Probabilistically Checkable Reconfiguration Proofs and Inapproximability of Reconfiguration Problems

Motivated by the inapproximability of reconfiguration problems, we present a new PCP-type characterization of PSPACE, which we call a probabilistically checkable reconfiguration proof (PCRP): Any PSPACE computation can be encoded into an exponentially long sequence of polynomially long proofs such that every adjacent pair of the proofs differs in at most one bit, and every proof can be probabilistically checked by reading a constant number of bits. Using the new characterization, we prove PSPACE-completeness of approximate versions of many reconfiguration problems, such as the Maxmin 3-SAT Reconfiguration problem. This resolves the open problem posed by Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno (ISAAC 2008; Theor. Comput. Sci. 2011) as well as the Reconfiguration Inapproximability Hypothesis by Ohsaka (STACS 2023) affirmatively. We also present PSPACE-completeness of approximating the Maxmin Clique Reconfiguration problem to within a factor of nε for some constant ε > 0.


Introduction
Reconfiguration problems ask to decide whether there exists a sequence of operations that transform one feasible solution to another.A canonical example is the 3-SAT RECON-FIGURATION problem, which is known to be PSPACE-complete [GKMP09].
It is natural to consider its approximate variant, whose complexity was posed as an open problem in [IDHPSUU11].
The main contribution of this paper is to prove PSPACE-completeness of approximating the MAXMIN 3-SAT RECONFIGURATION problem within a constant factor, which answers the open problem of [IDHPSUU11].In what follows, we present the background of this result and then the details of our results.

Background
Given a source problem that asks the existence of a feasible solution, reconfiguration problems are defined as a problem of deciding the existence of a reconfiguration sequence, that is, a step-by-step transformation between a pair of feasible solutions while always preserving the feasibility of solutions.For example, 3-SAT RECONFIGURATION [GKMP09] is defined from 3-SAT as a source problem.Many reconfiguration problems can be defined from Boolean satisfiability, constraint satisfaction problems, graph problems, and others.Studying reconfiguration problems may help elucidate the structure of the solution space [GKMP09], which is motivated by, e.g., the application to the behavior analysis of SAT solvers, such as DPLL [ABM04].From a different point of view, reconfiguration problems may date back to motion planning [HSS84] and classical puzzles, including 15 puzzles [JS79] and Rubik's Cube.
Typically, a reconfiguration problem becomes PSPACE-complete if its source problem is intractable (say, NP-complete); e.g., 3-SAT [GKMP09], INDEPENDENT SET [HD05,HD09], SET COVER [IDHPSUU11], and 4-COLORING [BC09].On the other hand, a source problem in P frequently leads to a reconfiguration problem in P, e.g., MATCHING [IDHPSUU11] and 2-SAT [GKMP09].Some exceptions are known: whereas 3-COLORING is NP-complete, its reconfiguration problem is solvable in polynomial time [CvJ11]; SHORTEST PATH on a graph is tractable, but its reconfiguration problem is PSPACE-complete [Bon13].We refer the readers to the surveys by Nishimura [Nis18] and van den Heuvel [van13] for algorithmic and hardness results and the Combinatorial Reconfiguration wiki [Hoa23] for an exhaustive list of related articles.
A common way to cope with intractable problems is to consider approximation problems.Relaxing the feasibility of intermediate solutions, we can formalize approximate variants for reconfiguration problems, which are also motivated by the situation wherein there does not exist a reconfiguration sequence for the original decision problem.For example, in MAXMIN 3-SAT RECONFIGURATION [IDHPSUU11], we are allowed to include any non-satisfying assignment in a reconfiguration sequence, but required to maximize the minimum fraction of satisfied clauses.Solving this problem may result in a reasonable reconfiguration sequence consisting of almost-satisfying assignments, e.g., each violating at most 1% of clauses.Intriguingly, a different trend regarding the approximability has been observed between a source problem and its reconfiguration analogue; e.g., SET COVER is NPhard to approximate within a factor better than ln n [DS14, Fei98, LY94], whereas MINMAX SET COVER RECONFIGURATION admits a 2-factor approximation algorithm [IDHPSUU11].Other reconfiguration problems whose approximability was investigated include: SUBSET SUM RECONFIGURATION has a PTAS [ID14]; SUBMODULAR RECONFIGURATION [OM22] and POWER SUPPLY RECONFIGURATION [IDHPSUU11] are constant-factor approximable.
Little is known about the hardness of approximation for reconfiguration problems.Using the fact that source problems (e.g. 2. it disproves the existence of a polynomial-length witness (in particular, a polynomiallength reconfiguration sequence) assuming NP ̸ = PSPACE; 3. it rules out any polynomial-time algorithm under the weak assumption that P ̸ = PSPACE.
In order to improve the NP-hardness of approximation to PSPACE-hardness of approximation, it is crucial to develop a reconfiguration analogue of the PCP theorem [ALMSS98,AS98].As indicated by the gap in the approximation factors of SET COVER and its reconfiguration counterpart, the required theory must be different and tailored to PSPACE.

