Approximation Algorithms for Union and Intersection Covering Problems

In a classical covering problem, we are given a set of requests that we need to satisfy (fully or partially), by buying a subset of items at minimum cost. For example, in the k-MST problem we want to find the cheapest tree spanning at least k nodes of an edge-weighted graph. Here nodes and edges represent requests and items, respectively. In this paper, we initiate the study of a new family of multi-layer covering problems. Each such problem consists of a collection of h distinct instances of a standard covering problem (layers), with the constraint that all layers share the same set of requests. We identify two main subfamilies of these problems: - in a union multi-layer problem, a request is satisfied if it is satisfied in at least one layer; - in an intersection multi-layer problem, a request is satisfied if it is satisfied in all layers. To see some natural applications, consider both generalizations of k-MST. Union k-MST can model a problem where we are asked to connect a set of users to at least one of two communication networks, e.g., a wireless and a wired network. On the other hand, intersection k-MST can formalize the problem of connecting a subset of users to both electricity and water. We present a number of hardness and approximation results for union and intersection versions of several standard optimization problems: MST, Steiner tree, set cover, facility location, TSP, and their partial covering variants

LP-Based Approximation Algorithms for Facility Location in Buy-at-Bulk Network Design

Abstract We study problems that integrate buy-at-bulk network design into the classical (connected) facility location problem. In such problems, we need to open facilities, build a routing network, and route every client demand to an open facility. Furthermore, capacities of the edges can be purchased in discrete units from K different cable types with costs that satisfy economies of scale. We extend the linear programming frame-work of Talwar [IPCO 2002] for the single-source buy-at-bulk problem to these variants and prove integrality gap upper bounds for both facility location and connected facility location buy-at-bulk problems. For the unconnected variant we prove an integrality gap bound of O(K), and for the connected version, we get an improved bound of O(1).

Approximation Algorithms for Connected Maximum Cut and Related Problems

An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V, E) and the goal is to find a subset of vertices S $\subseteq$ V that maximizes the number of edges in the cut \delta(S) such that the induced graph G[S] is connected. We present the first non-trivial \Omega(1/log n) approximation algorithm for the connected maximum cut problem in general graphs using novel techniques. We then extend our algorithm to an edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark contrast to the classical max-cut problem, we show that the connected maximum cut problem remains NP-hard even on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the connected maximum cut problem on planar graphs and more generally on graphs with bounded genus.Comment: 17 pages, Conference version to appear in ESA 201

Designing Overlapping Networks for Publish-Subscribe Systems

From the publish-subscribe systems of the early days of the Internet to the recent emergence of Web 3.0 and IoT (Internet of Things), new problems arise in the design of networks centered at producers and consumers of constantly evolving information. In a typical problem, each terminal is a source or sink of information and builds a physical network in the form of a tree or an overlay network in the form of a star rooted at itself. Every pair of pub-sub terminals that need to be coordinated (e.g. the source and sink of an important piece of control information) define an edge in a bipartite demand graph; the solution must ensure that the corresponding networks rooted at the endpoints of each demand edge overlap at some node. This simple overlap constraint, and the requirement that each network is a tree or a star, leads to a variety of new questions on the design of overlapping networks. In this paper, for the general demand case of the problem, we show that a natural LP formulation has a non-constant integrality gap; on the positive side, we present a logarithmic approximation for the general demand case. When the demand graph is complete, however, we design approximation algorithms with small constant performance ratios, irrespective of whether the pub networks and sub networks are required to be trees or stars

A constant-factor approximation algorithm for reconciliation k-median

In the reconciliation k-median problem we ask to cluster a set of data points by picking k cluster centers so as to minimize the sum of distances of the data points to their cluster centers plus the sum of pairwise distances between the centers. The problem, which is a variant of classic k-median, aims to find a set of cluster centers that are not too far from each other, and it has applications, or example, when selecting a committee to deliberate on a controversial topic. This problem was introduced recently (Ordozgoiti et al., 2019), and it was shown that a local-search-based algorithm is always within a factor O(k) of an optimum solution and performs well in practice. In this paper, we demonstrate a close connection of reconciliation k-median to a variant of the k-facility location problem, in which each potential cluster center has an individual opening cost and we aim at minimizing the sum of client-center distances and the opening costs. This connection enables us to provide a new algorithm for reconciliation k-median that yields a constant-factor approximation (independent of k). We also provide a sparsification scheme that reduces the number of potential cluster centers to O(k) in order to substantially speed up approximation algorithms. We empirically compare our new algorithms with the previous local-search approach, showing improved performance and stability. In addition, we show how our sparsification approach helps to reduce computation time without significantly compromising the solution quality