ -Y exchanges.In case of graphs, if T is triangle of a graph G, then we perform a ∆ -Y exchange on G by deleting the edges of T , adding a new vertex v and edges new edges joining v to vertices of T .More generally, consider a binary matroid M and let T be a coindependent triangle of M .Let N be a matroid isomorphic to the graphic matroid of K 4 on ground set T ∪ T ′ where T is a triangle of N , and the triad T ′ of N is disjoint from the ground set of M .We say that the matroid ∆ T (M ) := M ⊕ 3 N is obtained from M by performing a ∆ -Y exchange.McGuinness [35] gave a characterization of the dual of a 3-sum.
where the ∆ -Y exchanges ∆ T (M 1 ) and ∆ T (M 2 ) are performed using the same matroid N on ground set T ∪ T ′ and the 3-sum ∆ T (M 1 ) * ⊕ 3 ∆ T (M 2 ) is performed using the common triangle T ′ of ∆ T (M 1 ) * and ∆ T (M 2 ) * .
The reverse operation of a ∆ -Y exchange is called a Y -∆ exchange.If T is an independent triad of a binary matroid M , then we say that the matroid ∇ T (M ) := ∆ T (M * ) * is obtained from M by performing a Y -∆ exchange.The ∆ -Y and Y -∆ exchanges are indeed reverse operations of each other, see [41,Proposition 11.5.11].If u is a degree 3 vertex of a graph G with distinct adjacent vertices x, y and z, then we can perform the Y -∆ operation on the graphic matroid M (G) by deleting u and adding the edges xy, yz and xz.In particular, if T is an independent triad of a graphic matroid M (G) corresponding to the edges adjacent to a trivial cut G, then ∇ T (M (G)) is a graphic matroid.This implies the following.
Lemma 2.7.If T is a coindependent triangle of a cographic matroid M * (G) corresponding to a trivial cut of G, then ∆ T (M * (G)) is cographic.
Remark 2.8.We note that if T does not correspond to a trivial cut of G, then ∆ T (M * (G)) might not be a cographic matroid.As an example, if e is an edge of K 5 and T is the triangle of K 5 formed by the vertices not adjacent to e, then, ∆ T (M (K 5 − e)) = M (K 3,3 ), see also [41,Figure 11.20].Since K 5 − e is a planar and K 3,3 is a nonplanar graph, M (K 5 − e) is a cographic matroid while M (K 3,3 ) is not.

Algorithms and Oracles
In matroid algorithms, it is usually assumed that the matroid is given by an oracle and the running time is measured by the number of oracle calls and other conventional elementary steps.There are many different types of oracles that are often used, the independence, circuit and rank oracles probably being the most standard ones.For a matroid M = (E, I) and set X ⊆ E as an input, an independence oracle answers "Yes" if X is independent and "No" otherwise, a circuit oracle answers "Yes" if X is a circuit and "No" otherwise, and a rank oracle gives back r M (X).
In fact, these oracles have the same computational power.An oracle O 1 is polynomially reducible to another oracle O 2 if O 1 can be implemented by using a polynomial number of oracle calls to O 2 measured in terms of the size of the ground set.Two oracles are polynomially equivalent if they are mutually polynomially reducible to each other.It is not difficult to show that the independence, circuit and rank oracles are polynomially equivalent, see e.g.[42].
Let E denote the ground set of M , and X and Y be disjoint subsets of E. Then the rank function of the minor M/X\Y is r M/X\Y (Z) = r(Z ∪ X) − r(Z) for Z ⊆ E − (X ∪ Y ), and the rank function of the dual [41].That is, given an independence oracle access to the matroid M , independence oracles can be implemented for any minor and the dual of M by the polynomial equivalence of the rank and independence oracles.Therefore, we will use these basic matroid operations in our algorithm.
For a binary matroid M , it can be decided if M is not connected, connected but not 3-connected, or 3-connected using a polynomial number of oracle calls [49,Theorem 8.4.1].Moreover, the algorithm also provides a 1-sum decomposition in the first case, a 2-sum decomposition in the second case, and a 3-sum decomposition if it exists in the third case.When applied to a regular matroid recursively, the algorithm eventually gives a decomposition M into basic matroids each of which is either graphic, cographic or isomorphic to R 10 .If the matroid is 3-connected and is not R 10 , then [5] describes an algorithm how to modify this decomposition until it gives a decomposition tree with no bad nodes using a polynomial number of oracle calls.These observations together imply that for any 3-connected matroid different from R 10 , we can efficiently determine a decomposition tree not containing bad nodes.

