Tight Inapproximability of Target Set Reconfiguration

Given a graph $G$ with a vertex threshold function $\tau$, consider a dynamic process in which any inactive vertex $v$ becomes activated whenever at least $\tau(v)$ of its neighbors are activated. A vertex set $S$ is called a target set if all vertices of $G$ would be activated when initially activating vertices of $S$. In the Minmax Target Set Reconfiguration problem, for a graph $G$ and its two target sets $X$ and $Y$, we wish to transform $X$ into $Y$ by repeatedly adding or removing a single vertex, using only target sets of $G$, so as to minimize the maximum size of any intermediate target set. We prove that it is NP-hard to approximate Minmax Target Set Reconfiguration within a factor of $2-o\left(\frac{1}{\operatorname{polylog} n}\right)$, where $n$ is the number of vertices. Our result establishes a tight lower bound on approximability of Minmax Target Set Reconfiguration, which admits a $2$-factor approximation algorithm. The proof is based on a gap-preserving reduction from Target Set Selection to Minmax Target Set Reconfiguration, where NP-hardness of approximation for the former problem is proven by Chen (SIAM J. Discrete Math., 2009) and Charikar, Naamad, and Wirth (APPROX/RANDOM 2016).Comment: 13 page

Gap Preserving Reductions Between Reconfiguration Problems

Combinatorial reconfiguration is a growing research field studying problems on the transformability between a pair of solutions for a search problem. For example, in SAT Reconfiguration, for a Boolean formula ? and two satisfying truth assignments ?_? and ?_? for ?, we are asked to determine whether there is a sequence of satisfying truth assignments for ? starting from ?_? and ending with ?_?, each resulting from the previous one by flipping a single variable assignment. We consider the approximability of optimization variants of reconfiguration problems; e.g., Maxmin SAT Reconfiguration requires to maximize the minimum fraction of satisfied clauses of ? during transformation from ?_? to ?_?. Solving such optimization variants approximately, we may be able to obtain a reasonable reconfiguration sequence comprising almost-satisfying truth assignments. In this study, we prove a series of gap-preserving reductions to give evidence that a host of reconfiguration problems are PSPACE-hard to approximate, under some plausible assumption. Our starting point is a new working hypothesis called the Reconfiguration Inapproximability Hypothesis (RIH), which asserts that a gap version of Maxmin CSP Reconfiguration is PSPACE-hard. This hypothesis may be thought of as a reconfiguration analogue of the PCP theorem. Our main result is PSPACE-hardness of approximating Maxmin 3-SAT Reconfiguration of bounded occurrence under RIH. The crux of its proof is a gap-preserving reduction from Maxmin Binary CSP Reconfiguration to itself of bounded degree. Because a simple application of the degree reduction technique using expander graphs due to Papadimitriou and Yannakakis (J. Comput. Syst. Sci., 1991) does not preserve the perfect completeness, we modify the alphabet as if each vertex could take a pair of values simultaneously. To accomplish the soundness requirement, we further apply an explicit family of near-Ramanujan graphs and the expander mixing lemma. As an application of the main result, we demonstrate that under RIH, optimization variants of popular reconfiguration problems are PSPACE-hard to approximate, including Nondeterministic Constraint Logic due to Hearn and Demaine (Theor. Comput. Sci., 2005), Independent Set Reconfiguration, Clique Reconfiguration, and Vertex Cover Reconfiguration

