A Strongly Polynomial Algorithm for Linear Programs with At Most Two Nonzero Entries per Row or Column

We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Primal and dual feasibility were shown by Végh (MOR ’17) and Megiddo (SICOMP ’83), respectively. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale’s 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) by Allamigeon, Dadush, Loho, Natura, and Végh (FOCS ’22). The number of iterations needed by the IPM is bounded, up to a polynomial factor in the number of inequalities, by the straight line complexity of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By applying a reduction of Hochbaum (ORL ’04), the same bound applies to any linear program with at most two non-zeros per column or per row. To be able to run the IPM, one requires a suitable initial point. For this purpose, we develop a novel multistage approach, where each stage can be solved in strongly polynomial time given the result of the previous stage. Beyond this, substantial work is needed to ensure that the bit complexity of each iterate remains bounded during the execution of the algorithm. For this purpose, we show that one can maintain a representation of the iterates as a low complexity convex combination of vertices and extreme rays. Our approach is black-box and can be applied to any log-barrier path-following method.


INTRODUCTION
We consider linear programming (LP) in the following primal-dual form: min ⟨ , ⟩ where A ∈ R × , ∈ R , ∈ R , and A has rank .Our focus is on LP algorithms that nd exact primal and dual optimal solutions, or conclude infeasibility or unboundedness.We say that the dual progam is a two variable per inequality (2VPI) linear program if every row of A ⊤ includes at most two nonzero entries.In such a case, we refer to the pair of LPs as a 2VPI primal-dual pair.
The rst polynomial-time LP algorithms were the ellipsoid method by Khachiyan in 1979 [29] and interior point methods, introduced by Karmarkar in 1984 [28].However, it remains an outstanding open question to nd a strongly polynomial algorithm for linear programming.The question was listed by the Fields medalist Smale as one of the most prominent mathematical challenges for the 21st century [41].In such an algorithm, only poly( ) basic arithmetic operations and comparisons are allowed, and the algorithm uses polynomial space.
The notion of strongly polynomial algorithms was rst formally introduced by Megiddo [32], under the term 'genuinely polynomial'.
The same paper gave an algorithm for two variable per inequality feasibility systems, that is, for the dual feasibility problem in (LP) when all rows of A ⊤ have at most two nonzero entries.The corresponding primal feasibility problem can be reduced to the maximum generalized ow problem.For this, the rst strongly polynomial algorithm was given by Végh [50], followed by a faster and simpler algorithm by Olver and Végh [37].The minimum-cost generalized ow problem is the dual of a 2VPI LP, where the two nonzero entries in each column of A are a −1 entry and a positive entry.As discussed below, this naturally corresponds to a network ow model with multipliers on the arcs.As shown in [23], all 2VPI LPs are reducible to the dual of a minimum-cost generalized ow problem.The existence of a strongly polynomial algorithm for this problem has been a longstanding open question, mentioned e.g. in [1,10,11,20,24,36,37,50,51].Our main result resolves this question.
Theorem 1.1.There is a strongly polynomial algorithm for the minimum-cost generalized ow problem, and for two variable per inequality primal-dual pairs.

