Uniquely Coloring Graphs over Path Decompositions.
Andreas Björklund · arXiv (Cornell University) · 2015
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no $(k-\epsilon)^{\operatorname{pw}(G)}\operatorname{poly}(n)$ time algorithm for deciding if an $n$-vertex graph $G$ with pathwidth $\operatorname{pw}(G)$ admits a proper vertex coloring with $k$ colors unless the Strong Exponential Time Hypothesis (SETH) is false. We show here that nevertheless, when $k>\lfloor \Delta_h/2 \rfloor + 1$, where $\Delta_h$ is the maximum degree of a vertex in the graph $G$ after excluding the $h$ vertices of highest degree, there is a better algorithm, at least when the coloring is unique (up to color permutations). We present a Monte Carlo algorithm that given a graph $G$ along with a path decomposition of $G$ with pathwidth $\operatorname{pw}(G)$ runs in $(\lfloor \Delta_h/2 \rfloor + 1)^{\operatorname{pw}(G)}k^hn^{O(k^2)}$ time, and that distinguishes between uniquely $k$-colorable graphs and non-$k$-colorable graphs. Our algorithm falls short of disproving SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them. We exploit a new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than $k$ at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is $k$-colorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex, many orientations give a non-zero value if the graph has a unique $k$-coloring. Still, every orientation gives a zero value if the graph has no $k$-coloring, so a random orientation stands a good chance of being useful for separating the two cases.