Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence

We study the {edge-coloring} problem in the message-passing model of distributed computing. This is one of the most fundamental and well-studied problems in this area. Currently, the best-known deterministic algorithms for (2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01}, where Delta is the maximum degree of the input graph. Also, recent results of \cite{BE10} for vertex-coloring imply that one can get an O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an arbitrarily small constant epsilon > 0. In this paper we devise a drastically faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves the previous state-of-the-art {exponentially} in a wide range of Delta, specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In addition, for small values of Delta our deterministic algorithm outperforms all the existing {randomized} algorithms for this problem. On our way to these results we study the {vertex-coloring} problem on the family of graphs with bounded {neighborhood independence}. This is a large family, which strictly includes line graphs of r-hypergraphs for any r = O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to {general} graphs. Our main technical contribution is a subroutine that computes an O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq Delta

Generalized vertex coloring problems using split graphs

Graph theory experienced a remarkable increase of interest among the scientific community during the last decades. The vertex coloring problem (Min Coloring) deserves a particular attention rince it has been able to capture a wide variety of applications. For mathematicians, it is interesting for an additional reason: it is extremely hard to solve it in an efficient way. In this thesis, we introduce several problems generalizing the usual vertex coloring problem, and hence, extending its application domain. We say that a graph is (p, k)-colorable if its vertex set can be partitioned into p cliques and k stable sets. Then, for a given p (respectively k), one may ask the following questions: how to choose p cliques (respectively k stable sets) to be removed from the graph such that the number of stable sets (respectively cliques) partitioning the remaining vertices is minimized? These are called (p, k)-coloring problems. We also introduce Min Split-coloring which is, given a graph G, the problem of minimizing k such that G is (k, k)-colorable. Along the saine line, given a graph G, the objective of the problem Min Cocoloring is to minimize p + k such that G is (p, k)-colorable. All these problems, called together generalized coloring problems, are obviously at least as difficult as Min Coloring. The purpose of this dissertation is to study generalized coloring problems in nome restricted classes of graphs in order to bring a new insight on the relative difficulties of these problems. To this end, we detect in a more precise way the limits between NP-hard and polynomially solvable problems. Chapter 1 introduces generalized coloring problems by emphasizing nome preliminary results which will guide the questions to handle in the following chapters. Chapter 2 exposes the first clans of graphs, namely cacti, where Min Split-coloring is shown to be polynomially solvable. We also observe that generalized coloring problems can be polynomially solved in triangulated graphs. The main result of Chapter 3 is a new characterization of cographs: it is equivalent to say that G is a cograph, and to state that, for every subgraph G' ⊆ G, G' is (p, k)-colorable if and only if G' [V \ K] is (p – 1, k)-colorable, where K induces a maximum clique of G'. This result implies simple combinatorial algorithme to solve all generalized coloring problems; the one for Min Cocoloring improves the best time complexity known so far. In Chapter 4, we handle the recognition of polar graphs which can be seen as a particular (p, k)-coloring, where p cliques are independent (i.e., not linked at all) and k stable sets form a complete k-partite graph. It is known that the recognition of polar graphs is NP-complete. Here, we determine the first clans of graphs, namely cographs, where the polar graphs can be recognized in polynomial time, more precisely in time O(n log n). We also give a characterization by forbidden subgraphs. In the came manner, we characterize monopolar cographs, i.e., cographs admitting a polar partition with at most one clique or at most one stable set. Chapter 5 is devoted to generalized coloring problems in line graphs. Here, we detect the first classes of graphs, namely line graphs of trees, line graphs of bipartite graphs and line graphs of line-perfect graphs, where generalized coloring problems diverge in terms of NP-hardness. In Chapter 6 we study the approximability of generalized coloring problems in line graphs, in comparability graphs and in general graphs. We derive approximation algorithms with a performance guarantee using both the standard approximation ratio and the differential approximation ratio. We show that both Min Split-coloring and Min Cocoloring are at least as hard as Min Coloring to approximate from the standard approximation ratio point of view, whereas, they admit a polynomial time differential approximation scheme and Min Coloring only a constant differential approximation ratio. We also show that Min Cocoloring reduces to Min Split-coloring in all classes of graphs closed under addition of disjoint cliques and under join of a complete k-partite graph. In Chapter 7, we handle two different applications of Min Split-coloring in permutation graphs. They give birth to a new problem, called Min Threshold-coloring, that we study in the came spirit as the other generalized coloring problems. In the last chapter, we present several open questions arising from this thesis

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials $\mathcal C$ and show that if $p\in \cal C$ and there is a disk $D$ centered at zero in the complex plane such that $p(G)$ does not vanish on $D$ for all bounded degree graphs $G$, then for each $z$ in the interior of $D$ there exists a deterministic polynomial-time approximation algorithm for evaluating $p(G)$ at $z$. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated by. Barvinok, which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In particular a tiny error in Proposition 4.4 has been fixed. The introduction and concluding remarks have also been rewritten to incorporate the most recent developments. Accepted for publication in SIAM Journal on Computatio

An on-line competitive algorithm for coloring bipartite graphs without long induced paths

The existence of an on-line competitive algorithm for coloring bipartite graphs remains a tantalizing open problem. So far there are only partial positive results for bipartite graphs with certain small forbidden graphs as induced subgraphs. We propose a new on-line competitive coloring algorithm for $P_9$-free bipartite graphs

List coloring in the absence of a linear forest.

The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The Listk-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,…,k}. Let Pn denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that Listk-Coloring can be solved in polynomial time for graphs with no induced rP1+P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1+P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H