Reduction to 3-Connected Case Without Small Cocircuits
In this section, we focus on how the exchange distance behaves for basic matroid operations such as contraction and taking 1-or 2-sums.Furthermore, we identify structural properties of regular matroids such as the existence of a tight set or a triad that allow for reduction in the problem size.As mentioned in the introduction, these reduction steps will eventually make it possible to write up the matroid as the 3-sum of a regular matroid and the graphic matroid of a 4-regular graph.Algorithmic aspects of the preprocessing steps are discussed at the end of the section.
Let M be the 1-, 2-or 3-sum of binary matroids M where T is empty in case of 1-sums, it consists of a single element in case of 2-sums and of three elements in case of 3-sums.For any set X ⊆ E • ∪ E • , we define X • := X ∩ (E • − T ) and X • := X ∩ (E • − T ).In particular, for any basis B ∈ B(M ), we have B • = B ∩ E • and B • = B ∩ E • .Note that for 2-and 3-sums, B • and B • are not necessarily bases of M • and M • , respectively; see the characterization of bases in Section 2.
Since we will prove Theorem 1.1 and Theorem 1.2 in a stronger form for graphic matroids, we formulate some of the reductions for F -avoiding exchange sequences whose last step is partially fixed.This results in a series of rather technical lemmas, but this should not deter the interested reader from the later sections.We encourage first-time readers to skip these technical parts and only return to them after getting a general understanding of the structure of the proof.
Operations similar to those discussed in Section 3.1 and Section 3.2 were implicitly mentioned in [56], while an operation similar to the one discussed in Section 3.3 was considered in [48].However, we will deduce stronger properties of the reduction steps and also discuss the algorithmic aspects.

Making the Bases Disjoint
We start with the simple observation that it suffices to consider compatible pairs consisting of disjoint bases.
If there exists an F ′ -avoiding X ′ -Y ′ exchange sequence in M/(X 1 ∩ X 2 ) of width w and length ℓ, then there exists an F -avoiding X -Y exchange sequence in M of width w and length ℓ.Furthermore, if h ∈ E −(X 1 ∩X 2 ) is used in the last step of the X ′ -Y ′ exchange sequence, then it can be assumed to be used in the last step of the X -Y exchange sequence as well.
This implies that X ′ and Y ′ form compatible basis pairs of M/(X 1 ∩ X 2 ).As any sequence of symmetric exchanges that transforms X ′ into Y ′ also transforms X into Y, the lemma follows.
By the lemma, it suffices to consider instances where X 1 ∩ X 2 = Y 1 ∩ Y 2 = ∅.Furthermore, since the elements not contained in any of the bases cannot participate in exchanges and hence can be deleted, we can assume without loss of generality that E = X 1 ∪ X 2 = Y 1 ∪ Y 2 holds.

Excluding Tight Sets
Given a matroid M over ground set E, a set Nontrivial tight sets are special for the following reason: every partition E = X 1 ∪ X 2 into two disjoint bases necessarily satisfies |X i ∩ Z| = r M (Z) for i = 1, 2. In other words, pairs of disjoint bases of M are exactly the pairs of disjoint bases of the matroid M |Z ⊕ 1 M/Z.This observation allows us to reduce the size of the problem along a nontrivial tight set.
If there exists an F ′ -avoiding X ′ -Y ′ exchange sequence in M |Z of width w ′ and length ℓ ′ and an F ′′ -avoiding X ′′ -Y ′′ exchange sequence in M/Z of width w ′′ and length ℓ ′′ , then there exists an F -avoiding X -Y exchange sequence in M of width max{w ′ , w ′′ } and length ℓ ′ + ℓ ′′ .Furthermore, if h ∈ E is used in the last step of the X ′′ -Y ′′ exchange sequence, then it can be assumed to be used in the last step of the X -Y exchange sequence as well.
Proof.By the definition of contraction, the concatenation of the two exchange sequences results in an X -Y exchange sequence with the properties stated.
Furthermore, the bases of M are exactly the unions of a basis of M • and a basis of M • .Hence, for any pair (X 1 , X 2 ) of disjoint bases of M , the set X . Therefore, Lemma 3.2 implies the following.
. If there exists an X ′ -Y ′ exchange sequence in M • of width w ′ and length ℓ ′ and an X ′′ -Y ′′ exchange sequence in M • of width w ′′ and length ℓ ′′ , then there exists an X -Y exchange sequence in M of width max{w ′ , w ′′ } and length ℓ ′ + ℓ ′′ .

Reduction to 3-Connected Matroids
When the matroid happens to be the 2-sum of matroids, the problem admits a reduction similar to the one used for tight sets.However, while merging the solutions to the subproblems was trivial for tight sets, it becomes much more involved for 2-sums.To get a better understanding of this difficulty, let X 1 and X 2 be disjoint bases of Then, unfortunately, this step does not correspond to a feasible symmetric exchange between X 1 and X 2 in M , since both The main result of this section is to show that the exchanges can be scheduled on the two sides of the 2-sum in a way that avoids the problem described above.The next lemma, when used in conjunction with Lemma 3.2, eventually reduces the problem to the case of 3-connected matroids.
. If there exists an X ′ -Y ′ exchange sequence in M • of width w ′ and length ℓ ′ and an X ′′ -Y ′′ exchange sequence in M • of width w ′′ and length ℓ ′′ , then there exists an X -Y exchange sequence in M of width at most w ′ + w ′′ and length at most ℓ ′ + ℓ ′′ − 1 if both exchange sequences involve t and ℓ ′ + ℓ ′′ otherwise.

