The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, algebraic and probabilistic methods, and discuss several intriguing conjectures and open problems. This is mainly a survey of recent and less recent results in the area, but it contains several new results as well.

. The Lov'asz Local Lemma (LLL) is a powerful tool that is increasingly playing a valuable role in computer science. The original lemma was nonconstructive; a breakthrough of Beck and its generalizations (due to Alon and Molloy & Reed) have led to constructive versions. However, these methods do not capture some classes of applications of the LLL. We make progress on this, by providing algorithmic approaches to two families of applications of the LLL. The first provides constructive versions of certain applications of an extension of the LLL (modeling, e.g., hypergraph-partitioning and low-congestion routing problems); the second provides new algorithmic results on constructing disjoint paths in graphs. Our results can also be seen as constructive upper bounds on the integrality gap of certain packing problems. One common theme of our work is a "gradual rounding" approach.

The Lov'asz Local Lemma (LLL) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. We consider three classes of NP-hard integer programs: minimax, packing, and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan & Thompson to derive good approximation algorithms for such problems. We use our extended LLL to prove that randomized rounding produces, with non-zero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are sparse (e.g., VLSI routing using short paths, problems on hypergraphs with small dimension/degree). We also generalize the method of pessimistic estimators due to Raghavan, to constructivize our packing and covering results. 1

It is shown that the chromatic number of any graph with maximum degree d in which the number of edges in the induced subgraph on the set of all neighbors of any vertex does not exceed d 2 =f is at most O(d= log f): This is tight (up to a constant factor) for all admissible values of d and f .

We consider a canonical Ramsey type problem. An edge-coloring of a graph is called m-good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m-good edge-coloring of K n yields a properly edge-colored copy of G, and let g(m, G) denote the smallest n such that every m-good edgecoloring of K n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G � K t, we have c 1mt 2 /ln t � f(m, K t) � c 2mt 2, and c � 1mt 3 /ln t � g(m, K t) � c � 2mt 3 /ln t, where c 1, c 2, c � 1, c � 2 are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) � n for all graphs G with n vertices and

The choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). Applying probabilistic methods it is shown that there are two positive constants c1 and c2 such that for all m ≥ 2 and r ≥ 2 the choice number of the complete r-partite graph with m vertices in each vertex class is between c1r log m and c2r log m. This supplies the solutions of two problems of Erdős, Rubin and Taylor, as it implies that the choice number of almost all the graphs on n vertices is o(n) and that there is an n vertex graph G such that the sum of the choice number of G with that of its complement is at most O(n 1/2 (log n) 1/2).

We present a Garey/Johnson-style list of problems known to be complete for the second and higher levels of the polynomial-time Hierarchy (polynomial hierarchy, or PH for short). We also include the best-known hardness of approximation results. The list will be updated as necessary. Updates The compendium currently lists more than 80 problems. Latest changes include: • added [GT26] SUCCINCT k-KING, • added [GT25] SUCCINCT k-DIAMETER, • added [GT4] SUCCINCT k-RADIUS at third level, • added [GT24] MINIMUM VERTEX COLORING DEFINING SET, • added [GT23] GRAPH SANDWICH PROBLEM FOR Π, • added [L24] MINIMUM 3SAT DEFINING SET,

It is shown that for every ɛ> 0 and n> n0(ɛ), any complete graph K on n vertices whose edges are colored so that no vertex is incident with more than (1 − 1 √ 2 − ɛ)n edges of the same color, contains a Hamilton cycle in which adjacent edges have distinct colors. Moreover, for every k between 3 and n any such K contains a cycle of length k in which adjacent edges have distinct colors. 1

In this paper we prove the following result about vertex list colourings, which shows that a conjecture from [9] is asymptotically correct. Let G be a graph with the sets of lists S(v), satisfying that for every vertex jS(v)j = (1+o(1))d and for each colour c 2 S(v), the number of neighbours of v that have c in their list is at most d. Then there exist a proper colouring of G from these lists.

We give a deterministic polynomial time approximation algorithm for a minmax integer programming problem, achieving the best existential bound given by Srinivasan [17]. Such a minmax integer programming problem arises naturally from a classical problem on routing to minimize congestion. It also has a hypergraph coloring problem as its special case, which can be applied to support divide and conquer approaches for numerous problems. 1 Introduction Consider the following classical routing problem. The input is a graph with n nodes and m edges, and k pairs of nodes (s 1 ; t 1 ); : : : ; (s k ; t k ) . We are asked to connect each pair by a path, in such a way as to minimize the congestion --- the maximum number of paths using an edge. This problem is known to be NP-hard, so we settle for an approximation algorithm. Raghavan and Thompson [16] introduced the following approach. First, solve the multicommodity flow relaxation. That is, we want to send a unit flow between each pair, possibly...