Elsevier

Theoretical Computer Science

Volume 821, 12 June 2020, Pages 102-110
Theoretical Computer Science

List-coloring – Parameterizing from triviality

https://doi.org/10.1016/j.tcs.2020.02.022Get rights and content

Abstract

The classical graph coloring problem is given an undirected graph and the goal is to color the vertices of the graph with the minimum number of colors so that endpoints of each edge gets different colors. In list-coloring, each vertex is given a list of allowed colors with which it can be colored. Most versions of the problems are hard in several paradigms including approximation and parameterized complexity. We consider a few versions of the problems that are polynomial time solvable and try to extend the notion of feasible algorithms by parameterizing suitably in the paradigm of parameterized complexity. In particular, we provide extensions of these polynomial time solvable coloring problems that belong to the extreme ends of the parameterized complexity hierarchy including fixed-parameter tractable (FPT) class or the complexity class XP or W[1]-hard or para-NP -hard (i.e. NP -hard even for constant values of the parameter).

Specifically, we consider generalizations of odd cycle transversal and edge bipartization in the context of list-coloring and show them fixed-parameter tractable. We also look at list coloring where each list has nk colors and the goal is to color the vertices respecting the list. We show that the problem is in the parameterized complexity class XP, when parameterized by k.

Introduction

The graph coloring problem is one of the fundamental combinatorial optimization problems with applications in scheduling, register allocation, pattern matching, and many other active research areas. Given a graph G=(V,E), the graph coloring problem is a way to assign colors to vertices of a graph such that no two adjacent vertices share the same color. Such coloring is also known as a proper coloring. The smallest number of colors needed to color a graph G is called its chromatic number and is denoted by χ(G). Determining whether a graph is 3-colorable is NP -hard [9] while the 2-coloring problem has a linear time algorithm. It is even hard to approximate the chromatic number in polynomial time. The 3-coloring problem remains NP -complete even on 4-regular planar graphs [8]. There are some generalizations and variations of ordinary graph colorings that are motivated by practical applications. For example, sometimes a set of feasible colors L(v) is attached with each vertex vV and we may require that the vertex v be colored from a color from L(v). This takes us to the list coloring problem, and clearly, this is a generalization of standard graph coloring, and hence is considerably harder.

A list is -regular if each set contains exactly colors. The -regular list coloring problem is to decide whether G=(V,E) has a coloring that respects L, where L is -regular. A graph G is -choosable if for every −regular list L of G, there exists a coloring which respects L.

Cai [3] in one of the earliest papers on parameterizations of graph coloring studied various parameterizations. He showed surprisingly that it is NP -hard to determine whether a graph that is two vertices away from a bipartite graph is 3-colorable. A graph is k vertices away from a graph satisfying a property (say planar or bipartite), if there are k vertices in the graph whose removal results in a graph satisfying the property. We say that such a graph is planar +kv if it has k vertices whose deletion results in a planar graph. We consider such parameterizations in this paper for graph coloring and list coloring and give hardness and FPT (fixed-parameter tractable) results.

We begin with the notions of parameterized complexity before we explain our results.

The goal of parameterized complexity is to find ways of solving NP -hard problems more efficiently than brute force: here the aim is to restrict the combinatorial explosion to a parameter that is hopefully much smaller than the input size. A parameterization of a problem is assigning a positive integer parameter k to each input instance. Formally we say that a parameterized problem is fixed-parameter tractable (FPT ) if there is an algorithm A (called a fixed parameter algorithm), a computable function f, and a constant c such that, given problem instance (x,k), A correctly solves the problem in time bounded by f(k)|(x,k)|c, where x is the input and k is the parameter [6]. There is also an accompanying theory of parameterized intractability using which one can identify parameterized problems that are unlikely to admit FPT algorithms. These are essentially proved by showing that the problem is W-hard.

A parameterized problem is called slice-wise polynomial (XP ) if there exists an algorithm A, two computable functions f,g such that, given problem instance (x,k), the algorithm A correctly solves the problem in time bounded by f(k)|(x,k)|g(k), where x is the input and k is the parameter [6]. The complexity class containing all slice-wise polynomial problems is called XP. We say that a parameterized problem is para-NP -hard if the problem is NP -hard for some fixed constant value of the parameter. para-NP -hard problems are not in XP unless P=NP.

It is known [7] that 2-regular list coloring is polynomial-time solvable for general graphs. In a 2-regular list coloring instance, all vertices have lists of colors of size exactly 2. We consider the scenario when at most k vertices have lists of colors of size more than 2. We first define the problem formally. We show that this problem is W[1]-hard.

