TY - GEN

T1 - Computing Square Colorings on Bounded-Treewidth and Planar Graphs

AU - Agrawal, Akanksha

AU - Marx, Dániel

AU - Neuen, Daniel

AU - Slusallek, Jasper

N1 - Publisher Copyright:
Copyright © 2023.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - A square coloring of a graph G is a coloring of the square G2 of G, that is, a coloring of the vertices of G such that any two vertices that are at distance at most 2 in G receive different colors. We investigate the complexity of finding a square coloring with a given number of q colors. We show that the problem is polynomial-time solvable on graphs of bounded treewidth by presenting an algorithm with running time n2tw +4+O(1) for graphs of treewidth at most tw. The somewhat unusual exponent 2tw in the running time is essentially optimal: we show that for any ε > 0, there is no algorithm with running time f(tw)n(2-ε)tw unless the Exponential-Time Hypothesis (ETH) fails. We also show that the square coloring problem is NP-hard on planar graphs for any fixed number q ≥ 4 of colors. Our main algorithmic result is showing that the problem (when the number of colors q is part of the input) can be solved in subexponential time 2O(n2/3 log n) on planar graphs. The result follows from the combination of two algorithms. If the number q of colors is small (≤ n1/3), then we can exploit a treewidth bound on the square of the graph to solve the problem in time 2O(√qn log n).

AB - A square coloring of a graph G is a coloring of the square G2 of G, that is, a coloring of the vertices of G such that any two vertices that are at distance at most 2 in G receive different colors. We investigate the complexity of finding a square coloring with a given number of q colors. We show that the problem is polynomial-time solvable on graphs of bounded treewidth by presenting an algorithm with running time n2tw +4+O(1) for graphs of treewidth at most tw. The somewhat unusual exponent 2tw in the running time is essentially optimal: we show that for any ε > 0, there is no algorithm with running time f(tw)n(2-ε)tw unless the Exponential-Time Hypothesis (ETH) fails. We also show that the square coloring problem is NP-hard on planar graphs for any fixed number q ≥ 4 of colors. Our main algorithmic result is showing that the problem (when the number of colors q is part of the input) can be solved in subexponential time 2O(n2/3 log n) on planar graphs. The result follows from the combination of two algorithms. If the number q of colors is small (≤ n1/3), then we can exploit a treewidth bound on the square of the graph to solve the problem in time 2O(√qn log n).

UR - http://www.scopus.com/inward/record.url?scp=85161334439&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85161334439

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 2087

EP - 2110

BT - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

PB - Association for Computing Machinery

T2 - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

Y2 - 22 January 2023 through 25 January 2023

ER -