Hypergraph Representation via Axis-Aligned Point-Subspace Cover

We propose a new representation of $k$-partite, $k$-uniform hypergraphs (i.e. a hypergraph with a partition of vertices into $k$ parts such that each hyperedge contains exactly one vertex of each type; we call them $k$-hypergraphs for short) by a finite set $P$ of points in $\mathbb{R}^d$ and a parameter $\ell\leq d-1$. Each point in $P$ is covered by $k={d\choose\ell}$ many axis-aligned affine $\ell$-dimensional subspaces of $\mathbb{R}^d$, which we call $\ell$-subspaces for brevity. We interpret each point in $P$ as a hyperedge that contains each of the covering $\ell$-subspaces as a vertex. The class of $(d,\ell)$-hypergraphs is the class of $k$-hypergraphs that can be represented in this way, where $k={d\choose\ell}$. The resulting classes of hypergraphs are fairly rich: Every $k$-hypergraph is a $(k,k-1)$-hypergraph. On the other hand, $(d,\ell)$-hypergraphs form a proper subclass of the class of all $d\choose\ell$-hypergraphs for $\ell<d-1$. In this paper we give a natural structural characterization of $(d,\ell)$-hypergraphs based on vertex cuts. This characterization leads to a polynomial-time recognition algorithm that decides for a given $d\choose\ell$-hypergraph whether or not it is a $(d,\ell)$-hypergraph and that computes a representation if existing. We assume that the dimension $d$ is constant and that the partitioning of the vertex set is prescribed

On the Inapproximability of Vertex Cover on k-Partite k-Uniform Hypergraphs

Computing a minimum vertex cover in graphs and hypergraphs is a well-studied optimizaton problem. While intractable in general, it is well known that on bipartite graphs, vertex cover is polynomial time solvable. In this work, we study the natural extension of bipartite vertex cover to hypergraphs, namely finding a small vertex cover in k-uniform k-partite hypergraphs, when the k-partition is given as input. For this problem Lovász [16] gave a k 2 factor LP rounding based approximation, and a matching ( k 2 − o(1)) integrality gap instance was constructed by Aharoni et al. [1]. We prove the following results, which are the first strong hardness results for this problem (here ε&gt; 0 is an arbitrary constant): – NP-hardness of approximating within a factor of ( k 4 − ε) , and – Unique Games-hardness of approximating within a factor of ( k 2 − ε), showing optimality of Lovász’s algorithm under the Unique Games conjecture. The NP-hardness result is based on a reduction from minimum vertex cover in r-uniform hypergraphs for which NP-hardness of approximating within r−1−ε was shown by Dinur et al. [5]. The Unique Games-hardness result is obtained by applying the recent results of Kumar et al. [15], with a slight modification, to the LP integrality gap due to Aharoni et al. [1]. The modification is to ensure that the reduction preserves the desired structural properties of the hypergraph

On the Inapproximability of Vertex Cover on k-Partite k-Uniform Hypergraphs

Computing a minimum vertex cover in graphs and hypergraphs is a well-studied optimizaton problem. While intractable in general, it is well known that on bipartite graphs, vertex cover is polynomial time solvable. In this work, we study the natural extension of bipartite vertex cover to hypergraphs, namely finding a small vertex cover in k-uniform k-partite hypergraphs, when the k-partition is given as input. For this problem Lovász [16] gave a k2 factor LP rounding based approximation, and a matching (k2−o(1)) integrality gap instance was constructed by Aharoni et al. [1]. We prove the following results, which are the first strong hardness results for this problem (here ε> 0 is an arbitrary constant): NP-hardness of approximating within a factor of (k4−ε), and Unique Games-hardness of approximating within a factor of (k2−ε), showing optimality of Lovász’s algorithm under the Unique Games conjecture. The NP-hardness result is based on a reduction from minimum vertex cover in r-uniform hypergraphs for which NP-hardness of approximating within r–1–ε was shown by Dinur et al.[5]. The Unique Games-hardness result is obtained by applying the recent results of Kumar et al. [15], with a slight modification, to the LP integrality gap due to Aharoni et al. [1]. The modification is to ensure that the reduction preserves the desired structural properties of the hypergraph.</p

Point cover problem in 3D

We examine the problem of covering points with minimum number of axis-parallel lines in three dimensional space which is an NP-complete problem. We introduce Lagrangian based algorithms to approximate the point cover problem. We study the Lift-and-Project relaxation of the standard IP to obtain lower bounds. This method is used to strengthen the integrality gap of a problem. Our experimental results show that the Lagrangian relaxation method gives very good lower bounds at reasonable computational cost. We present a hybrid method where the Lift-and-Project LP is solved using the Subgradient Optimisation technique. We propose an approximation algorithm which iteratively uses the Lagrangian relaxation procedure. We also study a Branch-and-Bound method which gives an optimal solution. We use a drop-in accelerator while conducting the simulations on large instances

On Approximability of Bounded Degree Instances of Selected Optimization Problems

In order to cope with the approximation hardness of an underlying optimization problem, it is advantageous to consider specific families of instances with properties that can be exploited to obtain efficient approximation algorithms for the restricted version of the problem with improved performance guarantees. In this thesis, we investigate the approximation complexity of selected NP-hard optimization problems restricted to instances with bounded degree, occurrence or weight parameter. Specifically, we consider the family of dense instances, where typically the average degree is bounded from below by some function of the size of the instance. Complementarily, we examine the family of sparse instances, in which the average degree is bounded from above by some fixed constant. We focus on developing new methods for proving explicit approximation hardness results for general as well as for restricted instances. The fist part of the thesis contributes to the systematic investigation of the VERTEX COVER problem in k-hypergraphs and k-partite k-hypergraphs with density and regularity constraints. We design efficient approximation algorithms for the problems with improved performance guarantees as compared to the general case. On the other hand, we prove the optimality of our approximation upper bounds under the Unique Games Conjecture or a variant. In the second part of the thesis, we study mainly the approximation hardness of restricted instances of selected global optimization problems. We establish improved or in some cases the first inapproximability thresholds for the problems considered in this thesis such as the METRIC DIMENSION problem restricted to graphs with maximum degree 3 and the (1,2)-STEINER TREE problem. We introduce a new reductions method for proving explicit approximation lower bounds for problems that are related to the TRAVELING SALESPERSON (TSP) problem. In particular, we prove the best up to now inapproximability thresholds for the general METRIC TSP problem, the ASYMMETRIC TSP problem, the SHORTEST SUPERSTRING problem, the MAXIMUM TSP problem and TSP problems with bounded metrics