Complexity of Reconfiguration Problems for Constraint Satisfaction

Abstract

Constraint satisfaction problem (CSP) is a well-studied combinatorial search problem, in which we are asked to find an assignment of values to given variables so as to satisfy all of given constraints. We study a reconfiguration variant of CSP, in which we are given an instance of CSP together with its two satisfying assignments, and asked to determine whether one assignment can be transformed into the other by changing a single variable assignment at a time, while always remaining satisfying assignment. This problem generalizes several well-studied reconfiguration problems such as Boolean satisfiability reconfiguration, vertex coloring reconfiguration, homomorphism reconfiguration. In this paper, we study the problem from the viewpoints of polynomial-time solvability and parameterized complexity, and give several interesting boundaries of tractable and intractable cases.

Fixed-Parameter Algorithms for Graph Constraint Logic

“…In the reminder of this section, we give an overview of the proof of Theorem 1. Our strategy is to give an FPT-reduction from C2C on and/or constraint graphs to the binary constraint satisfiability reconfiguration problem (BCSR, for short) [5], which will be defined in Section 3.2. To do so, we first apply some preprocessing to a given instance of C2C on an and/or graph (in Section 3.1), and then give our FPT-reduction to BCSR (in Section 3.2).…”

“…In this way, from a preprocessed instance (G, A ini , A tar ) of NCL with the parameter |V or (G)| ≤ k, we have constructed the corresponding instance (H, D, C, Γ ini , Γ tar ) of BCSR such that d = max x∈X |D(x)| = 7 and p ≤ |V or (G)| ≤ k. In addition, we have shown that there is a one-to-one correspondence between proper solutions of H and feasible orientations of E. Since BCSR can be solved in time O * (d O(p) ) [5], the following lemma completes the proof of Theorem 1 for C2C.…”

“…We use the same reduction for C2E case. While an FPT-algorithm for C2C of BCSR is given in [5], that for C2E is not. However, we can simply improve the FPT-algorithm for C2C to C2E as follows: In the FPT-algorithm of [5], the authors first construct a contracted solution graph (CSG).…”

“…While an FPT-algorithm for C2C of BCSR is given in [5], that for C2E is not. However, we can simply improve the FPT-algorithm for C2C to C2E as follows: In the FPT-algorithm of [5], the authors first construct a contracted solution graph (CSG). Then they determine whether there is a path on CSG between two nodes corresponding to the initial and the target solutions.…”

Recolouring Homomorphisms to triangle-free reflexive graphs

“…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.…”