If a graph G=(V,E) is not colorable with respect to a 2-regular list L then a natural question is whether it is possible to color at least k vertices of the graph respecting L? This problem is known to be W[1] -hard [13]. In this paper, we address the second natural question (which is sometimes called ‘parametric dual’ defined below).

This is the same as asking whether we can delete k vertices and obtain a list coloring of the graph. If the lists (of size 2) of each vertex is the same, then note that this problem is the same as asking whether there are k vertices whose deletion results in a bipartite graph, which is the well-studied odd cycle transversal (OCT) problem [15], [18]. We show that Vertex Parameterized 2−regular List coloring, which is a generalization of OCT, is also FPT.

We also consider a closely related problem that is whether we can delete k edges such that resultant graph can be colored satisfying the given 2−regular list assignment. We define the problem formally as follows:

This problem is a generalization of the well-known edge bipartization problem. We show that this problem is also FPT.

Observe that for general graph k-regular list coloring is para-NP -hard as the problem is a generalization of k-coloring. It is known that if we ask whether the graph can be properly colored with (nk) colors, then the problem is FPT [6], [11]. We consider a similar variation on list coloring.

We show that (n-k)-regular list coloring is in XP which is the most technical part of the paper.

We end the paper with a couple of hardness results. Thomassen [19] showed that every planar graph is 5-choosable. Therefore, by definition of choosability, planar graph is always a Yes instance for 5-regular list coloring. Our first result is that 5-regular list coloring is para-NP -hard for planar+kv graphs.

Garey et al. [9] showed that coloring is polynomial-time solvable for graphs with maximum degree 3, actually, all such graphs other than 4-clique are 3-colorable. We show the following extension of the problem para-NP -hard.

In section 2 we show that the Vertex Parameterized 2-regular list coloring and Edge Parameterized 2-regular list coloring are FPT. Next we show that all-but-k 2-regular list coloring is W[1] -hard and that it becomes FPT with stronger parameterization or constant upper bound on the size of lists of colors. In section 3 we present the XP algorithm for (n-k)-regular list coloring. In section 4 we show the para-NP -hardness results on planar+kv graphs and G3+kv graphs.

Besides the results of Cai [3] mentioned in the introduction, there are other works on parameterizing the vertex coloring problems on graphs that are close to some special families. Formally if F is a graph family, then F+kv is another family of graphs, which have only k vertices whose deletion results into a F graph. Now for example, coloring problem, when parameterized by k, is W[1] -hard for chordal+kv graphs, interval+kv graphs and complete+kv graphs [16].

It is known that not every graph is k-choosable. Therefore a natural question is, given a graph G and a list can we find a coloring in polynomial time. In a recent paper [7] Dabrowski et al. have addressed many such list coloring problems. Jafke and Jansen [12] presented FPT algorithms for list coloring problems parameterized by k, where k is the vertex deletion distance to several graph classes for which coloring is known to be solvable in polynomial time.

Section snippets

2−regular list coloring

It is known [7] that 2-regular list coloring is polynomial time solvable for general graphs. We provide a proof here for completion, which will pop out as a corollary of the algorithm for the more general problem of the Vertex Parameterized 2−regular list coloring problem. Recall that here the question is whether it is possible to delete at most k vertices of the graph so that the rest of the graph can be colored respecting L.

Theorem 2.1

Vertex Parameterized 2−regular list coloring is fixed-parameter

(n-k)-regular list coloring is in XP

It is known that determining whether the vertices of a graph can be colored properly with nk colors is fixed-parameter tractable when parameterized by k [4], [6]. In this section we present an XP algorithm for (n-k)-regular list coloring – i.e. an algorithm that takes nO(k) time on an n vertex graph.

Let G=(V,E) be any graph and L be a (n-k)-regular list assignment such that L(v) denotes the list of colors corresponding to the vertex v. We begin by observing the following rule which can be

Hardness results

Theorem 4.1

5-regular list coloring is para-NP -hard on planar+kv graphs.

Proof

We will prove that 5-regular list coloring is para-NP -hard on planar+1v graphs. It is known that 4-regular list coloring is NP -complete for planar graphs even if every list contains four colors from {1,2,3,4,5} [7]. We will show a polynomial reduction from this problem to our problem.

Suppose we are given an instance G=(V,E) with the 4-regular list assignment. We create 5 copies of this instance and call them G1, G2, G3, G4 and G5.

Conclusions

We have considered some interesting generalizations of polynomial time solvable cases of coloring and list coloring, and provided parameterizations that encompass the extreme variety of parameterized complexity classes. The main open problem we leave with is whether (n-k)-regular list coloring is fixed-parameter tractable.1

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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A preliminary version of the paper appeared in the proceedings of 12th Frontiers in Algorithms Workshop (FAW) 2018, Springer LNCS 10823; Work done while the first and the third author were at IIT Jodhpur.

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