Covering Problems via Structural Approaches

The minimum set cover problem is, without question, among the most ubiquitous and well-studied problems in computer science. Its theoretical hardness has been fully characterized--logarithmic approximability has been established, and no sublogarithmic approximation exists unless P=NP. However, the gap between real-world instances and the theoretical worst case is often immense--many covering problems of practical relevance admit much better approximations, or even solvability in polynomial time. Simple combinatorial or geometric structure can often be exploited to obtain improved algorithms on a problem-by-problem basis, but there is no general method of determining the extent to which this is possible. In this thesis, we aim to shed light on the relationship between the structure and the hardness of covering problems. We discuss several measures of structural complexity of set cover instances and prove new algorithmic and hardness results linking the approximability of a set cover problem to its underlying structure. In particular, we provide: - An APX-hardness proof for a wide family of problems that encode a simple covering problem known as Special-3SC. - A class of polynomial dynamic programming algorithms for a group of weighted geometric set cover problems having simple structure. - A simplified quasi-uniform sampling algorithm that yields improved approximations for weighted covering problems having low cell complexity or geometric union complexity. - Applications of the above to various capacitated covering problems via linear programming strengthening and rounding. In total, we obtain new results for dozens of covering problems exhibiting geometric or combinatorial structure. We tabulate these problems and classify them according to their approximability

New Approximability Results for the Robust k-Median Problem

We consider a robust variant of the classical $k$-median problem, introduced by Anthony et al. \cite{AnthonyGGN10}. In the \emph{Robust $k$-Median problem}, we are given an $n$-vertex metric space $(V,d)$ and $m$ client sets $\set{S_i \subseteq V}_{i=1}^m$. The objective is to open a set $F \subseteq V$ of $k$ facilities such that the worst case connection cost over all client sets is minimized; in other words, minimize $\max_{i} \sum_{v \in S_i} d(F,v)$. Anthony et al.\ showed an $O(\log m)$ approximation algorithm for any metric and APX-hardness even in the case of uniform metric. In this paper, we show that their algorithm is nearly tight by providing $\Omega(\log m/ \log \log m)$ approximation hardness, unless ${\sf NP} \subseteq \bigcap_{\delta >0} {\sf DTIME}(2^{n^{\delta}})$. This hardness result holds even for uniform and line metrics. To our knowledge, this is one of the rare cases in which a problem on a line metric is hard to approximate to within logarithmic factor. We complement the hardness result by an experimental evaluation of different heuristics that shows that very simple heuristics achieve good approximations for realistic classes of instances.Comment: 19 page

Stabbing line segments with disks: complexity and approximation algorithms

Computational complexity and approximation algorithms are reported for a problem of stabbing a set of straight line segments with the least cardinality set of disks of fixed radii $r>0$ where the set of segments forms a straight line drawing $G=(V,E)$ of a planar graph without edge crossings. Close geometric problems arise in network security applications. We give strong NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel graphs and other subgraphs (which are often used in network design) for $r\in [d_{\min},\eta d_{\max}]$ and some constant $\eta$ where $d_{\max}$ and $d_{\min}$ are Euclidean lengths of the longest and shortest graph edges respectively. Fast $O(|E|\log|E|)$-time $O(1)$-approximation algorithm is proposed within the class of straight line drawings of planar graphs for which the inequality $r\geq \eta d_{\max}$ holds uniformly for some constant $\eta>0,$ i.e. when lengths of edges of $G$ are uniformly bounded from above by some linear function of $r.$Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International Conference on Analysis of Images, Social Networks and Texts (AIST-2017

Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems