Approximation Algorithms for Union and Intersection Covering Problems

TL;DR

This paper introduces and analyzes multi-layer covering problems, where multiple instances of classical covering problems are combined with union or intersection constraints, providing new approximation algorithms and hardness results.

Contribution

It formalizes the union and intersection multi-layer covering problems and offers the first approximation algorithms and hardness results for these new problem families.

Findings

Approximation algorithms are developed for union and intersection covering problems.

Hardness results establish computational limits for these problems.

Applications include network design and resource connection problems.

Abstract

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.