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

A Super-Fast Distributed Algorithm for Bipartite Metric Facility Location

The \textit{facility location} problem consists of a set of \textit{facilities} $\mathcal{F}$, a set of \textit{clients} $\mathcal{C}$, an \textit{opening cost} $f_i$ associated with each facility $x_i$, and a \textit{connection cost} $D(x_i,y_j)$ between each facility $x_i$ and client $y_j$. The goal is to find a subset of facilities to \textit{open}, and to connect each client to an open facility, so as to minimize the total facility opening costs plus connection costs. This paper presents the first expected-sub-logarithmic-round distributed O(1)-approximation algorithm in the $\mathcal{CONGEST}$ model for the \textit{metric} facility location problem on the complete bipartite network with parts $\mathcal{F}$ and $\mathcal{C}$. Our algorithm has an expected running time of $O((\log \log n)^3)$ rounds, where $n = |\mathcal{F}| + |\mathcal{C}|$. This result can be viewed as a continuation of our recent work (ICALP 2012) in which we presented the first sub-logarithmic-round distributed O(1)-approximation algorithm for metric facility location on a \textit{clique} network. The bipartite setting presents several new challenges not present in the problem on a clique network. We present two new techniques to overcome these challenges. (i) In order to deal with the problem of not being able to choose appropriate probabilities (due to lack of adequate knowledge), we design an algorithm that performs a random walk over a probability space and analyze the progress our algorithm makes as the random walk proceeds. (ii) In order to deal with a problem of quickly disseminating a collection of messages, possibly containing many duplicates, over the bipartite network, we design a probabilistic hashing scheme that delivers all of the messages in expected-$O(\log \log n)$ rounds.Comment: 22 pages. This is the full version of a paper that appeared in DISC 201