We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small. VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a ge...

We prove new results for approximating Graphic TSP. Specifically, we provide a polynomial-time 9 7 -approximation algorithm for cubic bipartite graphs and a ( 9 7 + 1 21(k−2) )-approximation algorithm for k-regular bipartite graphs, both of which are improved approximation factors compared to previous results. Our approach involves finding a cycle cover with relatively few cycles, which we are ...

In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for graphs with small intersection number and random intersection graphs (a model in which each one of m labels is chosen independently with probability p by each ...

We prove new results for approximating Graph-TSP. Speci cally, we provide a polynomial-time 9 7 approximation algorithm for cubic bipartite graphs and a ( 9 7 + 1 21(k−2) )-approximation algorithm for k-regular bipartite graphs, both of which are improved approximation factors compared to previous results. Our approach involves nding a cycle cover with relatively few cycles, which we are able t...

An edge dominating set of a graph G = (V,E) is a subset M ⊆ E of edges such that each edge in E \ M is incident to at least one edge in M . In this paper, we consider the parameterized edge dominating set problem which asks us to test whether a given graph has an edge dominating set with size bounded from above by an integer k or not, and we design an O(2.2351)-time and polynomial-space algorit...

We address the problem of covering points with orthogonal polygons. Specifically, given a set of n grid-points in the plane each designated in advance with either a horizontal or vertical reading, we investigate the existence of an orthogonal polygon covering these n points in such a way that each edge of the polygon covers exactly one point and each point is covered by exactly one edge with th...

An edge cover of a graph is a set of edges such that every vertex has at least an adjacent edge in it. We design a very simple deterministic fully polynomial-time approximation scheme (FPTAS) for counting the number of edge covers for any graph. Previously, approximation algorithm is only known for 3 regular graphs and it is randomized. Our main technique is correlation decay, which is a powerf...

A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs some problems that are NP-complete on general graphs, such as the maximum...

A cycle cover of a directed graph is a collection of node disjoint cycles such that every node is part of exactly one cycle. A k-cycle cover is a cycle cover in which every cycle has length at least k. While deciding whether a directed graph has a 2-cycle cover is solvable in polynomial time, deciding whether it has a 3-cycle cover is already NP-complete. Given a directed graph with nonnegative...

Generally, a graph G, an independent set is a subset S of vertices in G such that no two vertices in S are adjacent (connected by an edge) and a vertex cover is a subset S of vertices such that each edge of G has at least one of its endpoints in S. Again, the minimum vertex cover problem is to find a vertex cover with the smallest number of vertices. Consider a k-partite graph G = (V, E) with v...