The minimum clique partition (MCP) problem is that of partitioning the vertex set of a given graph into a minimum number of cliques. Given n points in the plane, the corresponding unit disk graph (UDG) has these points as vertices, and edges connecting points at distance at most 1. MCP in unit disk graphs is known to be NP-hard and several constant factor approximations are known, including a r...

The problem of covering edges and vertices in a graph or in a hypergraph was motivated by a problem arising in the context of component assembly problem The problem is given a graph and a clique size k nd the minimum number of k cliques such that all edges and vertices of the graph are covered by included in the cliques This paper provides a collection of approximation algorithms for various cl...

In (k, r)-Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically: • For any r ≥ 1, we show an algorithm that solves ...

We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a polynomial time approximation scheme (PTAS) for this problem on UDG. In fact, we present a robust algorithm that given a graph G (not nece...

We develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes weighted satisfiability, dominating set, hitting set, set cover, clique, and independent set. For example, although a trivial enumeration can easily test in time O(nk) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely coll...

A graph G is an interval graph if there is a one-one correspondence between its vertices and a family I of intervals, such that two vertices in G are adjacent if and only if their corresponding intervals overlap. In this context, the family I of intervals is referred to as an interval model of G. Recently, a powerful architecture called the reconngurable mesh has been proposed: in essence, a re...

We study the problem of counting the number of isomorphic copies of a given template graph, say H , in the input base graph, say G. In general, this is (provably) very hard to solve efficiently. So, a lot of work has gone into designing efficient approximation schemes, especially, when H is a perfect matching. In this work, we present schemes to count k-cliques and k-clique covers. Our embeddin...

Censor-Hillel et al. [PODC’15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in the past few years. This paper develops further algebraic techniques for designing algorithms in this model. We present deterministic and randomized algorit...

In this article we survey algorithmic lower bound results that have been obtained in the field of exact exponential time algorithms and parameterized complexity under certain assumptions on the running time of algorithms solving CNF-Sat, namely Exponential time hypothesis (ETH) and Strong Exponential time hypothesis (SETH).

In this paper we analyze three well-known preprocessors for Max-SAT. The first preprocessor is based on the so-called variable saturation. The second preprocessor is based on the resolution mechanism incorporated in modern branch and bound solvers. The third preprocessor is specific for the Maximum Clique problem and other problems with similar encoding in WCNF such as minimum vertex covering a...