Proof. Using the description of B(M
is a tight set in M and the statement follows from Lemma 3.2.Otherwise, We prove the lemma in two steps.First, consider the case when the X ′ -Y ′ exchange sequence in M • does not involve the element t.Note that in this case . We construct an X -Y exchange sequence as follows.We start with the steps of the X ′ -Y ′ exchange sequence, which transform X = (X 1 , X 2 ) into the basis pair (Y . By the symmetric exchange axiom, there exists e ∈ Y From this point, we perform the steps of the X ′′ -Y ′′ exchange sequence, but whenever a symmetric exchange uses t and some other element f , then exchange e and f instead.Formally, if a symmetric exchange transforms (Z In both cases, the pair obtained consists of disjoint bases of M due to the choice of e.It is not difficult to check that at the end of the procedure, we arrive at the basis pair (Y 1 , Y 2 ).The X -Y exchange sequence thus obtained has width at most w ′ + w ′′ and length ℓ ′ + ℓ ′′ .
By symmetry, it remains to consider the case when both the X ′ -Y ′ exchange sequence in M • and the X ′′ -Y ′′ exchange sequence in M • use the element t at least once.Let m ′ and m ′′ denote the number of occurrences of t in these sequences; we may assume that m ′ < m ′′ .We construct an X -Y exchange sequence as follows.We perform the steps of both the X ′ -Y ′ and X ′′ -Y ′′ exchange sequences, but we align the exchanges involving t on both sides, see Figure 2. Formally, we always perform the steps of the X ′ -Y ′ exchange sequence until we reach the next step that involves t, say, transforms a basis pair (Z From this point, we perform the steps of the X ′′ -Y ′′ exchange sequence until we reach the next step that involves t, say, transforms (Z Then these two steps are replaced by the symmetric exchange that transforms (Z Once there are no more steps using t on the side of M • , the exchange sequence can be finished as discussed in the previous case.The X -Y exchange sequence thus obtained has width at most w ′ + w ′′ and length

Excluding Cocircuits of Size Three
Recall that a triad is a cocircuit of size three.Let M be a matroid, X = (X 1 , X 2 ) and Y = (Y 1 , Y 2 ) be compatible pairs of disjoint bases of M , and that is, the bases in X partition the elements of T the same way as the bases in Y.We will use the following simple technical claim.
Proof.Without loss of generality, we may assume that t 1 , t 2 ∈ X 1 .Since X 1 is a basis of M , X 1 + t 3 contains a unique circuit C. By Lemma 2.1, the intersection of C and T has size different from one, hence C ∩ {t 1 , t 2 } ̸ = ∅.We may assume that t 1 ∈ C, implying X 1 − t 1 + t 3 being a basis.It remains to show that X 2 + t 1 − t 3 is also a basis.Suppose to the contrary that this does not hold, that is, First, we show that if the basis pairs X , Y are not consistent on a triad T , then one can obtain another pair X ′ , Y ′ that are consistent on T at the cost of at most two symmetric exchanges.Lemma 3.6.Let X = (X 1 , X 2 ) and Y = (Y 1 , Y 2 ) be compatible pairs of disjoint bases of a matroid M that are not consistent on a triad T = {t 1 , t 2 , t 3 } ⊆ X 1 ∪ X 2 .Then, there exists compatible pairs of disjoint bases that are consistent on T and are obtained by applying at most one symmetric exchange to X and to Y, respectively.

