Station Location - Complexity and Approximation

We consider a geometric set covering problem. In its original form it consists of adding stations to an existing geometric transportation network so that each of a given set of settlements is not too far from a station. The problem is known to be NP-hard in general. However, special cases with certain properties have been shown to be efficiently solvable in theory and in practice, especially if the covering matrix has (almost) consecutive ones property. In this paper we are narrowing the gap between intractable and efficiently solvable cases of the problem. We also present an approximation algorithm for cases with almost consecutive ones property

Multiple sink positioning in sensor networks

We study the problem of positioning multiple sinks, or data collection stops for a mobile sink, in a sensor network field. Given a sensor network represented by a unit disc graph G = ( V, E ), we say a set of points U (sink node locations) is an h -hop covering set for G if every node in G is at most h hops away from some point in U . Placing sink nodes at the points of a covering set guarantees that every sensor node has a short path to some sink node. This can increase network lifetime, reduce the occurrence of errors, and reduce latency. We also study variations of the problem where the sink locations are restricted to be at points of a regular lattice (lattice-based covering set), or at network nodes (graph-based covering set). We give the first polynomial time approximation scheme ( PTAS ) for the h -hop covering set problem, the h -hop lattice-based covering set problem, and the h-hop graph-based covering set problem. We give a new PTAS for the lattice-based disc cover problem, based on a new approach deriving from recent results on dominating sets in unit disc graphs. We show that this gives a (3 + f)-approximation algorithm for the disc cover problem, and gives the first distributed algorithm for this problem. We give a (5 + f)-approximation algorithm for the h -hop covering set problem in unit disc graphs, that does not require a geometric representation of the graph. Finally, we give a (3+f)-approximation algorithm for the h -hop covering set problem for unit disc graphs that runs in time quadratic in the number of nodes in the graph, for any constant f and h . In addition to showing how well a lattice-based approach for a disc cover problem approximates the optimal solution, we prove a geometric theorem that gives an exact relationship between the side of a triangular lattice and the number of lattice discs that are necessary and sufficient to cover an arbitrary disc on the plane

Approximation Algorithms for Geometric Covering Problems for Disks and Squares

Geometric covering is a well-studied topic in computational geometry. We study three covering problems: Disjoint Unit-Disk Cover, Depth-(≤ K) Packing and Red-Blue Unit-Square Cover. In the Disjoint Unit-Disk Cover problem, we are given a point set and want to cover the maximum number of points using disjoint unit disks. We prove that the problem is NP-complete and give a polynomial-time approximation scheme (PTAS) for it. In Depth-(≤ K) Packing for Arbitrary-Size Disks/Squares, we are given a set of arbitrary-size disks/squares, and want to find a subset with depth at most K and maximizing the total area. We prove a depth reduction theorem and present a PTAS. In Red-Blue Unit-Square Cover, we are given a red point set, a blue point set and a set of unit squares, and want to find a subset of unit squares to cover all the blue points and the minimum number of red points. We prove that the problem is NP-hard, and give a PTAS for it. A "mod-one" trick we introduce can be applied to several other covering problems on unit squares

Minimum-Cost Coverage of Point Sets by Disks

We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (t_j) and radii (r_j) that cover a given set of demand points Y in the plane at the smallest possible cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha is the cost of transmission to radius r. Special cases arise for alpha=1 (sum of radii) and alpha=2 (total area); power consumption models in wireless network design often use an exponent alpha>2. Different scenarios arise according to possible restrictions on the transmission centers t_j, which may be constrained to belong to a given discrete set or to lie on a line, etc. We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t_j on a given line in order to cover demand points Y in the plane; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NP-hardness for a discrete set of transmission points in the plane and any fixed alpha>1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on Computational Geometry 200

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

Algorithms for covering multiple submodular constraints and applications

We consider the problem of covering multiple submodular constraints. Given a finite ground set N, a weight function $w: N \rightarrow \mathbb {R}_+$, r monotone submodular functions $f_1,f_2,\ldots ,f_r$ over N and requirements $k_1,k_2,\ldots ,k_r$ the goal is to find a minimum weight subset $S \subseteq N$ such that $f_i(S) \ge k_i$ for $1 \le i \le r$. We refer to this problem as Multi-Submod-Cover and it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR. arxiv:abs1808.03260Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with $r=1$ Multi-Submod-Cover generalizes the well-known Submodular Set Cover problem (Submod-SC), and it can also be easily reduced to Submod-SC. A simple greedy algorithm gives an $O(\log (kr))$ approximation where $k = \sum _i k_i$ and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for Multi-Submod-Cover that covers each constraint to within a factor of $(1-1/e-\varepsilon )$ while incurring an approximation of $O(\frac{1}{\epsilon }\log r)$ in the cost. Second, we consider the special case when each $f_i$ is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover (Partial-SC), covering integer programs (CIPs) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems.publishedVersio