Alphabet Reduction for Reconfiguration Problems

We present a reconfiguration analogue of alphabet reduction \`a la Dinur (J. ACM, 2007) and its applications. Given a binary constraint graph $G$ and its two satisfying assignments $\psi^\mathsf{ini}$ and $\psi^\mathsf{tar}$, the Maxmin Binary CSP Reconfiguration problem requests to transform $\psi^\mathsf{ini}$ into $\psi^\mathsf{tar}$ by repeatedly changing the value of a single vertex so that the minimum fraction of satisfied edges is maximized. We demonstrate a polynomial-time reduction from Maxmin Binary CSP Reconfiguration with arbitrarily large alphabet size $W \in \mathbb{N}$ to itself with universal alphabet size $W_0 \in \mathbb{N}$ such that 1. the perfect completeness is preserved, and 2. if any reconfiguration for the former violates $\varepsilon$-fraction of edges, then $\Omega(\varepsilon)$-fraction of edges must be unsatisfied during any reconfiguration for the latter. The crux of its construction is the reconfigurability of Hadamard codes, which enables to reconfigure between a pair of codewords, while avoiding getting too close to the other codewords. Combining this alphabet reduction with gap amplification due to Ohsaka (SODA 2024), we are able to amplify the $1$ vs. $1-\varepsilon$ gap for arbitrarily small $\varepsilon \in (0,1)$ up to the $1$ vs. $1-\varepsilon_0$ for some universal $\varepsilon_0 \in (0,1)$ without blowing up the alphabet size. In particular, a $1$ vs. $1-\varepsilon_0$ gap version of Maxmin Binary CSP Reconfiguration with alphabet size $W_0$ is PSPACE-hard only assuming the Reconfiguration Inapproximability Hypothesis posed by Ohsaka (STACS 2023), whose gap parameter can be arbitrarily small. This may not be achieved only by gap amplification of Ohsaka, which makes the alphabet size gigantic depending on the gap value of the hypothesis.Comment: 25 page

On Approximate Reconfigurability of Label Cover

Given a two-prover game $G$ and its two satisfying labelings $\psi_\mathsf{s}$ and $\psi_\mathsf{t}$, the Label Cover Reconfiguration problem asks whether $\psi_\mathsf{s}$ can be transformed into $\psi_\mathsf{t}$ by repeatedly changing the value of a vertex while preserving any intermediate labeling satisfying $G$. We consider an optimization variant of Label Cover Reconfiguration by relaxing the feasibility of labelings, referred to as Maxmin Label Cover Reconfiguration: we are allowed to transform by passing through any non-satisfying labelings, but required to maximize the minimum fraction of satisfied edges during transformation from $\psi_\mathsf{s}$ to $\psi_\mathsf{t}$. Since the parallel repetition theorem of Raz (SIAM J. Comput., 1998), which implies NP-hardness of Label Cover within any constant factor, produces strong inapproximability results for many NP-hard problems, one may think of using Maxmin Label Cover Reconfiguration to derive inapproximability results for reconfiguration problems. We prove the following results on Maxmin Label Cover Reconfiguration, which display different trends from those of Label Cover and the parallel repetition theorem: (1) Maxmin Label Cover Reconfiguration can be approximated within a factor of nearly $\frac{1}{4}$ for restricted graph classes, including slightly dense graphs and balanced bipartite graphs. (2) A naive parallel repetition of Maxmin Label Cover Reconfiguration does not decrease the optimal objective value. (3) Label Cover Reconfiguration on projection games can be decided in polynomial time. The above results suggest that a reconfiguration analogue of the parallel repetition theorem is unlikely.Comment: 11 page