Proof. Define P
Observe that each member of P 1 can be obtained from X by exchanging at most one pair of elements; however, this might not be a feasible symmetric exchange.Similarly, define Observe that each member of P 2 can be obtained from Y by exchanging at most one pair of elements; again, this might not be a feasible symmetric exchange.
By Claim 3.5, at least two members of P 1 and at least two members of P 2 consist of disjoint bases.Therefore, there exist X ′ ∈ P 1 and Y ′ ∈ P 2 that are consistent on T , concluding the proof of the lemma.
Once the basis pairs are consistent on a triad, the problem size can be decreased by contracting and deleting appropriate elements of the triad.
otherwise.If there exists an F -avoiding X ′ -Y ′ exchange sequence in M/t 2 \t 3 of width w and length ℓ, then there exists an F -avoiding X -Y exchange sequence in M of width w and length at most ℓ + w.Furthermore, if h ∈ E − (T ∪ F ) is used in the last step of the X ′ -Y ′ exchange sequence, then it can be assumed to be used in the last step of the X -Y exchange sequence as well.
Proof.For any pair of disjoint bases Z = (Z 1 , Z 2 ) of M/t 2 \t 3 , we denote by Fix an F -avoiding X ′ -Y ′ exchange sequence in M/t 2 \t 3 of length ℓ and width w.The idea is to add certain extra steps to obtain a solution to the original instance.Consider a symmetric exchange in the sequence that transforms (Z 1 , Z 2 ) into (Z 1 − e + f, Z 2 − f + e).Without loss of generality, we may assume that , hence these pairs also differ in a single symmetric exchange in M .However, if , and these pairs cannot be obtained from each other by a single symmetric exchange.In this case, consider the pairs (Z 1 + t 3 , Z 2 + t 2 ) and (Z 1 − t 1 + {t 2 , t 3 }, Z 2 + t 1 ).By Claim 3.5, at least one of these pairs consists of disjoint bases of M .Furthermore, any of them can be obtained from both (Z 1 , Z 2 ) + and (Z 1 − t 1 + f, Z 2 − f + t 1 ) + by using a single symmetric exchange.
Summarizing the above, a symmetric exchange of elements e and f in the X ′ -Y ′ exchange sequence is left unchanged if e, f ̸ = t 1 .Otherwise, if, say, e = t 1 , it is replaced by two steps: the first exchanging t 1 and t i and the second exchanging t j and f for some appropriate choice of i and j satisfying {i, j} = {2, 3}.Observe that these modifications do not increase the usage of an element in E − {t 2 , t 3 }, hence the width of the new sequence is also w.Furthermore, the length of the sequence increases by the number of symmetric exchanges involving t 1 , hence the length of the new sequence is at most ℓ + w.Finally, note that the new sequence is F -avoiding as well, and its last step uses the elements of E − (T ∪ F ) that were involved in the last step of the X ′ -Y ′ exchange sequence, thus concluding the proof of the lemma.
The two lemmas allow us to reduce the problem size if the matroid contains a triad.Indeed, the basis pairs can be made consistent on any triad with the help of Lemma 3.6, which requires at most two symmetric exchanges.Once the basis pairs are consistent on a triad, we can decrease the number of elements as in Lemma 3.7.If the exchange sequence in the reduced instance has width w and length ℓ, then we get an exchange sequence of width w + 2 and length ℓ + w + 2 for the original instance.

Algorithmic Aspects
The preprocessing steps discussed in the previous subsections do not only reduce the problem size in a theoretical sense, but are also algorithmically tractable if the matroid M is given by an independence oracle.Recall that a 1-sum or 2-sum decomposition of M can be determined, if exists, efficiently.Thus it suffices to show that one can find a triad or a tight set of a matroid using a polynomial number of oracle calls.
By definition, a triad is a cocircuit of size three, or equivalently, a circuit of size three of the dual matroid.Since an independence oracle of the dual matroid can be implemented using the independence oracle of M , the existence of such a circuit can be decided by checking every 3-elements subset of the ground set.
Assume now that the ground set of M is the disjoint union of two bases, say X 1 and X 2 .Then for any set Z, we have 2 • r M (Z) ≥ |X 1 ∩ Z| + |X 2 ∩ Z| = |Z|, and equality holds if and only if Z is tight.Hence to decide whether X 1 ∪ X 2 properly contains a nonempty tight set of M , it suffices to minimize the submodular function f (Z) := r M (Z) − |Z|/2 over the sets ∅ ̸ = Z ⊊ X 1 ∪ X 2 , which can be performed in strongly polynomial time if given access to the independence oracle [16].
Finally, we show that the width and length bounds of Lemma 3.1, Lemma 3.2, Lemma 3.4, Lemma 3.6 and Lemma 3.7 are consistent with the statement of Theorem 1.1.Before that, we need the following simple observation.
Then there exists an F -avoiding X -Y exchange sequence of width at most 1 and length at most r.Furthermore, if h ∈ (X 1 ∪ X 2 ) − F , then the last step of the sequence can be assumed to use h.
Proof.The claim is straightforward to check for matroids of rank at most two.
With the help of the claim, we are now ready to prove that the inverse operations of the reduction steps preserve the quadratic running time.Since 2-sum behaves differently for F -avoiding exchange sequences than the other operations, we do this in the form of two corollaries.Moreover, we state the corollaries parameterized by a constant c ≥ 1; the reason is that we will choose c to be 1 for graphic matroids and 2 for general regular matroids.For nondisjoint bases, tight sets and triads, we get the following.Corollary 3.9.Let X = (X 1 , X 2 ) and Y = (Y 1 , Y 2 ) be compatible pairs of bases of a matroid M = (E, I) of rank r ≥ 3 and let F ⊆ (X 1 ∩ Y 1 ) ∪ (X 2 ∩ Y 2 ).Assume that for any minor M ′ = (E ′ , I ′ ) of M and for any pair X ′ , Y ′ of compatible pairs of disjoint bases of M ′ , there exists an F ′ -avoiding X ′ -Y ′ exchange sequence in M ′ of width at most 2 • c • (r ′ − 1) and length at most c • r ′ 2 , where . Note that r ′ < r holds.Our assumption, Claim 3.8, and Lemma 3.1 then imply the existence of an F -avoiding X -Y exchange sequence of width at most max{1, 2 a tight set, then let r ′ and r ′′ denote the ranks of M |Z and M/Z, respectively.Note that r = r ′ + r ′′ holds.Our assumption, Claim 3.8, and Lemma 3.2 then imply the existence of an F -avoiding X -Y exchange sequence of width at most max {max{1, 2 Our assumption, Claim 3.8, Lemma 3.6 and Lemma 3.7 then imply the existence of an F -avoiding X -Y exchange sequence of width at most 2 For 2-sums, we get the following.

