Finding Shortest Reconfiguration Sequences on Independent Set Polytopes

2026-04-27

AI Summary

This study addresses the problem of finding shortest reconfiguration sequences between two vertices in the 1-skeleton of the independent set polytope, where each step must induce a connected subgraph via the symmetric difference of consecutive independent sets. The work establishes, for the first time, a connection between this reconfiguration problem and shortest-path computation on 0/1-polytopes. By integrating tools from graph-theoretic decomposition, polyhedral geometry, and computational complexity theory, the authors prove that the problem is NP-hard, W[2]-hard, and inapproximable within any constant factor—even on planar bounded-degree graphs and split graphs—and further establish a logarithmic lower bound on the inapproximability of the optimal path length. Conversely, they present polynomial-time algorithms for block graphs, co-graphs, and bipartite chain graphs, and exactly characterize the optimal reconfiguration distance on paths and cycles.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science

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 $ε>0$, 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.

Problem

• reconfiguration
• independent set
• shortest path
• polytope
• computational complexity

Innovation

• reconfiguration
• independent set polytope
• shortest path
• parameterized complexity
• approximability