Background and Previous Work
Strongly polynomial algorithms for well-conditioned LPs.In a seminal, Fulkerson-prize winning paper [42], Tardos obtained the rst strongly polynomial algorithm for minimum-cost circulations.A particularly important technique in this paper was variable xing: by solving an approximate version of the LP with rounded costs, one can deduce that a certain variable is at the lower or upper capacity bound in an optimal solution.
Towards general LP, Tardos [43] extended this approach to obtain a strongly polynomial algorithm for 'combinatorial LPs'.More precisely, for (LP) with an integer constraint matrix A ∈ Z × , this algorithm runs in poly( , log Δ A ) iterations, where Δ A is the maximum subdeterminant of A. The running time is independent of and .In particular, this bound is strongly polynomial if all entries of A are at most poly( ), such as for multicommodity ows and other combinatorial problems.Using an interior point approach discussed below, Vavasis and Ye [49] obtained an algorithm with poly( , log ¯ A ) arithmetic operations, where ¯ A is the Dikin-Stewart-Todd condition number of the matrix A. For integer matrices, ¯ A = ( Δ A ), thus, this strengthens Tardos's result.A similar dependence, using a black-box approach extending Tardos's work [43] was obtained by Dadush, Natura and Végh [15].Further, Dadush, Huiberts, Natura and Végh [13] strengthened this dependence to poly( , log ¯ * A ), where ¯ * A is the optimized value of ¯ A under column rescalings.
Prior results on 2VPI and generalized ows.2VPI LPs are a natural class of LP that does not fall into the above 'well-conditioned' classes: even ¯ * A may be unbounded for the constraint matrix.At the same time, they form an interesting intermediate class, as it is easy to see that solving an arbitrary LP is reducible to solving one with at most three nonzero entries per row in A ⊤ .
For nding a feasible solution to a 2VPI system, Megiddo's [32] approach relied on parametric search.A faster parametric search algorithm was given by Cohen and Megiddo [10].Hochbaum and Naor [24] used an e cient Fourier-Motzkin elimination to obtain what is still the fastest deterministic approach.Dadush, Koh, Natura and Végh [14] used a variant of the discrete Newton method.Recently, Karczmarz [27] gave an improved randomized strongly polynomial algorithm, also using parametric search.
Consider now monotone 2VPI (M2VPI) systems, where each inequality has at most one positive and at most one negative entry.If such an LP is bounded, then there exists a unique pointwise minimal solution and a unique pointwise maximal solution.Already the algorithm in [32] can be used to nd these solutions.As noted by Adler and Cosares [1], an M2VPI linear program is strongly polynomially solvable if ≥ 0 or ≤ 0. Norton, Plotkin and Tardos [36] gave a strongly polynomial algorithm for a constant number of nonzero demands.
The generalized ow problem is (after normalization) the dual of the M2VPI problem.In this problem, we are given a directed graph = ( , ) with node demands , ∈ and arc costs and gain factors > 0 for ∈ .While traversing the arc = ( , ), the ow value gets multiplied to .In the minimum-cost generalized ow problem, we need to exactly satisfy all node demands at a minimum cost.The maximum generalized ow problem is the special case when the objective is to maximize the net ow reaching a special sink node .This is a fundamental network optimization model that traces back to Kantorovich's 1939 paper [26] introducing linear programming.Generalized ow networks can be used to model transportation of a commodity through a network with leakages, or conversions between various equities in nancial networks, as well as generalized assignment problems.We refer the reader to [2,Chapter 15] for further applications.
Goldberg, Plotkin, and Tardos [20] gave the rst weakly polynomial combinatorial algorithm for the maximum generalized ow problem.This was followed by a signi cant number of further such algorithms, such as [11,21,38,39,44,51], see further references in [37].In particular, [11] gave a strongly polynomial approximation scheme, i.e., a strongly polynomial algorithm that achieves a xed fraction of the optimum ow value in a capacitated generalized ow network.The strongly polynomial algorithms by Végh [50] and Olver and Végh [37] rely on the variable xing technique, however, in a new, 'continuous' scaling framework.While the original LP can be ill-conditioned, variable xing is still possible, since the dual solutions can be used to 'relabel' the ow to make it 'locally' amenable to classical network ow arguments.
However, relabelling heavily relies on the special cost function of the ow maximization problem, and does not seem to be extendable to the minimum-cost version.For solving the minimum-cost generalized ow problem, the only known (weakly polynomial) combinatorial approach is the ratio-circuit cancelling algorithm by Wayne [51].The fastest previous weakly polynomial algorithms can be obtained using interior point methods; an early such example is by Vaidya [45].Daitch and Spielman [16], and Lee and Sidford [31] gave fast algorithms for obtaining an additive -approximation; however, such approximation cannot be used to obtain exact optimal solutions.We also note that the latter results only apply for lossy ows, i.e., with gain factors ≤ 1.
Interior Point Methods and their limitations.Interior point methods (IPMs) give the fastest current weakly polynomial algorithms for general LP, see [12,25,46,47] as well as for special classes such as minimum-cost circulations [8,9] and multicommodity ows [48].
They are also a potent approach in the context of strongly polynomial computability, and form the basis of our result.
The algorithms discussed next fall into the class of primal-dual path-following algorithms.A key concept here is the central path, the algebraic curve formed by minimizers of ⟨ , ⟩ − =1 log( ) for > 0. As → 0, the limit of the central path is an optimal solution.Path-following methods maintain iterates in a certain neighborhood of the central path while geometrically decreasing , and thus, the optimality gap.The logarithmic barrier function above can be replaced by more general barrier functions.The a ne scaling step is a standard way to nd a movement direction.This can be interpreted as a least square computation in the local norm induced by the Hessian of the logarithmic barrier function.
Layered least squares (LLS) IPMs were introduced in the in uential work of Vavasis and Ye [49].The LLS step in the Vavasis-Ye algorithm decomposes the variables into di erent layers based on the values of the current iterate.The step direction is determined as a sequence of least squares computations that prioritizes decreasing variables at lower layers.Roughly speaking, such steps enable to traverse arbitrarily long but relatively straight segments of the central path in a single iteration.Combinatorial progress is measured by crossover events, where two variables get reordered consistently with their order in the limit optimal solution.This is very di erent from the variable xing technique prevalent in the combinatorial approaches discussed above.In particular, while we can infer the occurrence of a new crossover event within a certain number of iterations, the argument only shows existence, and we cannot identify the participating variables.The condition number ¯ A appearing in the running time is a bound on the norms of oblique projections.
This led to a line of research on improved combinatorial IPMs [30,[33][34][35].The paper [13] revealed that ¯ A is closely related to the circuit imbalance measure A that bounds the maximum ratio of two nonzero entries of an elementary vector in the kernel of A. Moreover, they obtained an LLS algorithm invariant under column rescaling, thus improving the dependence to the best * A value achievable under column rescalings.
The above results may raise hopes to nding a strongly polynomial IPM.However, the papers by Allamigeon, Benchimol, Gaubert and Joswig [3], Allamigeon, Gaubert and Vandame [6], and Zong, Lee and Yue [53] yield a surprising negative answer.By analyzing the tropical limits of linear programs, these papers exhibit parametric families of LPs such that for suitably large parameter values, no path-following method can be strongly polynomial.This was rst shown for the standard logarithmic barrier [3], and later for arbitrary self-concordant barriers [6].