Bounding the Number of Exchanges for Graphs
White's conjecture was settled for graphic matroids in [21] for sequences of length two, and in [11] for sequences of arbitrary length.Both results rely on the same algorithm, and in fact imply Gabow's conjecture as well for the graphic case.However, neither discusses the length of the resulting exchange sequence.
The goal of this section is to prove strengthenings of Theorem 1.1 and Theorem 1.2 for graphs.Due to the fact that most of the work has already been done in Section 3, the proofs are simple and compact.Throughout the section, we use the fact that every minor of a graphic matroid is graphic again without explicitly mentioning it.

Quadratic Upper Bound
We give the first polynomial bound on the exchange distance of compatible basis pairs in graphic matroids.In addition, through an analysis of the degree sequences of graphs that can be partitioned into two forests, we show how to exclude certain edges to participate in the exchange sequence.This observation plays a key role in the proof of Theorem 1.1: when considering the 3-sum of a regular and a graphic matroid along a triad T , one can solve the two subproblems corresponding to the two sides of the 3-sum while restricting the usage of the elements of T , which in turn allows for an efficient merging of the sequences.Proof.We prove the theorem by induction on the rank.For r = 2, the statement holds by Claim 3.8.Therefore, we consider the case r ≥ 3, implying that |V | ≥ 4. Since the elements not contained in any of the bases cannot participate in an X -Y exchange sequence, we may assume without loss of generality that That is, G is a graph whose edge set can be partitioned into two forests.Furthermore, by the induction hypothesis and Corollary 3.9, we may assume that If there are isolated vertices in the graph, then those can be deleted without changing the problem.Since E can be partitioned into two forests, we have . This implies that G contains a vertex u of degree at most 3. Since both forests are bases in the graphic matroid, those are maximal forests, implying that d(u) ≥ 2.
Assume first that G has a vertex u of degree 2. Then u is a leaf vertex in both X 1 and X 2 , implying

Strictly Monotone Sequences
Given compatible basis pairs X , Y of a matroid, an X -Y exchange sequence is called strictly monotone if each step decreases the difference between the first members of the pairs.Using this terminology, Conjecture 2 states that for any pair of disjoint bases X 1 , X 2 of a matroid, there exists a strictly monotone exchange sequence between (X 1 , X 2 ) and (X 2 , X 1 ).Theorem 4.3.Let X 1 , X 2 be disjoint bases of a graphic matroid M .Then, for any h ∈ X 1 ∪X 2 , there exists a sequence of symmetric exchanges that transforms (X 1 , X 2 ) into (X 2 , X 1 ), has length r and exchanges h in the last step.
Proof.Similarly to the proof of Theorem 4.3, we can assume that X 1 ∪ X 2 = Y 1 ∪ Y 2 = E, and hence G contains a vertex u of degree at most 3. Furthermore, if u has degree 2 then E − δ(u) is a nonempty proper tight set of M , while if d(u) = 3 then δ(u) is a triad of M .Therefore, the statement follows by the induction hypothesis, Claim 3.8, Lemma 3.2 and Lemma 3.7.
As a relaxation, a matroid has the k-serial exchange property for some positive integer k if for any two bases X 1 , X 2 and any subset A 1 ⊆ X 1 of size k, there is a subset A 2 ⊆ X 2 for which A 1 and A 2 are serially exchangeable.It was shown in [33] that every matroid has the 2-serial exchange property.Kotlar [32] further verified that for matroids of rank at least three, for any two bases X 1 , X 2 there exist A 1 ⊆ X 1 and A 2 ⊆ X 2 such that |A 1 | = |A 2 | = 3 and A 1 and A 2 are serially exchangeable.Recently, McGuiness [37] showed that all binary matroids of rank at least three have the 3-serial exchange property.However, it is still unknown whether all matroids of rank at least three have the 3-exchange property.
Using this terminology, the statement of Theorem 4.3 is equivalent to the (r − 1)-serial exchange property of graphic matroids.

