Finding a reconfiguration sequence between longest increasing subsequences

TL;DR

This paper introduces polynomial-time algorithms for reconfiguring between longest increasing subsequences, with implications for permutation graphs and bipartite cases, advancing understanding of sequence transformations.

Contribution

It provides the first polynomial-time algorithms for reconfiguration between longest increasing subsequences and related problems on permutation graphs.

Findings

Polynomial-time algorithm for reconfiguration sequence existence.

Polynomial-time algorithm for shortest reconfiguration sequence in bipartite permutation graphs.

Implications for Independent Set Reconfiguration and Token Sliding problems.

Abstract

In this note, we consider the problem of finding a step-by-step transformation between two longest increasing subsequences in a sequence, namely Longest Increasing Subsequence Reconfiguration. We give a polynomial-time algorithm for deciding whether there is a reconfiguration sequence between two longest increasing subsequences in a sequence. This implies that Independent Set Reconfiguration and Token Sliding are polynomial-time solvable on permutation graphs, provided that the input two independent sets are largest among all independent sets in the input graph. We also consider a special case, where the underlying permutation graph of an input sequence is bipartite. In this case, we give a polynomial-time algorithm for finding a shortest reconfiguration sequence (if it exists).