The Subspace Layered Least Squares Interior Point Method and Straight Line Complexity
The primal central path has a natural dual counterpart.The primaldual central path point ( cp ( ), cp ( )) is the unique pair of primal and dual feasible solutions to (LP) such that cp ( ) cp ( ) = for all 1 ≤ ≤ .Thus, the duality gap between cp ( ) and cp ( ) is .
The lower bounds in [3,6] are ultimately based on the following insight.The trajectory of any path-following IPM is a piecewise linear curve in the neighborhood of the central path; the number of pieces correspond to the number of iterations.Thus, a lower bound on the number of any piecewise linear curve in the neighborhood provides a lower bound on the number of iterations.For the examples in these papers, exponential lower bounds are shown.
The recent algorithm designed by Allamigeon, Dadush, Loho, Natura and Végh [4] complements these negative results by a positive algorithmic bound.Namely, they provide an IPM whose number of iterations matches such a lower bound within a strongly polynomial factor.Let us elaborate on the lower bound.
Assume (LP) is feasible and bounded with optimum value ★ .Given ≥ 0, we denote by the feasible sublevel sets.They correspond to the sets of the primal and dual feasible points ( , ) with objective value within from the optimum ★ , respectively.Assuming that (LP) has strictly feasible primal and dual solutions, the max central path is de ned as the parametric curve ↦ → ( ) ≔ ( ( ), ( )) ∈ R2 + , where ( ) ≔ max{ : ∈ P } , ( ) ≔ max{ : for all ∈ [ ].The max central path can be seen as a combinatorial proxy to the central path.In particular, for = , cp ( ) ∈ P and cp ( ) ∈ D , and it is easy to see that ( )/ ≤ cp ( ) ≤ ( ) and ( )/ ≤ cp ( ) ≤ ( ).
For each 1 ≤ ≤ , the function ↦ → ( ) is a piecewise linear concave function.It corresponds to a trajectory of the shadow simplex algorithm that interpolates between the objective functions ⟨ , ⟩ and − .The breakpoints correspond to basic feasible solutions, and therefore the number of linear pieces is at most the number of vertices, that is, at most 2 .We will use the following de nition.
De nition 1.2.Let : R + → R + be a function and ∈ [0, 1].The straight line complexity of with respect to , denoted SLC ( ), is the in mum number of pieces of a continuous piecewise linear function ℎ : R + → R + where ≤ ℎ ≤ . 1he number of iterations taken by the Subspace Layered Least Squares (SLLS) IPM in [4] can be bounded by the sum of straight line complexities of each coordinate of the (primal) max central path.We now state the guarantees of SLLS IPM as given in [5], which strengthens the main result of the conference version [4] by ensuring that each iteration of the IPM can be implemented in strongly polynomial time.

Theorem 1.3 ([4, 5]).
There is an interior-point method that given an instance of (LP) and strictly feasible solutions , nds a pair of primal and dual optimal solutions in min many iterations. 2 The algorithm can be implemented in the real RAM model, moreover, each iteration runs in strongly polynomial time in the Turing model.
We explain the real RAM and strongly polynomial computational models in Section 1.3 below.The SLLS IPM requires at most a ˜ ( 1.5 ) factor more iterations than any path-following IPM for any selfconcordant barrier function.This is because it can be shown that each SLC ( ) gives a lower bound on the number of piecewise linear segments traversing a corresponding wide neighborhood.Moreover, as noted above, SLC 1 ( ) ≤ 2 , thus, the number of iterations is always at most singly exponential.
We note that the theorem could be equivalently written in terms of the dual straight line complexities (see [5,Lemma 4.5]).Further, we note that the neighborhood parameter is not important for the overall bound.It is not di cult to show that for 0 According to Theorem 1.3, analyzing the number of iterations of SLLS IPM boils down to upper-bounding the straight line complexities of the variables.Note that this is a purely geometric question about understanding the structure of univariate piecewise linear functions ( ).

Computational Models
There are multiple related, yet distinct notions of a strongly polynomial computational model.Smale's question was posed in the Blum-Shub-Smale (BSS) real model of computation [7].In this model, the input can be given by arbitrary real numbers, and one step may compute a rational polynomial function of the previously computed quantities with real coe cients, or make comparisons between two quantities.In the more restrictive real RAM model, one can perform a sequence of elementary arithmetic operations (+, −, ×, /) and comparisons (≥) on real numbers.In this paper, we say that an algorithm is polynomial in the real RAM model if the number of elementary arithmetic operations and comparisons is bounded polynomially in the dimension of the input; in the case of LP, this is = × + + .We now turn to the Turing model.Consider a problem where the input is given by integers; for LP, the input (A, , ) is described by = 2( × + + ) integers representing the rational entries.An algorithm is strongly polynomial in the Turing model (see [22]), if it only performs poly( ) (in the LP case, this means poly( , )) elementary arithmetic operations and comparisons as in the real model.Additionally, the bit-complexity of the numbers during the computations must remain polynomially bounded in the encoding length of the input.Equivalently, the algorithm must be PSPACE.The model has an ambiguity regarding how divisions can be implemented, see discussion of variants in [22,Section 1.3].The results of this paper work with the most restricted setting: we maintain rational representations ( , ) of all numbers during the computation, and division / ′ ′ corresponds to computing the representation ( ′ , ′ ).
While a strongly polynomial algorithm in the Turing model implies a polynomial algorithm in the real RAM model, the converse is not necessarily the case: enforcing PSPACE may be challenging.For example, Gaussian elimination needs to be done carefully to keep the sizes of numbers under control, see [17] and [22,Section 1.4].The LLS interior point methods [4,13,[33][34][35]49] are polynomial in the real RAM model3 whenever log( ¯ A ) = poly( ).However, we are not aware of any strongly polynomial implementation of such algorithms in the Turing model.The principal di culty in this regard is keeping the bit-complexity of the iterates produced by the IPM uniformly bounded using only the allowed operations (+, −, ×, /).In particular, truncating the bit representation of the current iteration to -bits of precision, where depends on the bit-length of the input cannot be achieved using (1) basic operations.
In the weakly polynomial model, i.e., when running time dependence on the total encoding length is allowed, the bit complexity of the algorithms can be controlled by approximately solving linear systems and roundings.Recent work by Ghadiri, Peng, and Vempala [19] developed general tools that enable to keep the bit complexity of recent fast IPMs under control.However, these techniques are not applicable in the strongly polynomial model.In particular, they require estimates on parameters such as the total bit length of the input or the condition number of the matrix.They also require rounding to a xed number of bits depending on such numerical parameters.As mentioned above, in the most stringent de nition of strongly polynomial time, this cannot be done.
The implementation of the SLLS IPM given in [5], which we rely on, guarantees that each iteration of the IPM is strongly polynomial in the Turing model.More precisely, given the constraint matrix and the current iterate ( , ) as input, the IPM computes the next iterate ( ′ , ′ ) in time strongly polynomial in the input ( , , ).This in particular implies that the bit complexity of the iterates grows at most by a polynomial factor in each iteration.This is however insu cient for controlling the bit complexity over many iterations.We resolve this issue by providing a combinatorial rounding scheme, which maintains a representation of the current iterate as a low complexity convex combination of vertices and extreme rays.We describe this in further detail in Section 1.4.3.
We note that the algorithms presented in this paper are fully deterministic.

