2022/09/12 by Agrawal, Akanksha, Hait, Soumita, Mouawad, Amer E. · 1 citation
#Combinatorics (math.CO) #Computational Complexity (cs.CC) #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2209.05145
Assume we are given a graph G, two independent sets S and T in G of size k ≥ 1, and a positive integer ℓ ≥ 1. The goal is to decide whether there exists a sequence ⟨ I0, I1, ..., I_ℓ ⟩ of independent sets such that for all j ∈ \0,…,ℓ-1\ the set Ij is an independent set of size k, I0 = S, I_ℓ = T, and Ij+1 is obtained from Ij by a predetermined reconfiguration rule. We consider two reconfiguration rules. Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the Token Sliding Optimization (TSO) problem asks whether there exists a sequence of at most ℓ steps that transforms S into T, where at each step we are allowed to slide one token from a vertex to an unoccupied neighboring vertex. In the Token Jumping Optimization (TJO) problem, at each step, we are allowed to jump one token from a vertex to any other unoccupied vertex of the graph. Both TSO and TJO are known to be fixed-parameter tractable when parameterized by ℓ on nowhere dense classes of graphs. In this work, we show that both problems are fixed-parameter tractable for parameter k + ℓ + d on d-degenerate graphs as well as for parameter |M| + ℓ + Δ on graphs having a modulator M whose deletion leaves a graph of maximum degree Δ. We complement these result by showing that for parameter ℓ alone both problems become W[1]-hard already on 2-degenerate graphs. Our positive result makes use of the notion of independence covering families introduced by Lokshtanov et al. Finally, we show that using such families one can obtain a simpler and unified algorithm for the standard Token Jumping Reachability problem parameterized by k on both degenerate and nowhere dense classes of graphs.