Our Results
Our contribution is to present a new PCP-type characterization of PSPACE, which we call a probabilistically checkable reconfiguration proof (PCPR), and thereby affirmatively resolve the open problem posed by Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno [IDHPSUU11] and confirm RIH of Ohsaka [Ohs23b].
Our characterization of PSPACE encodes any PSPACE computation into an exponentially long reconfiguration sequence of polynomial-length proofs, each of which can be probabilistically checked by reading a constant number of bits.A reconfiguration sequence from π start to π goal over {0, 1} n is a sequence (π (1) , ) , and π (t) and π (t+1) differ in at most one bit for every t ∈ {1, Here, V π (t) (x) denotes the output of V on input x given oracle access to π (t) , and the probabilities are over the O(log n) random bits of the verifier V .
The verifier V can be regarded as a ∀•coRP-type verifier: The verifier co-nondeterministically guesses t ∈ {1, • • • , T} and probabilistically checks the t-th proof π (t) .This verifier should be compared with the standard coRP-type PCP verifier V ′ for PSPACE-complete problems, which can be obtained from the PCP theorem for NEXP ⊇ PSPACE [BFL91].The number of random bits used by V ′ is n Θ(1) , whereas the number of random bits of our verifier V is O(log n).The latter is crucial for the application to inapproximability of reconfiguration problems.The standard verifier V ′ uses only random bits, whereas our verifier V co-nondeterministically guesses t.Given that V ′ does not use any nondeterministic choice, it is natural to wonder whether V can be improved to a coRP-type verifier that chooses t ∈ {1, • • • , T} randomly; however, such an extension is impossible (see Observation 5.9), and thus our characterization is one of the "best" characterizations in this direction.
As a corollary of Theorem 5.1 and [Ohs23a,Ohs23b], we obtain that a host of reconfiguration problems are PSPACE-complete to approximate.
Moreover, we improve an inapproximability factor of MAXMIN CLIQUE RECONFIGURA-TION to a polynomial; that is, MAXMIN CLIQUE RECONFIGURATION is PSPACE-hard to approximate within a factor of n ε for some constant ε > 0, where n is the number of vertices (Theorem 6.2).This is the first polynomial-factor inapproximability result for approximate variants of reconfiguration problems (to the best of our knowledge). 1