Our Contributions
We prove Theorem 1.1, i.e., give a strongly polynomial algorithm for the minimum-cost generalized ow problem by showing that the total number of iterations of the SLLS IPM by [4] is strongly polynomially bounded, and that the SLLS IPM can be implemented in strongly polynomial time in the Turing model.Our result has three main ingredients: (1) Straight line complexity bound: We establish in Theorem 3.1 a strongly polynomial bound SLC ( ) = ( ) on the straight line complexities of the variables in the minimumcost generalized ow problem, with = Ω(1/( 2 )), where is the number of arcs and is the number of nodes of the graph.This bound applies for the uncapacitated version of the problem as described above; if in addition arcs have capacities, the bound becomes ( 2 ).
(2) Initialization: IPMs require a strictly feasible and well centered starting point ( 0 , 0 ), even though a strictly feasible (or even a feasible) solution may not exist.We present a careful initialization scheme that solves linear programs in three stages, and preserves the straight line complexity bounds.
(3) Implementation in the Turing model: We show that the bitlength of the computations can be controlled in a model using only basic arithmetic operations and comparisons.
The straight line complexity is established via a combinatorial argument using structural properties of generalized ows.In contrast, the initialization and implementation tasks are applicable for general LP, and can be seen as a direct strengthening of the result in [4,5].We now elaborate on each of these parts, and highlight the main technical ideas.

