We study the APPROXIMATE-COLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2-to-1 conjecture [Khot’02]. For q = 3, we base our hardness result on a certain ‘⊲< shaped ’ variant of his conjecture. We also prove that the problem ALMOST-3-COLORINGε is hard for any constant ε> 0, assuming Khot’s Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3-color all but a ε fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least ε. Our result is based on bounding various generalized noise-stability quantities using the invariance principle of Mossel et al [MOO’05].

Abstract — Applications in complex systems such as the Internet have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. In this paper, we introduce an algorithmic framework for studying combinatorial problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems (Vertex Cover, Shortest Path, Perfect Matching, and Spanning Tree) in this setting and establish tight upper and lower bounds for the approximability of these problems. 1.

Many approximation algorithms have been presented in the last decades for hard search problems. The focus of this paper is on cryptographic applications, where it is desired to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms – efficient algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information. For functions, [Feigenbaum et al., ICALP 2001] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to search problems is not straight forward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms – applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. Although our privacy requirement seems minimal, we show that for well studied problems, as vertex cover and maximum exact 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similar to [Halevi et al., STOC 2001], we define a relaxed notion of approximation algorithms that leak (little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for maximum exact 3SAT which leak little information.

We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: • Vertex Cover is Unique Games-hard to approximate log log d to within a factor 2−(2+od(1)). This exactly log d matches the algorithmic result of Halperin [1] up to the od(1) term. • Independent Set is Unique Games-hard to approxi-d mate to within a factor O( log2). This improves the d d logO(1) Unique Games hardness result of Samorod-

We develop a refinement of a forbidden submatrix characterization of 0/1-matrices fulfilling the Consecutive Ones Property (C1P). This novel characterization finds applications in new polynomial-time approximation algorithms and fixed-parameter tractability results for the problem to find a maximum-size submatrix of a 0/1-matrix such that the submatrix has the C1P. Moreover, we achieve a problem kernelization based on simple data reduction rules and provide several search tree algorithms. Finally, we derive inapproximability results.

Abstract. Let G = (V, E) be a connected multigraph, whose edges are associated with labels specified by an integer-valued function L: E → N. In addition, each label ℓ ∈ N to which at least one edge is mapped has a non-negative cost c(ℓ). The minimum label spanning tree problem (MinLST) asks to find a spanning tree in G that minimizes the overall cost of the labels used by its edges. Equivalently, we aim at finding a minimum cost subset of labels I ⊆ N such that the edge set {e ∈ E: L(e) ∈ I} forms a connected subgraph spanning all vertices. Similarly, in the minimum label s-t path problem (MinLP) the goal is to identify an s-t path minimizing the combined cost of its labels, where s and t are provided as part of the input. The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP. As a secondary objective, we make a concentrated effort to relate the algorithmic methods utilized in approximating these problems to a number of well-known techniques, originally studied in the context of integer covering. 1

A d-dimensional simplex is a collection of d+1 sets with empty intersection, every d of which have nonempty intersection. A k-uniform d-cluster is a collection of d + 1 sets of size k with empty intersection and union of size at most 2k. We prove the following result which simultaneously addresses an old conjecture of Chvátal [7] and a recent conjecture of the second author [28]. For d ≥ 2 and ζ> 0 there is a number T such that the following holds for sufficiently large n. Let G be a k-uniform set system on [n] = {1, · · · , n} with ζn < k < n/2 − T, and suppose either that G contains no d-dimensional simplex or that G contains no d-cluster. Then |G | ≤ � � n−1 k−1 with equality only for the family of all k-sets containing a specific element. In the non-uniform setting we obtain the following exact result that generalises an old question of Erdős and a result of Milner [11], who proved the case d = 2. Suppose d ≥ 2 and G is a set system on [n] that does not contain a d-dimensional simplex, with n sufficiently large. Then |G | ≤ 2n−1 + �d−1 � � n−1 i=0 i, with equality only for the family consisting of all sets that either contain some specific element or have size at most d − 1. Each of these results is proved via the corresponding stability result, which gives structural information on any G whose size is close to maximum. These in turn rely on a stability result that we obtain for intersecting families, which supersedes a result of Friedgut [17] that was proved using spectral techniques, and is based on a purely combinatorial result of Frankl.

We consider one-round games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are ‘unique ’ constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only algorithm known previously was for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things, our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false. Our proof is based on a novel ‘quantum rounding technique’, showing how to take a solution to an SDP and transform it to a strategy for entangled provers. 1

Abstract. We study the complexity of combinatorial problems that consist of competing infeasibility and optimization components. In particular, we investigate the complexity of the connection subgraph problem, which occurs, e.g., in resource environment economics and social networks. We present results on its worst-case hardness and approximability. We then provide a typical-case analysis by means of a detailed computational study. First, we identify an easy-hard-easy pattern, coinciding with the feasibility phase transition of the problem. Second, our experimental results reveal an interesting interplay between feasibility and optimization. They surprisingly show that proving optimality of the solution of the feasible instances can be substantially easier than proving infeasibility of the infeasible instances in a computationally hard region of the problem space. We also observe an intriguing easy-hard-easy profile for the optimization component itself. 1

The class of graphs where the size of a minimum vertex cover equals that of a maximum matching is known as König-Egerváry graphs. König-Egerváry graphs have been studied extensively from a graph theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding maximum König-Egerváry subgraphs of a given graph. More specifically, we look at the problem of finding a minimum number of vertices or edges to delete to make the resulting graph König-Egerváry. We show that both these versions are NP-complete and study their complexity from the points of view of approximation and parameterized complexity. En route, we point out an interesting connection between the vertex deletion version and the Above Guarantee Vertex Cover problem where one is interested in the parameterized complexity of the Vertex Cover problem when parameterized by the ‘additional number of vertices ’ needed beyond the matching size. This connection is of independent interest and could be useful in establishing the parameterized complexity of Above Guarantee Vertex Cover problem.