Proof Overview
Here, we present a proof sketch of Theorem 1.5.A naïve attempt for the proof of Theorem 1.5 would be to develop reconfiguration counterparts for the simple proof of the PCP theorem by Dinur [Din07].The proof of the PCP theorem consists of repeated applications of the three steps -a degree reduction (the preprocessing lemma [Din07, Lemma 1.9]), gap amplification and an alphabet reduction.Counterparts of some of the steps have been developed in the recent literature of reconfiguration problems [Ohs23a,Ohs23b].For example, Ohsaka [Ohs23b] presented a degree reduction for reconfiguration problems, i.e., a reduction that converts a graph that represents a PCRP (probabilistically checkable reconfiguration proof) system with soundness error 1 − ε into another graph whose degree ∆ is small.However, the parameter achieved in [Ohs23b] is weaker than that of [Din07,PY91]: , the degree ∆ can be ω(1), which is not sufficient for Dinur's proof to go through.It appears to be very difficult to construct PCRPs based on this approach.
Our actual approach is much simpler.We use existing machinery developed in the literature of PCP theorems in a black-box way.The main ingredient for our proof is the PCP of Proximity (PCPP) [BGHSV06,DR06].A PCPP for a language L ∈ NP allows us to approximately verify that x ∈ L by reading a constant number of bits from x and a proof.In particular, by encoding x by an error-correcting code, we can reliably check whether x ∈ L efficiently.
To construct a PCRP for every problem in PSPACE, it suffices to construct a PCRP for some PSPACE-complete problem.We consider the PSPACE-complete problem called SUC-CINCT GRAPH REACHABILITY.In what follows, we first explain how this problem can be regarded as a reconfiguration problem, and then explain how to construct a PCRP system for SUCCINCT GRAPH REACHABILITY.

SUCCINCT GRAPH REACHABILITY as Reconfiguration Problems
SUCCINCT GRAPH REACHABILITY is the following problem.The input consists of a circuit which succinctly represents an exponentially large graph G = (V , E) and two vertices v start and v goal ∈ V , and the task is to decide whether there exists a path from v start to v goal in G.Each vertex is represented by an n-bit string; i.e., V = {0, 1} n .For simplicity of notation, throughout this section, we assume that every vertex in G has a self-loop, i.e., (x, x) ∈ E for every x ∈ V .For two strings x and y, we denote by x • y the concatenation of x and y.
SUCCINCT GRAPH REACHABILITY can be naturally regarded as the following reconfiguration problem.Given a (succinctly described) graph G and two vertices v start , v goal ∈ V , the task is to decide whether there exists a sequence ( 1. every configuration x t • y t ∈ {0, 1} 2n satisfies the constraint that (x t , y t ) ∈ E, and 2. each adjacent pair of configurations satisfy x t = x t+1 or y t = y t+1 .
In other words, this is the reconfiguration problem which asks to decide whether the token that initially placed at the edge (v start , v start ) can be moved to the edge (v goal , v goal ) by a sequence of operations that move the token from an edge to one of its adjacent edges.
In Theorem 1.5, each adjacent pair of proofs differs in at most one bit.In terms of reconfiguration problems, this means that the operations which we are allowed to perform are to change one bit of a configuration instead of one vertex of the token placed at an edge.By introducing a special symbol "⊥", we can regard SUCCINCT GRAPH REACHABILITY as the following reconfiguration problem in which operations are restricted to changing one bit of configurations: Given a (succinctly described) graph G and two vertices v start , v goal ∈ V , the task is to decide whether there exists a sequence ( n and y t ∈ {0, 1} n ), and 2. each adjacent pair (x t • y t , x t+1 • y t+1 ) differs in at most one position.
Informally, the existence of the symbol ⊥ indicates that we are on the way of the transition, and we are allowed to include ⊥ in at most one of x t or y t (that is, we do not allow to change both vertices of the token simultaneously).This reconfiguration problem "simulates" SUC-CINCT GRAPH REACHABILITY in the following sense: If a token placed at an edge (x, y 1 ) ∈ E is moved to another edge (x, y 2 ) ∈ E, then in the new reconfiguration problem, we may consider a sequence of operations that first transform x • y 1 into x • ⊥ n by replacing each bit of y 1 with ⊥ one by one, and then transform x • ⊥ n into x • y 2 by replacing ⊥ with a bit of y 2 one by one.