Straight Line Complexity
Bound for Generalized Flows.Theorem 1.3 enables to bound the number of iterations in SLLS IPM by bounding the straight line complexities SLC ( ) for a suitable > 0. In the rst step, we reduce this to an even more concrete combinatorial question of circuit covers as explained next.
Circuit covers.For the purposes of analyzing straight line complexities, we can assume that a pair of primal and dual optimal solutions ( ¯ , ¯ ) to (LP) is provided.
For any vector ℎ ∈ ker(A), with ⟨ , ℎ⟩ ≥ 0 we can de ne the function ¯ ℎ ( ) : R + → (R + ∪ {∞}) by moving from ¯ in the direction of ℎ; this is called the ℎ-curve from ¯ .Namely, we de ne ¯ ℎ ( ) = ¯ + ( )ℎ, where ( ) is chosen maximally so that ¯ ℎ ( ) is feasible, and has cost at most larger than the cost of ¯ .For every ∈ [ ], the -th coordinate ¯ ℎ ( ) can easily be seen to be a piecewise linear concave function with two pieces, the rst with slope ℎ /⟨ , ℎ⟩ and the second constant.Note that In the linear space ker(A), an elementary vector is a support minimal nonzero vector, and the support of an elementary vector is called a circuit.Note that the latter coincides with the notion of circuits of the linear matroid of A; each circuit corresponds to a onedimensional subspace of elementary vectors.We let E (A) denote the set of all elementary vectors.
Given an optimal solution ¯ , let us consider the augmentations from ¯ by an elementary vector ℎ.Noting that ¯ ℎ ( ) is invariant under rescaling ℎ to ℎ for > 0, this gives one function per circuit.For any coordinate ∈ [ ], let us now consider the pointwise maximum at the -th coordinate ˆ ( ) = max{ ¯ ℎ ( ) : ℎ ∈ E (A)}.This is a piecewise linear function, but is not concave.Note also that the number of pieces can be exponential.Nevertheless, using standard circuit decomposition techniques, it is not di cult to show that ˆ ( ) approximates ( ) up to a factor : ( )/ ≤ ˆ ( ) ≤ ( ).Our strategy to obtain SLC bounds is by constructing circuit covers.Given a primal optimal solution ¯ , an index ∈ [ ] and > 0, we say that a set of vectors ⊆ ker(A) is an -circuit cover of with respect to ¯ if for every ℎ ′ ∈ E (A), there is a ℎ ∈ that -dominates ℎ ′ on .Then, the piecewise-linear function Circuit covers for generalized ows.We work with minimum-cost generalized ow in its capacitated form, where arcs may have capacities, and all demands are zero; this reduction yields only capacitated arcs when starting from the demand form.The circuits in this version of the generalized ow problem correspond to simple combinatorial structures: namely, a circuit either corresponds to a conservative directed cycle, where the product of the gain factors is one, or a 'bicycle', namely, a ow generating cycle connected by a path to a ow absorbing cycle.The latter are cycles where the product of the gain factors is greater and less than one, respectively.These structures played a fundamental role in all prior works on generalized ows, see e.g., [20,37,50,51], as well as for 2VPI algorithms, e.g., [10,14,27,32].
We construct a circuit cover of size ( ) to bound the straight line complexity of the variables in the generalized ow problem.The basis of our construction is path domination.Let us x two nodes and in the graph.We demonstrate a small collection of -walks that "dominate" the collection of all -paths in a certain sense.The general cover will be constructed by combining such walks.We now highlight the main ideas of path domination.
Consider an -walk ; this induces an -ow ¯ on the walk.We de ne the function ì : R 2 + → R + such that ì ( , ) denotes the maximum amount of ow that can be sent from to if there are units available at , each step of the walk satis es the capacity bound, and the cost incurred in any step of the walk is at most .This corresponds to a certain maximal scaling of ¯ .This scaling may have total cost larger than , and moreover since an arc of the graph can be used multiple times, it may also violate arc capacities.But, as long as the walk is -recurrent, meaning it uses each edge at most times, scaling down by will yield a feasible ow with total cost at most .There are two possible bottleneck arcs that prevent a larger scaling of ¯ from being used: a cost bottleneck arc c where c ¯ c is maximal, and a ow bottleneck arc f where ¯ f / f is maximal, where and denote the cost and capacity of arc , respectively.We associate the combinatorial signature ( c , f , ⪯) or ( c , f , ≻) with , where ⪯ means that c precedes or equals f on the walk , and ≻ means that f precedes c .
We say that the -walk ′ dominates if ì ′ ≥ ì .Denoting the number of nite capacity arcs by ¯ , we are able to show the existence of an ( ¯ )-sized family of -recurrent -walks that dominate all -paths.The family is constructed by xing the signature, and from each signature, selecting the best walk for the three segments de ned by , , and the two bottleneck arcs, such that each segment is highest gain subject to recurrence bounds and not having other bottlenecks.
Once we have this path domination result, we can use this to demonstrate domination of more complicated collections of objects with small dominating sets, and eventually all circuits.It easily follows, for instance, that there is a small collection of -walks with the property that for any ow-generating cycle containing , there exists a walk in this collection that dominates , in the sense that for every choice of cost bound ∈ R + , at least as much excess can be created using than .From this, by 'composing' dominating sets for cycles and paths, we obtain a small dominating set for the collection of all bicycles in the graph.