Finding a Sequence of Exchanges in Polynomial Time
This section is dedicated to the proofs of Theorem 1.1 and Theorem 1.2.

Preparations
For proving the theorems, we need some preliminary observations.We first discuss the structure of bispanning graphs, and characterize their partitions into disjoint spanning trees in terms of intersections with a triangle.We then verify the theorems for the matroid R 10 , and prove an analogous result to F 7 as well.Recall that the matroid R 10 is one of the basic building blocks of regular matroids, while F 7 is considered here to extend our results to max-flow min-cut matroids, see Section 6.Finally, we show that it suffices to consider the problem for matroids that arise as the 3-sum of a regular matroid and the graphic matroid of a 4-regular graph.

Partitions of Bispanning Graphs
Binary matroids have distinguished structural properties, which implies the following.Lemma 5.1.Let T = {t 1 , t 2 , t 3 } be a triangle of a binary matroid M on ground set E. Then, the following are equivalent: (i) E − T partitions into a basis of M and a basis of M/T , (ii) E − t i partitions into two bases of M for each i ∈ {1, 2, 3}, It remains to show that (iii) implies (i).Since r , then the desired inequality follows from 2 • r M (X) ≥ |X|.It remains to consider the case r M (X ∪ T ) = r M (X) + 1.Then, M being binary implies that X spans at least one of t 1 , t 2 and t 3 .Indeed, if X does not span t 1 or t 2 , then r M (X + t 1 + t 2 ) = r M (X) + 1 implies that there is a circuit C ⊆ X + t 1 + t 2 containing both t 1 and t 2 , thus C△T ⊆ X + t 3 is a cycle containing t 3 , hence X spans t 3 .If X spans t i , then concluding the proof of the lemma.
Remark 5.2.We note that (ii) does not necessarily imply (i) for nonbinary matroids.For example, consider the matroid on ground set {e 1 , e 2 , t 1 , t 2 , t 3 } in which e 1 and e 2 are parallel and the matroid obtained by deleting e 1 is the rank-2 uniform matroid.Then, {e 1 , t j }, {e 2 , t k } is a partition of E − t i into two bases of M for any choice of indices satisfying {i, j, k} = {1, 2, 3}.However, E − T = {e 1 , e 2 } consists of parallel elements in M and of loops in M/T , hence it can not be decomposed into a basis of M and a basis of M/T .
We will use the following corollary of the lemma for graphic matroids.
Therefore, the statement follows from the equivalence of Lemma 5.1(i) and Lemma 5.1(ii) applied to the graphic matroid of G.
The proof of Theorem 1.1 will rely on the following lemma.Proof.Let v 1 , v 2 and v 3 denote the vertices of T such that We denote by a and b the neighbours of v 1 distinct from v 2 and v 3 , and by u 1 , u 2 and u 3 the neighbours of a distinct from v 1 .Let f i be a new edge between vertices u i+1 u i+2 , where indices are meant in a cyclic order.Let . This shows that the conditions of Corollary 5.3 are satisfied, hence there exists a partition and F 1 − e + t 3 are spanning trees of G. Since F 2 is a forest such that v 1 a is one of its two components, we get that F 2 + t 3 , F 2 + t 2 and F 2 + e are also spanning trees of G.
Figure 5: Exchange sequences starting from a basis pair (X 1 , X 2 ) where X 1 and X 2 are 5-cycles.Basis pairs in the dashed set show a sequence of length 5 to (X 2 , X 1 ).From each basis pair of R 10 , one can obtain a basis pair shown on the figure with reflections and rotations, thus each pair of bases can be obtained from (X 1 , X 2 ) with at most 5 exchanges by its symmetry.
Proof.Let X = (X 1 , X 2 ) be a basis pair of R 10 .It follows from Theorem 2.4 that each proper minor of R 10 is graphic or cographic, hence we may assume that X 1 and X 2 are disjoint by Lemma 3.1.
We will use the representation of R 10 as the even-cycle matroid of the complete graph K 5 on vertices {v 1 , v 2 , v 3 , v 4 , v 5 }, see e.g.[41, page 238].The ground set of this matroid is the edge set of K 5 and the circuits are the cycles of length four and the unions of two triangles having exactly one vertex in common.
To get an isomorphism between this matroid and the binary matroid represented by A, map an edge v i v j to the column of A in which the ith and jth entries are zero, see Figure 1b.The bases are the sets of five edges containing no even cycles and exactly one odd cycle of K 5 .This implies that (Z 1 , Z 2 ) is a basis pair for a partition E = Z 1 ∪ Z 2 if and only if Z 1 is a 5-cycle of K 5 or Z 1 = {v i v j , v i v k , v j v k , v j v l , v k v m } for some {i, j, k, l, m} = {1, 2, 3, 4, 5}.Observe that there is an isomorphism that maps a basis of the latter form into a 5-cycle, e.g.(v 1 v 2 , v 1 v 3 , v 2 v 3 , v 2 v 4 , v 3 v 5 ) can be mapped to (v 1 v 3 , v 1 v 2 , v 4 v 5 , v 2 v 4 , v 3 v 5 ) by mapping (v 1 v 4 , v 1 v 5 , v 2 v 5 , v 3 v 4 , v 4 v 5 ) to (v 1 v 5 , v 1 v 4 , v 2 v 5 , v 3 v 4 , v 2 v 3 ).Therefore, to prove Theorems 1.1 and 1.2 for R 10 , we may assume that X 1 is a 5-cycle.Figure 5 illustrates that each pair of disjoint bases of R 10 is reachable from this basis with at most 5 exchanges.
Though it is not needed in the proof of Theorem 1.1 and Theorem 1.2, we verify analogous statements for the Fano matroid as well.This will allow us to extend our results to max-flow min-cut matroids.
Proposition 5.6.For any pair X , Y of compatible basis pairs of the Fano matroid, there exists an X -Y exchange sequence of width at most 4 and length at most 9.For disjoint bases X 1 , X 2 of the Fano matroid, there exists an (X 1 , X 2 )-(X 2 , X 1 ) exchange sequence of length 3. Furthermore, such sequences can be determined in polynomial time.
Proof.The matroid F 7 is an excluded minor of graphic matroids, see [41,Thoerem 10.3.1].For any pair of bases X = (X 1 , X 2 ), the restriction F 7 |(X 1 ∪ X 2 ) is a proper minor of F 7 , thus it is graphic.Since the rank of F 7 is 3, the statements follow from Theorems 4.1 and 4.3.