PCRP System for SUCCINCT GRAPH REACHABILITY
The main idea for constructing a PCRP for SUCCINCT GRAPH REACHABILITY is to probabilistically check item 1, i.e., the condition that (x t , y t ) ∈ E or (x t ∈ {0, 1} n and y t ∈ {0, 1} n ), by reading a constant number of bits.To this end, we encode each vertex by a locally testable error-correcting code Enc: {0, 1} n → {0, 1} ℓ and use the PCPP to check whether the encoded pair of vertices (x t , y t ) satisfies (x t , y t ) ∈ E. Specifically, let V PCPP be a PCPP verifier for the language L G = {Enc(x) • Enc(y) | (x, y) ∈ E}.This verifier takes random access to f • g and a proof π ∈ {0, 1} p and checks whether 1. Using the local tester for Enc, we check that both f and g are close to some codewords Enc(x) and Enc(y), respectively.If both are far from codewords, then we reject.
2. By random sampling, we test whether either f or g contains many ⊥ symbols.If so, we accept.
3. Finally, by running V PCPP for ( f • g, π), we check that (x, y) ∈ E. We accept if and only if V PCPP accepts.
The first item ensures that either f or g is close to some codewords Enc(x) and Enc(y).The second item checks whether we are on the way of the transition from one edge to another, in which case we accept.We run the test of the third item only if either f or g is close to some codewords, and both f and g do not contain many ⊥ symbols.Using the PCPP, we check that f and g encode x and y such that (x, y) ∈ E.
We note that the size of alphabets {0, 1, ⊥} of the PCRP system is 3.This can be reduced to 2 by using a simple alphabet reduction of Ohsaka [Ohs23b], which transforms any PCRP system with perfect completeness over alphabets of constant size into a PCRP system over the binary alphabets {0, 1}.
It is thus important to make sure that the PCRP system has perfect completeness, i.e., in the YES case, the verifier accepts with probability 1.For this reason, in the actual proof, we need to modify the PCPP verifier V PCPP so that it immediately accepts if a ⊥ symbol in f or g is queried by V PCPP .Details can be found in Section 5.

Related Work
Another characterization of PSPACE is probabilistically checkable debate systems due to Condon, Feigenbaum, Lund, and Shor [CFLS95], which can be used to show PSPACEhardness of approximating QUANTIFIED BOOLEAN FORMULA and the problem of selecting as many finite-state automata as possible that accept a common string.These results are incomparable to our PCRP because the underlying structure of the problems is different from each other.
We summarize approximate variants of reconfiguration problems whose inapproximability was investigated.MAXMIN CLIQUE RECONFIGURATION and MAXMIN SAT RECON-FIGURATION are NP-hard to approximate [IDHPSUU11].SHORTEST PATH RECONFIGU-RATION is PSPACE-hard to approximate with respect to its objective value [GJKL22].The obejctive value called the price is determined based on the number of vertices in a path changed at a time, which is fundamentally different from those of reconfiguration problems listed in Corollary 1.6.SUBMODULAR RECONFIGURATION is constant-factor inapproximable [OM22], whose proof resorts to inapproximability results of SUBMODULAR FUNC-TION MAXIMIZATION [FMV11].
We note that approximability of reconfiguration problems frequently refers to that of the shortest sequence [BHIKMMSW20, IKKKO22, KMM11, MNORTU16], which seems orthogonal to the present study.
The pebble game [PH70] is a single-player game, which models the trade-off between the memory usage and running time of a computation and is recently used in the context of proof complexity [Nor13].This game can be thought of as a reconfiguration problem, whose objective function, called the pebbling price, is defined as the maximum number of pebbles at any time required to place a pebble to the unique sink.The pebbling price is known to be PSPACE-hard to approximate within an additive n

Notations
For a nonnegative integer n ∈ ℕ, let [n] := {1, 2, . . ., n}.A sequence E of a finite number of elements E (1) , . . ., E (T) is denoted by (E (1) , . . ., E (T) ), and we write E ∈ E to indicate that E appears in E. The symbol • stands for a concatenation of two strings.Let Σ be a finite set called alphabet.For a length-n string f ∈ Σ n and index set I ⊆ [n], we use f | I to denote the restriction of f to I. We write 0 n and 1 n for 0 , respectively.The relative distance between two strings f , g ∈ Σ n , denoted ∆( f , g), is defined as the fraction of positions on which f and g differ; namely, For a set of strings S ⊆ Σ n , analogous notions are defined; e.g., ∆( f , S)

Probabilistically Checkable Proofs of Proximity
We formally define the notion of verifier.