Initialization.
A strongly polynomial straight line complexity bound implies a strongly polynomial iteration bound of the SLLS IPM; however, it requires an initial point ( 0 , 0 ) ∈ P ++ × D ++ near the central path.Such a point may not even exist; in fact, the primal or dual programs in (LP) may be infeasible.Whereas one could use the combinatorial algorithms to decide primal [37] and dual [24] feasibility, these algorithms do not directly yield strictly feasible solutions (which may again not exist).
The situation is analogous to Simplex, where Stage I can be used to nd a feasible solution by solving an auxiliary LP.Various initialization methods have been developed for IPMs, but none of these is directly applicable for our purposes: only solving auxiliary systems with small straight line complexity, while remaining in the strongly polynomial model.
A common initialization technique is the self-dual homogenous formulation [52].However, writing the self-dual formulation of a generalized ow LP results in a more complicated problem and it is not clear if the straight line complexity admits a similar bound.(Note also that in the simpler case of (standard) network ows, the constraint matrix is totally unimodular, while the combined matrix does not have this property.) We present two initialization methods.Our rst approach uses a 'big-' method, as in [49].Let us create a negative copy of each variable with a large penalty cost.That is, one can replace the primal system using variables ( , ′ , ′′ ) ∈ R 3 in the form Here, ′ represents a negative copy of each variable and ′′ corresponds to a slack variable for the box constraint 0 ≤ ≤ 2 1 .Such a system, along with its dual, is easy to initialize for su ciently large .Moreover, the constraint matrix remains 'nice', e.g., it can be still interpreted as a (capacitated) generalized ow problem, where the ′ variables correspond to expensive reverse arcs.As long as there exists a pair of primal and dual optimal solutions ( ★ , ★ ) to (LP) with ∥ ★ ∥ ∞ , ∥ ★ ∥ ∞ < 1 , these will also be optimal solutions to the extended formulation.
However, nding a suitable large becomes challenging.In [49], such a bound is derived based on ¯ A .This is hard to compute in general; one could use a repeated guessing of this condition number, but this would lead to a log log ¯ A running time dependence.Bounds on the norms of optimal solutions are routinely derived using bitcomplexity arguments, see e.g.[22]; however, this is also not possible in the strongly polynomial model.
To address this, we use the existing strongly polynomial algorithms of e.g., [37] and [24] to solve up to primal and dual feasibility problems to rst obtain maximum support primal and dual solutions.We then reduce the problem to a system with a pair of strictly positive primal and dual solutions.The reduction is achieved by deleting some variables and projecting out some others.In the generalized ow problem, these amount to graphical operations of deletions and contractions, and thus preserve the generalized ow structure.Given the strictly positive primal and dual solutions ( ˆ , ˆ ), choosing larger than ⟨ ˆ , ˆ ⟩ divided by the smallest entry of ( ˆ , ˆ ) guarantees that ∥ ★ ∥ ∞ , ∥ ★ ∥ ∞ < 1 for any pair of primal and dual optimal solutions.Whereas the above approach can implement the big-method, it is only applicable to the particular minimum-cost generalized ow setting as it requires feasibility solvers.Also, it needs to solve 2 systems as preprocessing.We also develop a more principled, multistage initialization strategy that is applicable to general LP, preserves straight line complexity, and only requires solving four IPM problems.Since we will need to solve di erent LPs derived from (LP), one needs to clarify what 'preserving straight line complexity' means.We de ne SLC (A) as the maximum value of SLC ( ) for any variable ∈ [ ] in any LP of the form (LP) with constraint matrix A, but taking any possible right-hand side ∈ R and cost ∈ R .All our auxiliary LPs will have SLC bounded by SLC (B), is the matrix also used in the big-M formulation (3).Our strategy can be interpreted as a facial reduction strategy, we carefully x or project out variables that yield an equivalent LP to the original one, and where strictly feasible primal and dual solutions in fact exist.Throughout the process, the solutions from a previous stage provide a starting point near the central path.
1.4.3Implementation in the Turing model.As discussed in Section 1.3, to obtain a strongly polynomial algorithm in the Turing model one needs to devise a new rounding approach, as the previous ones rely on bit-complexity information and rounding that are not implementable in the strongly polynomial model.
The main ingredient is a general strongly polynomial technique to keep the bit-complexity of all iterations polynomially bounded in the input encoding length.This technique is not particular to the SLLS algorithm but can be used for any path-following method.The main subroutine takes an iterate ( , ) in the central path neighborhood, and computes ( ˜ , ˜ ) that is in a slightly larger neighborhood, may have slightly worse optimality gap, but its encoding length is polynomially bounded in the length of the input.
To argue about the encoding length, we 'anchor' the point ( ˜ , ˜ ) to vertices of the primal and dual polytopes.In strongly polynomial time, we can write a Minkowski-Weyl decomposition of and using vertices and extreme rays.However, we cannot simply round the coe cients.In particular, it is possible that ( , ) is written as a 'highly unstable' convex combination of primal and dual vertices such that either ⟨ , ⟩ < ⟨ , ⟩ /2 or ⟨ , ⟩ < 2 ⟨ , ⟩ for each pair of primal and dual vertices ( , ).We proceed in two stages.First, we try to nd a value ★ ≈ ⟨ , ⟩ / such that ★ has small encoding length.This is easy as long as the combination contains primal and dual vertices ( , ) with ⟨ , ⟩ /2 ≤ ⟨ , ⟩ ≤ 2 ⟨ , ⟩.In the 'highly unstable' situation as above, it turns out that the direction from ( , ) pointing towards a pair of primal and dual vertices ( , ) with much better gap is a very good movement direction of the IPM.Hence, we can replace ( , ) during the rounding step by a much better iterate that is also numerically more stable.In the second stage, we add a cost bound to our feasible region according to ★ .On this bounded polytope, we can now nd a Minkowski-Weyl decomposition and simply round the coe cients.The guarantees of this rounding are based on the near-monotonocity property of the central path.

Organization
The rest of the paper is structured as follows.Section 2 introduces some necessary background, in particular, regarding straight line complexity and circuits.Section 3 analyzes the straight line complexity of minimum cost generalized ows.It contains an overview of the proof strategy for obtaining a weaker bound of ( 4 ).The stronger bound of ( 2 ) and all the proofs can be found in the full version.
The initialization procedure for the IPM, as well as the rounding procedure needed to control the bit-complexity, are also deferred to the full version.
We denote the primal and dual feasible regions of (LP) by respectively.We let P ++ ≔ P ∩ R ++ and D ++ ≔ D ∩ R ++ denote the strictly feasible regions.Interior point methods require P ++ , D ++ ≠ ∅.We do not make this assumption in general; in the full version, we show how one can use a sequence of reductions to simpler IPM problems to rst either nd a suitable initial point ( 0 , 0 ), or conclude infeasibility or unboundedness of the input LP.
Recall the de nitions of the sublevel sets P , D in (1) as the set of primal and dual solutions with objective value within from the optimum value ★ .
Our main tool for analyzing SLC are circuits.

De nition 2.2 (Elementary vectors and circuits)
. Let A ∈ R × and assume ker(A) ≠ {0 }.A vector ∈ ker(A) is an elementary vector in ker(A) if is a support-minimal nonzero vector in ker(A).
We let E (A) denote the set of all elementary vectors.A set ⊆ [ ] is a circuit of A if it is the support of some elementary vector; we let C(A) ⊆ 2 [ ] denote the set of circuits.We say that a vector ∈ R conforms to ∈ R if > 0 whenever ≠ 0. A conformal circuit decomposition of a vector ∈ ker(A) is a decomposition of the form = ℓ =1 ( ) , where (1) , . . ., (ℓ ) ∈ E (A), ℓ ≤ , and each ( ) conforms to .This notion can be seen as a generalization of the cycle decomposition of circulations for networks ows.The existence of such a decomposition is well-known, see e.g., [18,40].Proposition 2.3.For every A ∈ R × , every vector ∈ ker(A) admits a conformal circuit decomposition.

