Greedy coloring | EPFL Graph Search

In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not, in general, use the minimum number of colors possible. Different choices of the sequence of vertices will typically produce different colorings of the given graph, so much of the study of greedy colorings has concerned how to find a good ordering. There always exists an ordering that produces an optimal coloring, but although such orderings can be found for many special classes of graphs, they are hard to find in general. Commonly used strategies for vertex ordering involve placing higher-degree vertices earlier than lower-degree vertices, or choosing vertices with fewer available colors in preference to vertices that are less constrained. Variations of greedy coloring choose the colors in an online manner, without any knowledge of the structure of the uncolored part of the graph, or choose other colors than the first available in order to reduce the total number of colors. Greedy coloring algorithms have been applied to scheduling and register allocation problems, the analysis of combinatorial games, and the proofs of other mathematical results including Brooks' theorem on the relation between coloring and degree. Other concepts in graph theory derived from greedy colorings include the Grundy number of a graph (the largest number of colors that can be found by a greedy coloring), and the well-colored graphs, graphs for which all greedy colorings use the same number of colors. The greedy coloring for a given vertex ordering can be computed by an algorithm that runs in linear time. The algorithm processes the vertices in the given ordering, assigning a color to each one as it is processed. The colors may be represented by the numbers and each vertex is given the color with the smallest number that is not already used by one of its neighbors.

How to Match in Modern Computational Settings

Buddhima Ruwanmini Gamlath Gamlath Ralalage

This thesis focuses on the maximum matching problem in modern computational settings where the algorithms have to make decisions with partial information.First, we consider two stochastic models called query-commit and price-of-information where the algorithm only knows the distribution from which the edges are sampled.In the query-commit model, the algorithm must query edges to know if they exist and is committed to adding all queried edges that exist to its output.In the price-of-information model, the algorithm incurs costs for querying edges, and the total query cost is subtracted from the output matching's weight.For maximum weighted matching in these models, previously known best algorithms were greedy algorithms that achieve 1/2 approximations. We improve the approximation ratio to 1 - 1/e in both models. Next, we consider situations where the input graphs do not fit into the space available for an algorithm instance. We consider two such models: the semi-streaming model where the algorithm receives the input as a stream of edges and the algorithm has only sub-linear (in the number of edges) space, and the massively parallel computation (MPC) model where the input is distributed among several machines, each of which has sub-linear space, and algorithm instances running on different machines must communicate in synchronous rounds.We start with a particular case of the semi-streaming model where the edges arrive in uniformly random order, and the algorithm goes over the stream only once. For this setting, we give the first algorithm that finds a (1/2 + c)-approximate maximum weighted matching in expectation; such algorithms were previously known only for the unweighted graphs.We then show how to efficiently find (1 - epsilon)-approximate weighted matchings for any epsilon > 0 in multi-pass semi-streaming and MPC models by extending our algorithmic ideas used in the single-pass semi-streaming model with random order edge arrivals.Finally, we study online algorithms for matching, where the input graph is gradually revealed over time. In the online edge-arrival setting, the graph is revealed one edge at a time, and an algorithm is forced to make irrevocable decisions on whether to add each edge to the output matching upon their arrival. We show that no online algorithm can achieve a competitive ratio of 1/2 + c for any constant c > 0 in this setting.In the online vertex-arrival setting, the graph is revealed one vertex at a time, together with its incident edges to already revealed vertices, and the algorithm must irrevocably decide to ignore the revealed vertex or match it to one of the available neighbors.In this setting, we show how to round a previously known fractional online matching algorithm to get an integral online matching algorithm with a competitive ratio of 1/2 + c for some constant c > 0.

EPFL2021

Tight Bounds for Online Edge Coloring

David Wajc

Vizing's celebrated theorem asserts that any graph of maximum degree Delta admits an edge coloring using at most Delta + 1 colors. In contrast, Bar-Noy, Motwani and Naor showed over a quarter century ago that the trivial greedy algorithm, which uses 2 Delta - 1 colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with Delta = O(log n), and they conjectured the existence of online algorithms using Delta (1+o(1)) colors for Delta = omega(log n). Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS'03 and Bahmani et al., SODA'10). We resolve the above conjecture for adversarial vertex arrivals in bipartite graphs, for which we present a (1+o(1))Delta-edge-coloring algorithm for Delta = omega(log n) known a priori. Surprisingly, if Delta is not known ahead of time, we show that no (e/e-1 - Omega(1)) Delta-edge-coloring algorithm exists. We then provide an optimal, (e/e-1 + o(1))Delta-edge-coloring algorithm for unknown Delta = omega(log n). To obtain our results, we study a nonstandard fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms.

