Finding Optimal Solutions to Token Swapping by Conflict-Based Search and Reduction to SAT

Abstract

We study practical approaches to solving the token swapping (TSWAP) problem optimally in this paper. In TSWAP, we are given an undirected graph with colored vertices. A colored token is placed in each vertex. A pair of tokens can be swapped between adjacent vertices. The goal is to perform a sequence of swaps so that token and vertex colors agree across the graph. The minimum number of swaps is required in the optimization variant of the problem. We observed similarities between the TSWAP problem and multi-agent path finding (MAPF) where instead of tokens we have multiple agents that need to be moved from their current vertices to given unique target vertices. The difference between both problems consists in local conditions that state transitions (swaps/moves) must satisfy. We developed two algorithms for solving TSWAP optimally by adapting two different approaches to MAPF - conflict-based search (CBS) and SAT-based approach that uses multi-value decision diagrams (MDD-SAT). This constitutes the first attempt to design optimal solving algorithms for TSWAP. Experimental evaluation on various types of graphs shows that the reduction to SAT scales better than CBS in optimal TSWAP solving. It has been also demonstrated that TSWAP instances are easier to solve than corresponding similar MAPF instances.