Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

Abstract

a b s t r a c tSuppose we are given a graph G together with two proper vertex k-colourings of G, α and β. How easily can we decide whether it is possible to transform α into β by recolouring vertices of G one at a time, making sure we always have a proper k-colouring of G? This decision problem is trivial for k = 2, and decidable in polynomial time for k = 3. Here we prove it is PSPACE-complete for all k ≥ 4. In particular, we prove that the problem remains PSPACE-complete for bipartite graphs, as well as for: (i) planar graphs and 4 ≤ k ≤ 6, and (ii) bipartite planar graphs and k = 4. Moreover, the values of k in (i) and (ii) are tight, in the sense that for larger values of k, it is always possible to recolour α to β.We also exhibit, for every k ≥ 4, a class of graphs {G N,k : N ∈ N * }, together with two k-colourings for each G N,k , such that the minimum number of recolouring steps required to transform the first colouring into the second is superpolynomial in the size of the graph: the minimum number of steps is Ω(2 N ), whereas the size of G N is O(N 2 ). This is in stark contrast to the k = 3 case, where it is known that the minimum number of recolouring steps is at most quadratic in the number of vertices. We also show that a class of bipartite graphs can be constructed with this property, and that: (i) for 4 ≤ k ≤ 6 planar graphs and (ii) for k = 4 bipartite planar graphs can be constructed with this property. This provides a remarkable correspondence between the tractability of the problem and its underlying structure.

Toward Cereceda's conjecture for planar graphs

“…See [4,8,9] for some examples. Computational work has focused on deciding whether there is a path in the reconfiguration graph between a given pair of colorings [5,10,14]. Other structural considerations of the reconfiguration graph have also been investigated in [1,2].…”

Introduction to Reconfiguration

“…Polynomial-time reachability algorithms have been developed for tractable source problems such as 2-COLORING [31][32][33], MATCHING [4], MINIMUM SPANNING TREE [4], and 2-SATISFIABILITY [34]. Reachability has been shown to be PSPACE-complete for many NP-complete source problems, such as CLIQUE [4], 4-COLORING [35], INDEPENDENT SET [4,17], 3-SATISFIABILITY [34], VERTEX COVER [4], DOMINATING SET [4], LIST EDGE-COLORING [36], LIST L(2, 1)-LABELING [37], INTEGER PROGRAMMING [4], STEINER TREE [38], and SET COVER [4].…”

“…For k = 3, it was shown that the diameter is in O(|V(G)| 2 ) for each connected component (and that this bound is tight, as there exist configurations at distance Ω(|V(G)| 2 )), and that both reachability and shortest transformation can be solved in polynomial time [32]. In contrast, for k ≥ 4, there are both yes-instances and no-instances for connectivity [31], and there exists a family of graphs and a k ≥ 4 such that for every graph in the family there exist components of diameter superpolynomial in |V(G)| [35]. Moreover, for k ≥ 4, reachability is strongly NP-hard [79] and PSPACE-complete, even for bipartite graphs, planar graphs for 4 ≤ k ≤ 6, and bipartite planar graphs for k = 4 [35].…”

“…In contrast, for k ≥ 4, there are both yes-instances and no-instances for connectivity [31], and there exists a family of graphs and a k ≥ 4 such that for every graph in the family there exist components of diameter superpolynomial in |V(G)| [35]. Moreover, for k ≥ 4, reachability is strongly NP-hard [79] and PSPACE-complete, even for bipartite graphs, planar graphs for 4 ≤ k ≤ 6, and bipartite planar graphs for k = 4 [35].…”

“…In LIST COLORING RECONFIGURATION [35], a generalization of k-COLORING RECONFIGURATION, each vertex v is supplied with a list L(v) of allowable colors. Early results in the area were obtained by generalizing results on k-COLORING RECONFIGURATION: the polynomial-time reachability algorithm for k ≤ 3 can be extended to this variant [32], and the reconfiguration graph is connected when the size of each list is at least col(G) + 2 [83].…”