Straight Line Complexity and Circuits
In this section, we establish an intimate connection between the SLC of an LP and its circuits.Recall the de nition ( 2) of the max central path ( , ) from the introduction.
Note that ¯ ℎ = ¯ ℎ for all > 0. It is easy to see that with the convention that we omit the rst term from the minimum if ⟨ , ℎ⟩ = 0 ( ¯ ℎ is a constant function in this case).The next lemma shows that for every ∈ [ ] and ≥ 0, the th coordinate of the max central path at is upper bounded by a circuit augmentation from an optimal solution, up to a factor .
De nition 2.7 (Circuit cover).Let ¯ be a primal optimal solution to (LP).Let ∈ [ ] and ≥ 0. An -primal circuit cover of with respect to ¯ is a set ⊆ ker(A) which -dominates E (A) on with respect to ¯ .
The utility of a circuit cover is illustrated by the following lemma.Note that (0) is the maximum value of the -th coordinate in an optimal solution.Assuming (0) < ∞, there exists a (basic) optimal solution ¯ such that (0) = ¯ .

Let
= ( , ) be a directed multigraph with arc capacities ∈ (R ++ ∪ {∞}) and gain factors ∈ R ++ .A ow in is any nonnegative vector ∈ R + .Note that a ow is allowed to violate arc capacities.For a node ∈ , we denote in ( ) and out ( ) as the set of incoming and outgoing arcs of respectively.The net ow of at node is de ned as Let ∇ ∈ R denote the vector of net ows at every node in .A ow is a circulation if ∇ = 0.For , ∈ , we denote by , ⊆ the subset of arcs with tail and head .
An instance of the minimum-cost generalized ow problem is given by a directed multigraph = ( , ) with node demands ∈ R , arc costs ∈ R , capacities ∈ (R ++ ∪ {∞}) and gain factors ∈ R ++ .It can be formulated as the following LP: Throughout this section, we will use for the number of nodes of and for the number of arcs; note that applied to (MGF), this is the reverse of the convention used for general LPs.Let ⊆ denote the subset of arcs with nite capacities.We de ne := | | for the number of nite capacity arcs.
We assume that (MGF) has a nite optimum, since otherwise the max central path does not exist; our initialization procedure will ensure that we only consider such instances.For an arc ∈ , we denote as the coordinate of the primal max central path which corresponds to the ow variable .For a capacitated arc ∈ , we also denote ← as the coordinate of the primal max central path which corresponds to the slack variable − .Our goal in this section is to prove the following bound on the SLC of each coordinate of the primal max central path.
In the full version, we prove a stronger bound of ( ( + )) on the SLC.
It will be more convenient to work with the special case of (MGF) where = 0 and ≥ 0. Note that 0 is trivially an optimal solution.One can show that it su ces to bound SLC ( ) for every arc in this instance in order to prove Theorem 3.1.This is achieved by replacing the cost with any optimal reduced cost, and considering the residual graph with respect to any optimal solution * to (MGF).Let denote the set of nite capacity arcs in the reduced instance, and de ne ¯ ≔ | |.By picking * to be basic, one can further ensure that ¯ ≤ + .See the full version for more details.

