Two Generalizations of Proper Coloring: Hardness and Approximability

Abstract

We study two natural generalizations of q -Coloring. These problems can be seen as optimization problems and are mostly applied to graphs that are not properly colorable with q colors. One of them is known as Maximum q -Colorable Induced Subgraph, and asks to find the largest set of vertices inducing a q-colorable graph. While very natural, this generalization has a downside of that it does not assign any color to vertices outside of the solution, which limits its application.

To address this issue, we introduce another natural generalization of q -Coloring. The main concept of this new problem is properly colored vertex, which is a vertex that has no neighbour colored with the same color as itself. The Maximum Properly q -Colored Vertices asks to find a q-coloring of the input graph that maximizes the number of such vertices.

Our work focuses on similarities and differences between these two problems. The first part of our work is the computational hardness of Maximum Properly q -Colored Vertices in comparsion to Maximum q -Colorable Induced Subgraph. Here we first show that Maximum Properly q -Colored Vertices admits a $1.4391^n$ exact algorithm for $q=2$, and is NP-complete in this case even on unit-disk graphs. Following the parameterized complexity study of Maximum q -Colorable Induced Subgraph by Misra et al. [WG ’13], we then show that Maximum Properly q -Colored Vertices and Maximum $(q+1)$ -Colorable Induced Subgraph are basically the same problem when restricted to split graphs. In contrast to this, we show that Maximum Properly q -Colored Vertices and Maximum q -Colorable Induced Subgraph behave differently on perfect graphs, as Maximum Properly q -Colored Vertices is W[2]-hard on this graph class, while Maximum q -Colorable Induced Subgraph was known to be FPT.

The second part of our work is dedicated to efficient approximation of both problems on unit-disk graphs. Namely, we design several approximation algorithms for these problems restricted to this graph class. The first kind of algorithms is based on (now classical) shifting technique and achieves an $(1-\epsilon )$-approximate solution in $n^{\mathcal {O}(\frac{q}{\epsilon })}$ time. The second kind of algorithms that we obtain aims to get rid of the dependency in q in the exponent. These two algorithms run in $n^{\mathcal {O}(\frac{1}{\epsilon })}$, though the approximation ratio they achieve is only $(1-\frac{1}{e}-\epsilon )$. Here we use the greedy $(1-\frac{1}{e})$-approximation algorithm for the Maximum Coverage problem. These algorithms rely heavily on the given geometric representation of the graph, so we propose the third kind of algorithms that does not require the geometric representation. They allow to achieve the same approximation ratios under a tradeoff of additional $\frac{1}{\epsilon }\log ^2{\frac{q}{\epsilon }}$ multiplier in the exponent. While the three methods used to design these schemes are not novel, our research extends boundaries of their applicability while showing how they can be efficiently combined together to achieve new algorithms.

This research was supported by Russian Science Foundation (project 18-71-10042).