Finding Shortest Reconfiguration Sequences on Independent Set Polytopes

TL;DR

This paper studies the computational complexity of finding the shortest reconfiguration sequences between independent sets in graphs, revealing NP-hardness and inapproximability results, while also providing efficient algorithms for specific graph classes.

Contribution

It introduces the shortest reconfiguration problem for independent sets, proves its NP-hardness and inapproximability in general, and offers polynomial algorithms for certain graph classes.

Findings

NP-hardness on planar graphs of bounded degree

W[2]-hardness on split graphs when parameterized by steps

Polynomial algorithms for block graphs, cographs, and bipartite chain graphs

Abstract

We initiate the study of the shortest reconfiguration problem for independent sets under the adjacency relation derived from the independent set polytope. Given a graph and two independent sets, the problem asks for a shortest sequence transforming one into the other such that the subgraph induced by the symmetric difference of any two consecutive sets is connected. This is equivalent to finding a shortest path on the $1$-skeleton of the independent set polytope. We prove that the problem is NP-hard even on planar graphs of bounded degree, as well as on split graphs. Notably, the hardness for planar graphs of bounded degree still holds even when deciding whether the target can be reached in at most two steps. For split graphs, we further show the W[2]-hardness when parameterized by the number of steps, as well as the inapproximability of the optimal length. As a consequence, we prove that the length of a shortest path between two vertices of a 0/1 polytope in $\mathbb{R}^n$ described by $O(n)$ linear inequalities is hard to approximate within a factor of $(1-\varepsilon)\ln n$ for any constant $\epsilon >0$, unless $P=NP$. On the positive side, we provide polynomial-time algorithms for block graphs, cographs, and bipartite chain graphs. Moreover, for paths and cycles, we show that the optimal length of the shortest reconfiguration sequence exactly matches a trivial upper bound.