SLC Bounds via Domination
We will follow exactly the general plan discussed in Section 2.1: we demonstrate the existence of a small primal circuit cover.Recall the notion of an elementary vector from De nition 2.2.The following is precisely this same notion, in the context of generalized circulations.
In order to characterize elementary circulations, we need the following concepts.
De nition 3.3 (Walk, trail, path and cycle).A walk is a sequence = ( 0 , 1 , . . ., ℓ , ℓ ) where is an arc from −1 to for all ∈ and open otherwise.If ≠ for all ≠ , then it is called a trail.If 0 = ℓ and ≠ for all 0 ≤ < ≤ ℓ, then it is called a cycle at 0 .
The gain of is ( ) ≔ ℓ
For an -walk and an -walk , we denote ⊕ as the concatenated -walk.We use ( ) and ( ) to refer to the node set and arc set of a walk .
De nition 3.4 (Objects).A ow-generating object at ∈ is a pair ( , ) where is a ow-generating -walk for some ∈ and is an -walk.It is simple if is a cycle, is a path, and ( ) ∩ ( ) = { }.A ow-absorbing object at ∈ is a pair ( , ) where is an -walk for some ∈ and is a ow-absorbing -walk.It is simple if is a path, is a cycle, and ( ) ∩ ( ) = { }.
A conservative object is a triple ( , , ) where either (i) for some , ∈ , is a ow-generating -walk, is anwalk, and is a ow-absorbing -walk; or (ii) is a conservative closed -walk for some , is the trivial path at , and = . 4he object is simple in case (ii) if is a cycle, and in case (i) if ( , ) and ( , ) are simple, and -( ) ∩ ( ) = ∅ in the case that ( ) ≠ ∅; or -the intersection of and is a path, in the case that ( ) = ∅.
Since an arc can be used multiple times in a walk or an object, we introduce the notion of recurrence to keep track of its multiplicity.De nition 3.5 (Recurrence).A walk is called -recurrent if every arc appears at most times as a step in .Similarly, an object is -recurrent if every arc appears at most times in total as a step in some constituent walk of .
Note that -recurrent only upper bounds the number of repetitions of an arc; for ≤ ℓ, any -recurrent walk is also ℓ-recurrent.
When considering ows supported on the arc set of a walk, it will be important to be able distinguish between ow on di erent "steps" of the walk that involve the same arc of the graph.We do this formally by de ning the "splitting" of a walk, which simply makes parallel copies of arcs to turn the walk into a corresponding trail.
De nition 3.6 (Splitting).Let ˜ = ( , ˜ ) be the directed multigraph with the same node set as , but with 10 parallel copies of each arc, each with the same gain factor, cost and capacity as the corresponding arc in .(The choice of 10 is just to be su ciently large for our purposes.)For each ∈ , we use 1 , 2 , . . ., 10 to index the corresponding copies in ˜ .Given a 10 -recurrent walk = ( 0 , 1 , 1 , 2 , . . ., , ) in , we de ne a splitting of to be a trail ˜ = ( 0 , 1  1 , 2 , 2 2 , . . ., , ) in ˜ , where for each ≠ with = , ≠ .
Given an object , a splitting of is a tuple of trails in ˜ , each being a splitting of the corresponding walk in , and where in addition the trails are arc-disjoint.
Note that up to trivial relabelling of copies of arcs, the splitting of a walk or object is unique.
For an object or walk , we use ( ) and ( ) to denote its node set and arc set respectively, and also use ( ˜ ) ⊆ ˜ to denote the arc set of a splitting ˜ .
De nition 3.7 (Induced ows).Given an -walk with splitting ˜ , we say that ˜ ∈ R ˜ + is a ow induced by ˜ if ˜ is nonzero, supported on ( ˜ ), and ˜ = ˜ for any pair of consecutive arcs , ∈ ( ˜ ) where comes before in ˜ .We say that ¯ ∈ R + is a ow induced by if ¯ is the projection onto of a ow ˜ induced by a splitting of , that is, ¯ = ˜ .Given a ow-generating object = ( , ) at , a ow induced by a splitting ˜ = ( ˜ , ˜ ) is a vector ˜ ∈ R ˜ + that can be written as a sum of a ow induced by ˜ and a ow induced by ˜ , and where in addition ∇ ˜ = 0 for all ≠ .The de nition for owabsorbing objects is completely analogous; and for a conservative object = ( , , ), ˜ should satisfy ∇ ˜ = 0, and be a sum of ows induced by the components of a splitting of .
We are ready to characterize elementary circulations.Lemma 3.8.A ow is an elementary circulation if and only if it is induced by a simple conservative object.
We remark that all ows induced by an object are the same up to scaling.The following will be a crucial notion: it is the largest possible ow induced by a walk (or object), with the property that on each step of the walk or object, the ow does not exceed the capacity of the arc, and the cost of that step ( ow times arc cost) does not exceed a given bound .
De nition 3.9.Let be a walk, with ˜ a splitting of and ˜ a ow induced by ˜ .De ne ˜ : R + → R ˜ + to be the function that maps ˜ ( ) to the largest scaling of ˜ so that ˜ ≤ and ˜ ≤ for each ∈ ( ˜ ).Then let : R + → R + be the projection of ˜ onto , i.e., ( ) = ˜ ( ).
We de ne ˜ ( ) and ( ) for an object with splitting ˜ in identical fashion.

Note that
( ) and ( ) do not depend on the choice of splitting, and so are well-de ned.
Remark 3.10.This de nition is closely related to the de nition of ℎ-curves for general LPs provided in De nition 2.4.It is more general, in that we de ne for objects that are not conservative, and hence which do not lie in the kernel.If we consider a conservative object , and take ℎ to be a ow induced by , then and the ℎ-curve 0 ℎ are "close": if is -recurrent, then 1 ( ) ≤ 0 ℎ ( ) ≤ ( ).The reason that they are not identical, only within a factor , is simply because of the per-step nature of the capacity bounds (meaning ( ) might overload an arc by a factor ) and cost bounds (meaning ( ) could have total cost , given each arc could in principle contribute a cost of ).
Let E denote the collection of simple conservative objects.For any ∈ and collection U of conservative objects, we de ne U := ∈ U .The following is essentially Lemma 2.8 for this setting, taking into account the scaling necessary to make ( ) feasible for cost bound .Lemma 3.11.Fix any edge ∈ .Suppose that D is a collection of -recurrent conservative objects that -dominates E on , in that D ≥ E for some constant .Then SLC /( 2 ) ( ) ≤ |D |.
As such, our goal is now to demonstrate such a dominating collection D; we will do this with = 1, = ( ) and |D | = ( 2 ¯ 2 ).

Path Domination
While our goal is to dominate simple conservative objects, we build up to this in stages.Our rst step is to build a small collection ofwalks that dominate all -paths; these will become building blocks in the next section.
In words, ì ( , ) is the maximum amount of ow that can be sent to with a ow induced by , given that each step respects the cost and capacity bounds, and that there are only units available at to be sent.
The function ì for a given -walk has a very simple form.We can write it as ì ( , ) = min{ /cost( ), ( ) , limit( )}, where cost( ) is the largest cost of a step of the walk per unit of ow measured at ; ( ) is the gain of the walk; and limit( ) is the maximum amount of ow that can arrive at given that each step respects the capacity.
For walks, we use the following stronger "bivariate" notion of domination.This will be crucial when we come to use this to demonstrate domination, in the usual sense, for ow-generating/ ow-absorbing objects and eventually conservative objects in the following sections.
Write P ( , ) for the set of all -paths for any distinct , ∈ .Given a collection W of -walks, de ne The main theorem of this section is the following.It shows that the collection of -paths can be dominated by a small collection of -recurrent -walks: for any -path , any cost bound , and any amount of ow available at , there is a walk in the collection that does a better job at sending ow to under the same cost and ow restrictions.