IEEE COMPUTER SOC2019

Some coloring and walking problems in graphs

Benjamin Leroy-Beaulieu

Graph theory is an important topic in discrete mathematics. It is particularly interesting because it has a wide range of applications. Among the main problems in graph theory, we shall mention the following ones: graph coloring and the Hamiltonian circuit problem. Chapter 1 presents basic definitions of graph theory, such as graph coloring, graph coloring with color-classes of bounded size b, and Hamiltonian circuits and paths. We also present online algorithms and online coloring. Chapter 2 starts with some general remarks about online graph covering with sets of bounded sizes (such as online bounded coloring): we give a simple method for transforming an online covering algorithm into an online bounded covering algorithm, and to derive the performance ratio of the bounded algorithm from the performance ratio of the unbounded algorithm. As will be shown in later chapters, this method often leads to optimal results. Furthermore, some basic preliminary results on online graph covering with sets of bounded size are given: for every graph, the performance ratio is bounded above by 1/2 + b/2 and for b = 2, this bound is optimal. In the second part, online coloring of co-interval graphs is studied. Based on two industrial applications, two different versions of this problem are discussed. In the case where the intervals are presented in increasing order of their left ends, we show that the performance ratio is 1 in the unbounded case and 2 - 1/b in the bounded case. In the case where the intervals may be presented in any order, we show that the performance ratio is at most 3 in the bounded case. Chapter 3 deals with online coloring of permutation and comparability graphs. First, we give a tight analysis of the First-Fit algorithm on bipartite permutation graphs and we show that its performance ratio is O(√n), even for some simple presentation orders. For both classes of graphs, we show that the performance ratio is bounded above by (χ+1)/2 in the unbounded case and that the performance ratio of First-Fit is equal to 1/2 + b/2 in the bounded case. In the second part of this chapter, we study cocoloring of permutation graphs. We show that the performance ratio is n/4 + 1/2 and we give better bounds in some more restricted cocoloring problems. Chapter 4 deals with an application of online coloring: the online Track Assignment Problem. Depending on the assumptions that are made, the Track Assignment Problem can be reduced to coloring permutation or overlap graphs online. We show that when a permutation graph is presented on a latticial plane, from west to east, then the performance ratio is exactly 2 - (min{b,k})-1, where k is the best known upper bound on the bounded chromatic number. We also show that, when a permutation graph is presented on a latticial plan, starting from the origin and growing, simultaneously or not, towards west and east, then the performance ratio is exactly 2 - 1/χ. We also show that online coloring overlap graphs does not have a performance ratio bounded by a constant, even if the overlap graph is bipartite and presented in increasing order of the intervals left ends. In this special case, we show that First-Fit has a tight performance ratio of O(√n). We consider coloring overlap graphs online where the intervals have a bounded size between 1 and a given number M. In this case, we show that the performance ratio can be bounded above by 2√M if M ≤ M0, and by log M (⎡log M / log log M⎤ + 1) if M > M0, M0 being defined by the equation 2√M0 = 3 log(M0). For large values of M, the ratio is O(log2 M / log log M). Chapter 5 is about online coloring of trees, forests and split-graphs. For trees, we show that the performance ratio of online coloring is exactly ½log2(2n) in the unbouded case and at most 1 + ⎣log2(b)⎦/χb in the bounded case. For split-graphs, we show that the performance ratio of online coloring is exactly 1 + 1/χ in the unbounded case and is at most 2 + 1/χb + 3/b in the bounded case. In Chapter 6, we present a class of digraphs: the quasi-adjoint graphs. These are a super class of both the graphs used for a DNA sequencing algorithm in (Blazewicz, Kasprzak, "Computational complexity of isothermic DNA sequencing by hybridization", 2006) and the adjoints. A polynomial recognition algorithm in O(n3), as well as a polynomial algorithm in O(n2 + m2) for finding a Hamiltonian circuit in quasi-adjoint graphs are given. Furthermore, some results about related problems such as finding a Eulerian circuit while respecting some forbidden transitions (a sequence of two consecutive arcs) are discussed.

EPFL2008