A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

Given a $k$-uniform hyper-graph, the E$k$-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that E$k$-Vertex-Cover is NP-hard to approximate within factor $(k-1-\epsilon)$ for any $k \geq 3$ and any $\epsilon>0$. The result is essentially tight as this problem can be easily approximated within factor $k$. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of $s$-wise $t$-intersecting families of subsets

Lagrangian Relaxation and Partial Cover

Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover.Comment: 20 pages, extended abstract appeared in STACS 200

Integrality gaps of semidefinite programs for Vertex Cover and relations to ℓ1\ell_1 embeddability of Negative Type metrics

We study various SDP formulations for {\sc Vertex Cover} by adding different constraints to the standard formulation. We show that {\sc Vertex Cover} cannot be approximated better than $2-o(1)$ even when we add the so called pentagonal inequality constraints to the standard SDP formulation, en route answering an open question of Karakostas~\cite{Karakostas}. We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution is an $\ell_1$ metric, we get an exact relaxation (integrality gap is 1), and on the other hand if the solution is arbitrarily close to being $\ell_1$ embeddable, the integrality gap may be as big as $2-o(1)$. Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar \cite{Char02} to provide a family of simple examples for negative type metrics that cannot be embedded into $\ell_1$ with distortion better than 8/7-\eps. To this end we prove a new isoperimetric inequality for the hypercube.Comment: A more complete version. Changed order of results. A complete proof of (current) Theorem

On k-Column Sparse Packing Programs

We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an (ek+o(k))-approximation algorithm for k-column sparse PIPs, improving on recent results of $k^2\cdot 2^k$ and $O(k^2)$. We also show that the integrality gap of our linear programming relaxation is at least 2k-1; it is known that k-column sparse PIPs are $\Omega(k/ \log k)$-hard to approximate. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail

New Tools and Connections for Exponential-Time Approximation

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of 1. r for maximum independent set in O∗(exp(O~(n/rlog2r+rlog2r))) time, 2. r for chromatic number in O∗(exp(O~(n/rlogr+rlog2r))) time, 3. (2−1/r) for minimum vertex cover in O∗(exp(n/rΩ(r))) time, and 4. (k−1/r) for minimum k-hypergraph vertex cover in O∗(exp(n/(kr)Ω(kr))) time. (Throughout, O~ and O∗ omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O∗(2n/r) (Bourgeois et al. i

On the Approximability and Hardness of Minimum Topic Connected Overlay and Its Special Instances

In the context of designing a scalable overlay network to support decentralized topic-based pub/sub communication, the Minimum Topic-Connected Overlay problem (Min-TCO in short) has been investigated: Given a set of t topics and a collection of n users together with the lists of topics they are interested in, the aim is to connect these users to a network by a minimum number of edges such that every graph induced by users interested in a common topic is connected. It is known that Min-TCO is NP-hard and approximable within O(log t) in polynomial time. In this paper, we further investigate the problem and some of its special instances. We give various hardness results for instances where the number of topics in which an user is interested in is bounded by a constant, and also for the instances where the number of users interested in a common topic is constant. For the latter case, we present a first constant approximation algorithm. We also present some polynomial-time algorithms for very restricted instances of Min-TCO.Comment: 20 page