Constraint Satisfaction Problems
Here, we review constraint satisfaction problems.
For an assignment A : N → Σ, we say that A satisfies constraint ψ j if ψ j (A) = 1, and A satisfies Ψ if it satisfies all constraints of Ψ.Moreover, we say that Ψ is satisfiable if Ψ is satisfied by some assignment.For an assignment A : N → Σ, its value is defined as the fraction of constraints of Ψ satisfied by A; namely, We refer to the equivalence between a PCP system and GAP q-CSP (see, e.g., [AB09, Section 11.3]), whose proof is included for the sake of completeness.Proposition 4.9.Let V be a verifier with randomness complexity O(log n), query complexity O(1), and alphabet Σ, and let x ∈ {0, 1} * be an input.Then, one can construct in polynomial time a constraint system Ψ = (ψ j ) j∈[m] over poly(|x|) variables and alphabet Σ such that val On the other hand, for a q-ary constraint system Ψ over variable set N and alphabet Σ, one can construct in polynomial time a verifier V with randomness complexity O(log n), query complexity O(1), and alphabet Σ such that ℙ[V A = 1] = val Ψ (A) for every assignment A : N → Σ.
Proof.Let V be a verifier with randomness complexity r(n) = O(n), query complexity q(n) = q ∈ ℕ, and alphabet Σ.Given an input x ∈ {0, 1} * , we can assume the proof length for V to be poly(|x|).We construct a q-ary constraint system Ψ over variable set N := [poly(|x|)] and alphabet Σ as follows: • for every possible sequence R ∈ {0, 1} r(|x|) of r(|x|) random bits, we run V (x) to generate a query sequence I R = (i 1 , . . ., i q ) and a circuit D R : Σ q → {0, 1} in polynomial time.
• create a new constraint ψ j such that for every assignment A : N → Σ.
Note that the construction of Ψ completes in polynomial time; in particular, the size of Ψ is polynomial in |x|.Observe that for any proof π ∈ Σ N , which can be thought of as an assignment to Ψ, completing the proof of the first statement.The second statement is omitted as can be shown similarly.

PSPACE-completeness of SUCCINCT GRAPH REACHABILITY
We first introduce a canonical PSPACE-complete problem called SUCCINCT GRAPH REACH-ABILITY, for which we design a PCRP system.
We now show the PSPACE-completeness of SUCCINCT GRAPH REACHABILITY.Membership in PSPACE follows from the fact that SUCCINCT GRAPH REACHABILITY ∈ NPSPACE and Savitch's theorem [Sav70].Consider the following PSPACE-complete problem: Given a deterministic Turing machine M, input x ∈ {0, 1} * , and 1 n , does M accept x in space n?Note that all possible configurations of M having space n on input x are specified by {0, 1} αn for some constant α depending on M. Let c init ∈ {0, 1} αn denote the initial configuration of M on input x, and M(x, c) ∈ {0, 1} αn denote the next configuration of M following c ∈ {0, 1} αn .Define now a circuit S : {0, 1} 2+αn → {0, 1} 2+αn of polynomial size (in |M|, |x|, and n) as follows: (5.4) Observe easily that S