Reducing the Graphic Part to 4-regular Graphs
This section is devoted to proving our main structural observation.The proposition is based on a careful combination of the result of Aprile and Fiorini on decompositions trees of 3-connected matroids (Theorem 2.5), the result of McGuiness on the dual of 3-sums (Proposition 2.6), our observation on the cographicness of a matroid obtained from a cographic matroid by a ∆ -Y exchange using a coindependent triangle (Lemma 2.7), and the reduction steps introduced in Section 3. Proposition 5.7.Let M be a 3-connected regular matroid that is not graphic, cographic or isomorphic to R 10 .Then, there exists a regular matroid M • and a graphic matroid M • such that M • ⊕ 3 M • ∈ {M, M * }, and such a decomposition can be determined using a polynomial number of oracle calls.Moreover, if M contains no circuit or cocircuit of size at most 3, its ground set can be partitioned into two bases and it contains no nontrivial tight set, then M • is the graphic matroid of a simple 4-regular graph.
Proof.By Theorem 2.5, M can be written in the form M = M 1 ⊕ 3 M 2 , where M 1 is a regular matroid and M 2 is a graphic or cographic matroid.Moreover, if M 2 is not graphic, then it is the cographic matroid of a graph G such that the 3-sum is taken along a triangle T of the matroids which is a trivial cut of G.If Let G = (V, E • ) be a connected graph whose graphic matroid is M • and let v 1 , v 2 and v 3 denote the vertices of the triangle T .We define V ′ := V − {v (b) While we could verify White's conjecture for basis pairs in regular matroids, the problem remains open for longer sequences.We believe that such a result might follow by combining our techniques with Blasiak's approach for the graphic case.
4. A common generalization of White's conjecture for sequences of length two and Gabow's conjecture was proposed by Hamidoune [14], suggesting that the exchange distance of compatible basis pairs is at most the rank of the matroid.In [8], the conjecture was verified for strongly base orderable matroids, split matroids, spikes, and graphic matroids of wheel graphs.However, it remains open even for graphic matroids in general.
Consider a matroid M = (E, B) where E denotes the ground set and B is the family of bases of M .For a field K, let S M denote the polynomial ring K[y B | B ∈ B].The toric ideal associated to M is the kernel of the K-homomorphism φ M : S M → K[x e : e ∈ E] given by y B → e∈B x e .Assume now that the basis pair g. [41, Proposition 2.1.11].Lemma 2.1.Let C and T be a circuit and a cocircuit of a matroid M .Then |C ∩ T | ̸ = 1.

3 (
b) Representation of R10 as an evencycle matroid.The bases of F7 are the non-line 3-element sets of the Fano plane.

Figure 2 :
Figure 2: Illustration of Lemma 3.4, where steps of the X ′ -Y ′ and X ′′ -Y ′′ exchange sequences involving t are combined to obtain a step of the X -Y exchange sequence.

