Reconfiguration of satisfying assignments and subset sums: Easy to find, hard to connect

Cardinal, Jean; Demaine, Erik D.; Eppstein, David; Hearn, Robert A.; Winslow, Andrew

doi:10.1016/j.tcs.2019.05.028

Cited by 5 publications

(3 citation statements)

References 38 publications

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“…We show that Problem 4.3 is PSPACE-hard, whose proof is reminiscent of [28,Theorem 2] Proof. We demonstrate a polynomial-time reduction from Monotone Not-All-Equal 3-SAT Reconfiguration, which is PSPACEcomplete [12]. A 3-conjunctive normal form (3-CNF) formula 𝜙 is said to be monotone if no clause contains negative literals (e.g.,…”

Section: Hardnessmentioning

confidence: 99%

Reconfiguration Problems on Submodular Functions

Ohsaka

Matsuoka

2022

Proceedings of the Fifteenth ACM International Conference on Web Search and Data Mining

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Reconfiguration problems require finding a step-by-step transformation between a pair of feasible solutions for a particular problem. The primary concern in Theoretical Computer Science has been revealing their computational complexity for classical problems.This paper presents an initial study on reconfiguration problems derived from a submodular function, which has more of a flavor of Data Mining. Our submodular reconfiguration problems request to find a solution sequence connecting two input solutions such that each solution has an objective value above a threshold in a submodular function 𝑓 : 2 [𝑛] → R + and is obtained from the previous one by applying a simple transformation rule. We formulate three reconfiguration problems: Monotone Submodular Reconfiguration (MSReco), which applies to influence maximization, and two versions of Unconstrained Submodular Reconfiguration (USReco), which apply to determinantal point processes. Our contributions are summarized as follows:• We prove that MSReco and USReco are both PSPACE-complete.• We design a 1 2 -approximation algorithm for MSReco and a 1 𝑛approximation algorithm for (one version of) USReco.• We devise inapproximability results that approximating the optimum value of MSReco within a (1− 1+𝜖 𝑛 2 )-factor is PSPACE-hard, and we cannot find a ( 5 6 + 𝜖)-approximation for USReco.• We conduct numerical study on the reconfiguration version of influence maximization and determinantal point processes using real-world social network and movie rating data. CCS CONCEPTS• Information systems → Data mining; • Theory of computation → Approximation algorithms analysis.

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Section: Hardnessmentioning

confidence: 99%

Reconfiguration Problems on Submodular Functions

Ohsaka

Matsuoka

2022

Proceedings of the Fifteenth ACM International Conference on Web Search and Data Mining

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“…, C of exact covers of U from C = C 0 to C = C such that C i ⊆ D for all i and C i is obtained from C i−1 by a split or a merge for each i ∈ [ ]. It is known that Exact Cover Reconfiguration is PSPACE-complete [7].…”

Section: Pspace-completeness On Forests Of Depthmentioning

confidence: 99%

Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited

Gima¹,

Ito²,

Kobayashi³

et al. 2022

Preprint

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Given a graph and two vertex sets satisfying a certain feasibility condition, a reconfiguration problem asks whether we can reach one vertex set from the other by repeating prescribed modification steps while maintaining feasibility. In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic meta-theorem for reconfiguration problems that says if the feasibility can be expressed in monadic second-order logic (MSO), then the problem is fixed-parameter tractable parameterized by treewidth + , where is the number of steps allowed to reach the target set. On the other hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if is not part of the parameter, then the problem is PSPACE-complete even on graphs of bounded bandwidth.In this paper, we present the first algorithmic meta-theorems for the case where is not part of the parameter, using some structural graph parameters incomparable with bandwidth. We show that if the feasibility is defined in MSO, then the reconfiguration problem under the so-called token jumping rule is fixed-parameter tractable parameterized by neighborhood diversity. We also show that the problem is fixed-parameter tractable parameterized by treedepth + k, where k is the size of sets being transformed. We finally complement the positive result for treedepth by showing that the problem is PSPACE-complete on forests of depth 3.

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“…Analogously, the k-recolouring problem also generalizes to reconfiguration problems for H-colourings and CSP, both of which are well studied; see, e.g., [4-6, 13, 20] and [2,8,10,11,[14][15][16]18], respectively. In particular, Gopalan et al [10] proved a dichotomy theorem for the reconfiguration variation of CSP(H) for structures H with two vertices.…”

Section: Introductionmentioning

confidence: 99%

Recolouring Homomorphisms to triangle-free reflexive graphs

Lee¹,

Noel²,

Siggers³

2021

Preprint

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For a graph H, the H-recolouring problem Recol(H) asks, for two given homomorphisms from a given graph G to H, if one can get between them by a sequence of homomorphisms of G to H in which consecutive homomorphisms differ on only one vertex. We show that, if G and H are reflexive and H is triangle-free, then this problem can be solved in polynomial time. This shows, at the same time, that the closely related H-reconfiguration problem Recon(H) of deciding whether two given homomorphisms from a given graph G to H are in the same component of the Hom-graph Hom(G, H), can be solved in polynomial time for triangle-free reflexive graphs H.

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Reconfiguration of satisfying assignments and subset sums: Easy to find, hard to connect

Cited by 5 publications

References 38 publications

Reconfiguration Problems on Submodular Functions

Reconfiguration Problems on Submodular Functions

Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited

Recolouring Homomorphisms to triangle-free reflexive graphs

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