Verifier Description
Our verifier V is given a polynomial-size circuit S : {0, 1} n → {0, 1} n and oracle access to n) , and is designed as follows: 1. V ensures that f or g must be a codeword of Enc by running the local tester M on f and g separately.Note that M rejects whenever it reads ⊥ at least once, which still 2. V allows f • g to contain ⊥, enabling f or g to transform between different codewords of Enc.Specifically, V accepts with probability equal to the fraction of ⊥ in f or g, which can be done by testing whether f i = ⊥ or g j = ⊥ for independently and uniformly chosen i, j ∈ [ℓ(n)].During f = ⊥ n or g = ⊥ n , the contents of π can be modified arbitrarily without being rejected, which is essential in the perfect completeness (Lemma 5.4).
3. On the other hand, if neither f nor g contains "many" ⊥'s, V expects f • g to be close to L ckt (S); thus, it wants to execute the smooth PCPP verifier V ckt (S), whose behavior is, however, undefined if f • g • π contains ⊥.Instead, we run a modified verifier V ′ ckt (S), which accepts if and only if ( 3 This test is crucial for proving the soundness (Lemma 5.5).
The precise pseudocode of V (S) is presented below.Verifier V f •g•π (S) using local tester M for Enc and smooth PCPP verifier V ckt for L ckt .
We first show the completeness.
◁ By the following case analysis, V (S) turns out to accept every intermediate proof f • g •π with probability 1, as desired.
by replacing some symbols of the second Enc(α) by ⊥.Observe that the local tester M always accepts f = Enc(α).
We show that V ′ ckt (S) always accepts The local tester M always accepts Enc(α), and V (S) would not have run the modified verifier V ′ ckt (S); i.e., V (S) always accepts f • g • π.
We then show the soundness.
Proof of Theorem 5.1.We first prove the "only if " direction.Since SUCCINCT GRAPH REACH-ABILITY is PSPACE-complete, it is sufficient to create its verifier V and polynomial-time computable proofs π start and π goal .The verifier V is described in Section 5.2.1.For a polynomial-size circuit S : {0, 1} n → {0, 1}, the number of queries that V makes is bounded by 2 , and the number of random bits that V uses is bounded by 2 We reduce the alphabet size of V from three (i.e., {0, 1, ⊥}) to two.Using Proposition 4.9, we first convert V (S) into a constraint system Ψ = (ψ j ) j∈[m] over alphabet {0, 1, ⊥} such that ℙ[V (S) accepts π] is equal to val Ψ (π) for any proof π ∈ {0, 1, ⊥} poly (n) .By [Ohs23b], we obtain a constraint system Ψ ′ = (ψ ′ j ) j∈[m ′ ] over alphabet {0, 1} and its two satisfying assignments Using Proposition 4.9 again, we convert Ψ ′ into a verifier V ′ with randomness complexity O(log n), query complexity O(1), and alphabet {0, 1} such that ℙ[V ′ accepts π ′ ] is equal to val Ψ ′ (π ′ ) for any proof Consequently, if S is a YES instance, by Lemma 5.4, there exists a reconfiguration sequence A from A start to A goal over {0, 1} poly(n) such that V ′ accepts any proof in A with probability 1, whereas if S is a NO instance, by Lemma 5.5, for any reconfiguration sequence A from A start to A goal over {0, 1} poly(n) , A includes a proof that is rejected by V ′ with probability Ω(1), which can be amplified to 1 2 by a constant number of repetition, as desired.
We then prove the "if" direction.Suppose a language L admits a verifier V with randomness complexity r(n) = O(log n) and query complexity q(n) = O(1), associated with polynomialtime computable proofs π start and π goal .Consider then the following nondeterministic algorithm for finding a reconfiguration sequence from π start to π goal .Nondeterministic polynomial-space algorithm for finding a reconfiguration sequence.
Suppose the token is currently placed at (α, t) and just before at (β, t).Then, the token must be placed at (α, t + 1) in the next step.Similarly, if the token is placed at (β, t) and just before at (α, t); then, it must be at (β, t + 1) in the next step, which ensures that we eventually reach (α, T + 1) or (β, T + 1) to terminate.The latter statement is obvious from the construction.
Proof.We first show that ( f • g)| I contains ⊥ for (I, D) ∼ V ckt (S) with probability at most εq.Denote by I ckt the indices of f • g • π, where |I ckt | = 2ℓ(n) + p(n).By smoothness of V ckt , we have p ckt := ℙ (I,D) [i ∈ I] for all i ∈ I ckt .Since |I| = q for any (I, D) ∼ V ckt (S), we obtain . (5.13) Using a union bound and the assumption that each f and g contains ⊥ in at most ε • ℓ(n) positions, we derive (5.14) Subsequently, we show that f •g is δ ckt -far from L ckt (S), where (5.15) Similarly, ∆(g, Enc(β ⋆ )) ρ − ε if β ̸ = β ⋆ .Consequently, we obtain where we used the fact that ε ρ 3 .Taking a union bound, we derive (5.17) Let π be a proof obtained from π by replacing every occurrence of ⊥ by 0 (5.18) Accordingly, we get completing the proof.
Using Claim 5.8 and the definition of ε in Eq. (5.8), we derive (5.20) Consequently, we get max accomplishing the proof of Lemma 5.5.

