4 research outputs found
Computing Square Colorings on Bounded-Treewidth and Planar Graphs
A square coloring of a graph is a coloring of the square of ,
that is, a coloring of the vertices of such that any two vertices that are
at distance at most in receive different colors. We investigate the
complexity of finding a square coloring with a given number of colors. We
show that the problem is polynomial-time solvable on graphs of bounded
treewidth by presenting an algorithm with running time for graphs of treewidth at most . The somewhat
unusual exponent in the running time is essentially
optimal: we show that for any , there is no algorithm with running
time 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 of colors. Our main algorithmic result is showing
that the problem (when the number of colors is part of the input) can be
solved in subexponential time on planar graphs. The
result follows from the combination of two algorithms. If the number of
colors is small (), then we can exploit a treewidth bound on the
square of the graph to solve the problem in time . If
the number of colors is large (), then an algorithm based on
protrusion decompositions and building on our result for the bounded-treewidth
case solves the problem in time .Comment: 72 pages, 15 figures, full version of a paper accepted at SODA 202
Computing Square Colorings on Bounded-Treewidth and Planar Graphs
A {\em square coloring} of a graph is a coloring of the square of , that is, a coloring of the vertices of such that any two vertices that are at distance at most in receive different colors.
We investigate the complexity of finding a square coloring with a given number of colors.
We show that the problem is polynomial-time solvable on graphs of bounded treewidth by presenting an algorithm with running time n^{2^{\ttw + 4}+O(1)} for graphs of treewidth at most \ttw.
The somewhat unusual exponent 2^\ttw in the running time is essentially optimal: we show that for any , there is no algorithm with running time f(\ttw)n^{(2-\epsilon)^\ttw} 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 of colors.
Our main algorithmic result is showing that the problem (when the number of colors is part of the input) can be
solved in subexponential time on planar graphs. The result
follows from the combination of two algorithms. If the number
of colors is small (), then we can exploit a
treewidth bound on the square of the graph to solve the problem in
time . If the number of colors is large
(), then an algorithm based on protrusion
decompositions and building on our result for the bounded-treewidth case solves the problem in time