Corollary 3 . 10 .
Let X = (X 1 , X 2 ) and Y = (Y 1 , Y 2 ) be compatible pairs of disjoint bases of a matroid M of rank r ≥ 3 where M contains no nontrivial tight set.Assume that for any minor M ′ of M and for any pair X ′ , Y ′ of compatible pairs of disjoint bases of M ′ , there exists an X ′ -Y ′ exchange sequence in M ′ of width at most 2 • c • (r ′ − 1) and length at most c • r ′ 2 , where 2 ≤ r ′ < r is the rank of M ′ and c ≥ 1.If M is the 2-sum of two matroids, then there exists an X -Y exchange sequence in M of width at most 2 • c • (r − 1) and length at most c • r 2 .Proof.If M = M • ⊕ 2 M • ,then both M • and M • are minors of M [41, Proposition 7.1.21].Let r ′ and r ′′ denote the ranks of M • and M • , respectively.Note that r = r ′ + r ′′ − 1 holds.Moreover, we claim that r ′ , r ′′ ≥ 2. Indeed, e.g.r ′ ≥ 1 and |E • | ≥ 3 hold by the definition of 2-sums, and r ′ = 1 would imply that M contains a nontrival tight set.Our assumption and Lemma 3.4 then imply the existence of an X -Y exchange sequence of width at most 2•c•(r ′ −1)+2•c•(r ′′ −1) = 2•c•(r−1) and length at most c•r ′ 2 +c•r ′′ 2 ≤ c•r 2 .

Theorem 4 . 1 .
Let X = (X 1 , X 2 ) and Y = (Y 1 , Y 2 ) be compatible pairs of bases of a graphic matroid M of rank r ≥ 2 where the underlying graph is G = (V, E), and letF ⊆ (X 1 ∩ Y 1 ) ∪ (X 2 ∩ Y 2 ) be such that |V (F )| ≤ 3.Then, there exists an F -avoiding X -Y exchange sequence of width at most 2 • (r − 1) and length at most r 2 .
and the statement follows by the induction hypothesis and Corollary 3.9.If G contains no vertex of degree 2, then it has at least four vertices of degree 3.By condition |V (F )| ≤ 3, there exists a vertex u such that d(u) = 3 and δ(u) ∩ F = ∅.That is, δ(u) defines a triad of M disjoint from F , and the statement follows by the induction hypothesis and Corollary 3.9.The pair X = (X1, X2) of disjoint bases of M .The pair Y = (Y1, Y2) of disjoint bases of M .

Figure 3 :Remark 4 . 2 .
Figure 3: Example showing that Theorem 4.1 no longer holds if F consists of a pair of disjoint edges.For the choice F = {b, e} ⊆ (X 1 ∩ Y 1 ) ∪ (X 2 ∩ Y 2 ), every X -Y exchange sequence uses at least one of b and e.

5. 1 . 2
Solving the Problem for R 10 and F 7 We now verify Theorem 1.1 and Theorem 1.2 for the matroid R 10 .Recall that R 10 is the binary matroid represented by a matrix A ∈ GF (2) 5×10 in which the columns are different and each of them contains exactly two zero entries.Proposition 5.5.R 10 satisfies Theorem 1.1 and Theorem 1.2. v M 2 is graphic, then let M • := M 1 and M • := M 2 .If M 2 is not graphic, then M * = ∆ T (M 1 ) ⊕ ∆ T (M 2 ) by Proposition 2.6, where ∆ T (M 2 ) is a cographic matroid by Lemma 2.7, thus M • := ∆ T (M 1 ) * and M • := ∆(M 2 ) * satisfy the requirements of the first part of the statement.Assume now that M satisfies all the conditions of the proposition.Let E • and E • denote the ground sets of M • and M • , respectively, and let T := E • ∩ E • .The ground set of M can be partitioned into two bases and M contains no nontrivial tight set, hence the same holds for M * as well.Since M • ⊕ 3 M • ∈ {M, M * } contains no nontrivial tight set, the restrictions of M • ⊕ 3 M • and M • to E • − T are the same, and T is coindependent in M • , we get

Figure 6 :
Figure 6: Illustration of Remark 5.8, where M 0 is the cographic matroid of G and N i is the graphic matroid of H i for 1 ≤ i ≤ 4. The matroid M is obtained by taking the 3-sum of M 0 with the N i s in an arbitrary order along the triangles having the same color.
41, Theorem 9.1.2].Lemma 2.2.A matroid is binary if and only if C 1 △C 2 is a cycle for any cycles C 1 , C 2 .
Each of the graphs G−t 1 , G−t 2 , G−t 3 decompose into two spanning trees if and only if |E| = 2•|V |−1 and any nonempty subset U ⊆ V spans at most 2 • |U | − 2 edges in each of them.The latter condition is equivalent to |E