Impossibility of Extension to Average Case
Since the verifier of Theorem 5.1 co-nondeterministically guesses t ∈ [T] and probabilistically checks π (t) , one might think of extending it so as to choose t ∈ [T] randomly.The soundness case then requires that the verifier accepts "most" of but rejects a constant fraction of the proofs in any reconfiguration sequence.The resulting reconfiguration proof (π (1) , . . ., π (T) ) can be thought of as a (kind of) rectangular PCPs [BHPT20], whose column is of exponential length and row is of polynomial length, and we pick t ∈ [T] and run a verifier on π (t) to decide whether to accept.However, such relaxation is impossible, as formally stated below.

PSPACE-hardness of Approximation for Reconfiguration Problems
In this section, we show that many popular reconfiguration problems are PSPACE-hard to approximate, answering an open problem of [IDHPSUU11].Since Ohsaka [Ohs23b] has already shown gap-preserving reductions starting from the Reconfiguration Inapproximability Hypothesis (RIH), which asserts that a gap version of MAXMIN CSP RECONFIGURA-TION is PSPACE-hard, we prove that RIH is true.

Constant-factor Inapproximability of MAXMIN CSP RECONFIG-URATION
We first define reconfiguration problems on constraint satisfaction.For a q-ary constraint system Ψ = (ψ j ) j∈[m] over variable set N and alphabet Σ and its two satisfying assignments A start and A goal for Ψ, a reconfiguration sequence from A start to A goal over Σ N is • Vertex set: W is the set consisting of all length-(ℓ − 1) walks w = (w 1 , . . ., w ℓ ) over X .
Note that the number of vertices is equal to N := |W| = nd ℓ−1 , which is polynomial in n.
• Edge set: H contains an edge between w 1 ̸ = w 2 ∈ W if and only if a subgraph of G induced by w 1 ∪ w 2 forms a clique.We are now ready to accomplish the proof of Theorem 6.2.

13
−ε term for the graph size n[CLNV15,DL17].We leave open whether our PCRPs can be used to derive PSPACEhardness of approximating the pebbling price within a multiplicative factor.

|C
(t) ∆C (t+1) | = 1. 5 CLIQUE RECONFIGURATION asks if there is a reconfiguration sequence from C start to C goal made up of cliques only of size at least min{|C start |, |C goal |} − 1.For a reconfiguration sequence of cliques of G, denoted C = (C (1) , . . ., C (T) ), let val G (C) := min C (i) ∈C |C (i) |. (6.3)Then, for a pair of cliques C start and C goal of G, MAXMIN CLIQUE RECONFIGURATION requires to maximize val G (C) subject to C = (C start , . . ., C goal ).Subsequently, let val G (C start ↭ C goal ) denote the maximum value of val G (C) over all possible reconfiguration sequences C from C start to C goal ; namely, val G (C start ↭ C goal ) := max C=(C start ,...,C goal ) val G (C).(6.4)Reduction.We first describe a gap-amplification reduction from MAXMIN CLIQUE RE-CONFIGURATION to itself using the derandomized graph product[AFWZ95].Let (G, C start , C goal ) be an instance of MAXMIN CLIQUE RECONFIGURATION, where G = (V , E) is a graph on n vertices.By Theorem 6.1 and[Ohs23b], it is PSPACE-hard to distinguish whether valG (C start ↭ C goal ) ω(G) − 1 or val G (C start ↭ C goal ) (1 − ε)(ω(G) − 1)for some constant ε ∈ (0, 1) evenwhen |C start | = |C goal | = ω(G)and ω(G) then a new instance (H, D start , D goal ) of MAXMIN CLIQUE RECONFIGURATION as follows.Let ℓ = ⌈log n⌉, and X be a (d, λ)-expander graph over the same vertex as G.The precise value of d and λ will be determined later.Graph H = (W, F) is defined as follows:

