Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar.
Another model of reconfiguration is , introduced by Kami ' @cite_18 . In the token jumping model, we are also given an integer @math , and the vertex set of the configuration space consists of the matchings of size exactly @math . Two matchings @math and @math are adjacent if and only if they differ in only two edges. Kami ' [Theorem 1] DBLP:journals tcs KaminskiMM12 proved that the token jumping model is equivalent to the token addition removal model when @math . Thus, using the result by @cite_3 , the reachability can be checked in polynomial time also under the token jumping model [Corollary 2] DBLP:journals tcs KaminskiMM12 .
@cite_18 @cite_3
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We study problems of reconfigurability of independent sets in graphs. We consider three different models (token jumping, token sliding, and token addition and removal) and analyze relationships between them. We prove that independent set reconfigurability in perfect graphs (under any of the three models) generalizes the shortest path reconfigurability problem in general graphs and is therefore PSPACE-complete. On the positive side, we give polynomial results for even-hole-free graphs and P"4-free graphs. For given two independent sets ( I _b ) and ( I _r ) of a graph, the sliding token problem is to determine if there exists a sequence of independent sets which transforms ( I _b ) into ( I _r ) so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. The sliding token problem is one of the reconfiguration problems that attract the attention from the viewpoint of theoretical computer science. These problems tend to be PSPACE-complete in general, and some polynomial time algorithms are shown in restricted cases. Recently, the problems for finding a shortest reconfiguration sequence are investigated. For the 3SAT reconfiguration problem, a trichotomy for the complexity of finding the shortest sequence has been shown; it is in P, NP-complete, or PSPACE-complete in certain conditions. Even if it is polynomial time solvable to decide whether two instances are reconfigured with each other, it can be NP-complete to find a shortest sequence between them. We show nontrivial polynomial time algorithms for finding a shortest sequence between two independent sets for some graph classes. As far as the authors know, one of them is the first polynomial time algorithm for the shortest sliding token problem that requires detours of tokens. A token located at some vertex @math of a connected, undirected graph G on n vertices is said to be taking a “random walk” on G if, whenever it is instructed to move, it moves with equal probability to any of the neighbors of @math . The authors consider the following problem: Suppose that two tokens are placed on G, and at each tick of the clock a certain demon decides which of them is to make the next move. The demon is trying to keep the tokens apart as long as possible. What is the expected time M before they meet?The problem arises in the study of self-stabilizing systems, a topic of recent interest in distributed computing. Since previous upper bounds for M were exponential in n, the issue was to obtain a polynomial bound. The authors use a novel potential function argument to show that in the worst case @math . Consider a graph such that each vertex has a nonnegative integer capacity and each edge has a positive integer weight. Then, a b-matching in the graph is a multi-set of edges (represented by an integer vector on edges) such that the total number of edges incident to each vertex is at most the capacity of the vertex. In this paper, we study a reconfiguration variant for maximum-weight b-matchings: For two given maximum-weight b-matchings in a graph, we are asked to determine whether there exists a sequence of maximum-weight b-matchings in the graph between them, with subsequent b-matchings obtained by removing one edge and adding another. We show that this reconfiguration problem is solvable in polynomial time for instances with no integrality gap. Such instances include bipartite graphs with any capacity function on vertices, and 2-matchings in general graphs. Thus, our result implies that the reconfiguration problem for maximum-weight matchings can be solved in polynomial time for bipartite graphs.