Shortest Reconfiguration of Sliding Tokens on a Caterpillar

Abstract

Suppose that we are given two independent sets I b and Ir of a graph such that |I b | = |Ir|, and imagine that a token is placed on each vertex in I b . Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I b into Ir 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. The reconfiguration problems tend to be PSPACE-complete in general, and some polynomial time algorithms are shown in restricted cases. Recently, the problems that aim at finding a shortest reconfiguration sequence are investigated. For the 3SAT problem, a trichotomy for the complexity of finding the shortest sequence has been shown; that is, it is in P, NPcomplete, or PSPACE-complete in certain conditions. In general, 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. Namely, finding a shortest sequence between two independent sets can be more difficult than the decision problem of reconfigurability between them. In this paper, we show that the problem for finding a shortest sequence between two independent sets is polynomial time solvable for some graph classes which are subclasses of the class of interval graphs. More precisely, we can find a shortest sequence between two independent sets on a graph G in polynomial time if either G is a proper interval graph, a trivially perfect graph, or a caterpillar. As far as the authors know, this is the first polynomial time algorithm for the shortest sliding token problem for a graph class that requires detours.Recently, the reconfiguration problems attract the attention from the viewpoint of theoretical computer science. The problem arises when we wish to find a stepby-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible and each step abides by a fixed reconfiguration rule, that is, an adjacency relation defined on feasible solutions of the original problem. The reconfiguration problems have been studied extensively for several

Shortest Reconfiguration Sequence for Sliding Tokens on Spiders

“…The algorithm is constructive, and the sequence itself can be output in O(n 2 ) time. As mentioned in [26], the number of required token-slides in a sequence can be Ω(n 2 ), hence our algorithm is optimal for the number of token-slides. Note: Recently, it is announced that the Shortest Sliding Token problem on a tree can be solved in polynomial time by Sugimori [23].…”

“…The Sliding Token problem on a tree can be solved in linear time [6]. Polynomial-time algorithms for the Shortest Sliding Token problem were first investigated in [26]. In [26], the authors gave polynomial-time algorithms for solving Shortest Sliding Token when the input graph is either a proper interval graph, a trivially perfect graph, or a caterpillar.…”

“…Polynomial-time algorithms for the Shortest Sliding Token problem were first investigated in [26]. In [26], the authors gave polynomial-time algorithms for solving Shortest Sliding Token when the input graph is either a proper interval graph, a trivially perfect graph, or a caterpillar. We note that caterpillars is the first graph class that required detours to solve the Shortest Sliding Token problem.…”

Introduction to Reconfiguration

“…Results on INDEPENDENT SET RECONFIGURATION have not been confined to reachability. Polynomial-time algorithms have been developed for shortest transformation under TS for proper interval graphs, trivially perfect graphs, and caterpillars [58] (though NP-hard in general for all reconfiguration steps [17]), for connectivity under TS for interval graphs [56] and under TAR for cographs [55], and for finding an actual reconfiguration sequence under TS for trees (as an extension of the reachability algorithm) [59]. It has also been shown that there is an infinite family of instances on paths for which the length of the reconfiguration sequence is at least quadratic in the number of vertices [59]; in contrast, for every yes-instance of reachability on cographs, there is a bound on the length of the reconfiguration sequence [42] as well as a bound on the diameter for claw-free graphs [57].…”

Introduction to Reconfiguration

“…Results on INDEPENDENT SET RECONFIGURATION have not been confined to reachability. Polynomial-time algorithms have been developed for shortest transformation under TS for proper interval graphs, trivially perfect graphs, and caterpillars [58] (though NP-hard in general for all reconfiguration steps [17]), for connectivity under TS for interval graphs [56] and under TAR for cographs [55], and for finding an actual reconfiguration sequence under TS for trees (as an extension of the reachability algorithm) [59]. It has also been shown that there is an infinite family of instances on paths for which the length of the reconfiguration sequence is at least quadratic in the number of vertices [59]; in contrast, for every yes-instance of reachability on cographs, there is a bound on the length of the reconfiguration sequence [42] as well as a bound on the diameter for claw-free graphs [57].…”