Gap Amplification for Reconfiguration Problems

In this paper, we demonstrate gap amplification for reconfiguration problems. In particular, we prove an explicit factor of PSPACE-hardness of approximation for three popular reconfiguration problems only assuming the Reconfiguration Inapproximability Hypothesis (RIH) due to Ohsaka (STACS 2023). Our main result is that under RIH, Maxmin Binary CSP Reconfiguration is PSPACE-hard to approximate within a factor of $0.9942$. Moreover, the same result holds even if the constraint graph is restricted to $(d,\lambda)$-expander for arbitrarily small $\frac{\lambda}{d}$. The crux of its proof is an alteration of the gap amplification technique due to Dinur (J. ACM, 2007), which amplifies the $1$ vs. $1-\epsilon$ gap for arbitrarily small $\epsilon > 0$ up to the $1$ vs. $1-0.0058$ gap. As an application of the main result, we demonstrate that Minmax Set Cover Reconfiguration and Minmax Dominating Set Reconfiguratio} are PSPACE-hard to approximate within a factor of $1.0029$ under RIH. Our proof is based on a gap-preserving reduction from Label Cover to Set Cover due to Lund and Yannakakis (J. ACM, 1994). However, unlike Lund--Yannakakis' reduction, the expander mixing lemma is essential to use. We highlight that all results hold unconditionally as long as "PSPACE-hard" is replaced by "NP-hard," and are the first explicit inapproximability results for reconfiguration problems without resorting to the parallel repetition theorem. We finally complement the main result by showing that it is NP-hard to approximate Maxmin Binary CSP Reconfiguration within a factor better than $\frac{3}{4}$.Comment: 41 pages, to appear in Proc. 35th Annu. ACM-SIAM Symp. Discrete Algorithms (SODA), 202

A Critical Reexamination of Intra-List Distance and Dispersion

Diversification of recommendation results is a promising approach for coping with the uncertainty associated with users' information needs. Of particular importance in diversified recommendation is to define and optimize an appropriate diversity objective. In this study, we revisit the most popular diversity objective called intra-list distance (ILD), defined as the average pairwise distance between selected items, and a similar but lesser known objective called dispersion, which is the minimum pairwise distance. Owing to their simplicity and flexibility, ILD and dispersion have been used in a plethora of diversified recommendation research. Nevertheless, we do not actually know what kind of items are preferred by them. We present a critical reexamination of ILD and dispersion from theoretical and experimental perspectives. Our theoretical results reveal that these objectives have potential drawbacks: ILD may select duplicate items that are very close to each other, whereas dispersion may overlook distant item pairs. As a competitor to ILD and dispersion, we design a diversity objective called Gaussian ILD, which can interpolate between ILD and dispersion by tuning the bandwidth parameter. We verify our theoretical results by experimental results using real-world data and confirm the extreme behavior of ILD and dispersion in practice.Comment: 10 pages, to appear in 46th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 2023

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 $n^\epsilon$ for some constant $\epsilon > 0$.Comment: 31 page

Optimal PSPACE-hardness of Approximating Set Cover Reconfiguration

In the Minmax Set Cover Reconfiguration problem, given a set system $\mathcal{F}$ over a universe and its two covers $\mathcal{C}^\mathsf{start}$ and $\mathcal{C}^\mathsf{goal}$ of size $k$, we wish to transform $\mathcal{C}^\mathsf{start}$ into $\mathcal{C}^\mathsf{goal}$ by repeatedly adding or removing a single set of $\mathcal{F}$ while covering the universe in any intermediate state. Then, the objective is to minimize the maximize size of any intermediate cover during transformation. We prove that Minmax Set Cover Reconfiguration and Minmax Dominating Set Reconfiguration are $\mathsf{PSPACE}$-hard to approximate within a factor of $2-\frac{1}{\operatorname{polyloglog} N}$, where $N$ is the size of the universe and the number of vertices in a graph, respectively, improving upon Ohsaka (SODA 2024) and Karthik C. S. and Manurangsi (2023). This is the first result that exhibits a sharp threshold for the approximation factor of any reconfiguration problem because both problems admit a $2$-factor approximation algorithm as per Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno (Theor. Comput. Sci., 2011). Our proof is based on a reconfiguration analogue of the FGLSS reduction from Probabilistically Checkable Reconfiguration Proofs of Hirahara and Ohsaka (2024). We also prove that for any constant $\varepsilon \in (0,1)$, Minmax Hypergraph Vertex Cover Reconfiguration on $\operatorname{poly}(\varepsilon^{-1})$-uniform hypergraphs is $\mathsf{PSPACE}$-hard to approximate within a factor of $2-\varepsilon$.Comment: 28 page