ForZ
any clique C ⊆ V of G, define D C ⊆ W as D C := w ∈ W w ⊆ C , (6.5)which is a clique of H as well.Constructing D start := D C start and D goal := D C goal completes the reduction.We refer to the following property about random walks over expander graphs.Lemma 6.3 ([AFWZ95]).Let S be any vertex set of X , and Z := (Z 1 , . . ., Z ℓ ) a ℓ-tuple of random variables denoting the vertices of a uniformly chosen (ℓ − 1)-length random walk over X .Then, it holds that soundness are shown below.Lemma 6.4.If val G (C start ↭ C goal ) ω(G) − 1, then val H (D start ↭ D goal ) It suffices to consider the case that C start and C goal differ in exactly two vertices (i.e., C goal is obtained from C start by removing and adding a single vertex).There is a reconfiguration sequence (C start , C • , C goal ) from C start to C goal , where C • := C start ∩ C goal is a clique of size ω(G) − 1.Since D C start ⊃ D C • and D C goal ⊃ D C • by definition, we can reconfigure from D start = D C start to D goal = D C goal by first removing the vertices of D C start \ D C • one by one and then adding the vertices of D C goal \ D C • one by one.Thus, we have val H (D start ↭ D goal ) |D C • |.Lemma 6.3 derives that|D C • | |W| = ℙ Z ∀i ∈ [ℓ], Z i ∈ C • |C • If val G (C start ↭ C goal ) < (1 − ε)(ω(G) − 1), then val H (D start ↭ D goal ) < |W| • (1 − ε) Weshow the contrapositive.Suppose we are given a reconfiguration sequence D = (D (1) , . . ., D (T) ) from D start to D goal such that val H (D) |W| • (1 − ε) ω(G)−1 |V | + 2 λ d ℓ .For any clique D of H, define C D as C D := w∈D w, (6.10) which is a clique of G as well.Observe that val G (C D (t) ↭ C D (t+1) ) min{|C D (t) |, |C D (t+1) |} for anyt ∈ [T − 1] since C D (t) ⊂ C D (t+1) or C D (t) ⊃ C D (t+1) , implying further that val G (C start ↭ C goal ) min t∈[T−1] val G (C D (t) ↭ C D (t+1) ) min t∈[T−1] min |C D (t) |, |C D (t+1) | min D(t) ∈D |C D (t) |. ∈ D (t) ℙ Z ∀i ∈ [ℓ], Z i ∈ C D (t) |C D (t) have |C D (t) | (1 − ε)(ω(G) − 1) for all t ∈ [T]; thus, val G (C start ↭ C goal ) (1 − ε)(ω(G) − 1), as desired.
NP-hardness results are not optimal.It was left open to improve the NP-hardness results to PSPACE-hardness.We here stress the significance of showing PSPACE-hardness compared to NP-hardness:1.PSPACE-hardness is tight because most reconfiguration problems belong to PSPACE , MAX 3-SAT) are NP-hard to approximate [ALMSS98, AS98], Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno [IDHPSUU11] proved that several reconfiguration problems (e.g., MAXMIN 3-SAT RECONFIGURATION) are NPhard to approximate; however, most reconfiguration problems are in PSPACE, and thus their