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Approximability of the Unique Coverage Problem

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Approximability of the Unique Coverage Problem Combination Can Be Hard: Approximability of the Unique Coverage Problem Erik D. Demaine∗ Uriel Feige† MohammadTaghi Hajiaghayi∗ Mohammad R. Salavatipour‡ Abstract S = {S1,...,Sm} of subsets of U, and a cost We prove semi-logarithmic inapproximability for a maximization ci of each subset Si; and given a budget B. Find 0 problem called unique coverage: given a collection of sets, find a a subcollection S ⊆ S, whose total cost is at subcollection that maximizes the number of elements covered ex- most the budget B, to maximize the total profit of actly once. Specifically, we prove O(1/ logσ(ε) n) inapproxima- elements that are uniquely covered, i.e., appear in ε 0 bility assuming that NP 6⊆ BPTIME(2n ) for some ε > 0. We exactly one set of S . 1/3−ε also prove O(1/ log n) inapproximability, for any ε > 0, as- Motivation. Logarithmic inapproximability for mini- suming that refuting random instances of 3SAT is hard on average; mization problems is by now commonplace, starting in 1993 and prove O(1/ log n) inapproximability under a plausible hypoth- with a result for the celebrated set cover problem [36], which esis concerning the hardness of another problem, balanced bipar- has since been improved to the optimal constant [16] and to tite independent set. We establish matching upper bounds up to assume just P 6= NP [41], and has been used to prove other exponents, even for a more general (budgeted) setting, giving an tight (not necessarily logarithmic) inapproximability results Ω(1/ log n)-approximation algorithm as well as an Ω(1/ log B)- for a variety of minimization problems, e.g., [29, 22, 11]. approximation algorithm when every set has at most B elements. In contrast, for maximization problems, log n inapproxima- We also show that our inapproximability results extend to envy-free bility seems more difficult, and relatively few results are pricing, an important problem in computational economics. We de- known. The only two such results of which we know are scribe how the (budgeted) unique coverage problem, motivated by (1 + ε)/ ln n inapproximability for domatic number unless real-world applications, has close connections to other theoretical NP ⊆ DTIME(nO(log log n)) [18], and 1/ log1/3−ε n inap- problems including max cut, maximum coverage, and radio broad- proximability for the maximum edge-disjoint paths and cy- casting. cles problems unless NP ⊆ ZPTIME(npolylog n) [5, 42]. Although these problems are interesting, they are rather spe- 1 Introduction cific, and we lack a central maximization problem analogous In this paper we consider the approximability of the follow- to set cover to serve as a basis for further reduction to many ing natural maximization analog of set cover: other maximization problems. The unique coverage problem defined above is a natural Unique Coverage Problem. Given a universe maximization version of set cover which was brought to U = {e , . , e } of elements, and given a col- 1 n our attention from its applications in wireless networks. In lection S = {S ,...,S } of subsets of U. Find a 1 m one (simplified) application, we have a certain budget to subcollection S0 ⊆ S to maximize the number of build/place some transmitters at a subset of some specified elements that are uniquely covered, i.e., appear in set of possible locations. Our goal is to maximize the clients exactly one set of S0. that are “covered” by (i.e., are within the range of) exactly We also consider a generalized form of this problem that is one transmitter; these are the clients that receive signal useful for several applications (detailed in Section 2): without interference; see Section 2.1 for details. Another closely related application studied much earlier is the radio Budgeted Unique Coverage Problem. Given a broadcast problem, in which a message (starting from one universe U = {e1, . , en} of elements, and a node of the network) is to be sent to all the nodes in the profit pi for each element ei; given a collection network in rounds. In each round, some of the nodes that have already received the message send it to their neighbors, ∗Computer Science and Artificial Intelligence Laboratory, Mas- and a node receives a message only if it receives it from sachusetts Institute of Technology. {edemaine,hajiagha}@mit.edu exactly one of its neighbors. The goal is to find the minimum †Microsoft Research; and Department of Computer Science and Applied Mathematics, Weizmann Institute. [email protected] number of rounds to broadcast the message to all the nodes; ‡Department of Computing Science, University of Alberta. mreza@cs. see Section 2.5 for details. Therefore, every single round of ualberta.ca. Supported by NSERC and a faculty startup grant a radio broadcast can be seen as a unique coverage problem. These applications along with others are studied in more (as indeed is the case with unique coverage) these problems detail in Section 2. naturally decompose into Θ(log n) subproblems, where at Known results. To the best of our knowledge, there least an Ω(1/ log n) fraction of the optimum’s value comes is no explicit study in the literature of the unique coverage from one of these subproblems. In isolation, each subprob- problem and its budgeted version. However, the closely re- lem can be approximated up to a constant factor, leading to lated radio broadcast problem has been studied extensively an Ω(1/ log n)-approximation algorithm for the whole prob- in the past, and implicitly include an Ω(1/ log n) approxi- lem. It may appear that this isolation approach is too na¨ıve mation algorithm for the basic (unbudgeted) unique coverage to give the best possible approximation, and that by a clever problem; see Section 2.5 for details. combination of the subproblems, it should be possible to Concurrently and independently of our work, Gu- get an Ω(1)-approximation algorithm. Our hardness results ruswami and Trevisan [25] study the so called 1-in-k SAT show to the contrary that such intelligent combination can be problem, which includes the unique coverage problem (but hard, in the sense that the na¨ıve isolation approach cannot be not its budgeted version) as a special case. In particular, they substantially improved, and suggest how one might obtain show that there is an approximation algorithm that achieves better hardness results for these problems. an approximation ratio of 1/e on satisfiable instances (in which all items can be covered by mutually disjoint sets). 2 Applications and Related Problems Our results. On the positive side, we give an 2.1 Wireless Networks. Our original motivation for the Ω(1/ log n)-approximation for the budgeted unique cover- budgeted unique coverage problem is a real-world applica- age problem. We also show that, if each set has a bound tion arising in wireless networks.1 We are given a map of the B on the ratio between the maximum profit of a set and the densities of mobile clients throughout a service region (e.g., minimum profit of an element, then budgeted unique cov- the plane with obstacles). We are also given a collection of erage has an Ω(1/ log B)-approximation. Section 4 proves candidate locations for wireless base stations, each with a these results. specified building cost and a specified coverage region (typi- The main focus of this paper is proving the following cally a cone or a disk, possibly obstructed by obstacles). This inapproximability results. We show that it is hard to ap- collection may include multiple options for base stations at proximate the unique coverage problem within a factor of σ the same location, e.g., different powers and different ori- Ω(1/ log n), for some constant σ depending on ε, assum- entations of antennae. The goal is to choose a set of base nε ing that NP 6⊆ BPTIME(2 ) for some ε > 0. This inap- stations and options to build, subject to a budget on the to- 1/3−ε proximability can be strengthened to Ω(1/ log n) (for tal building cost, in order to maximize the density of served any ε > 0) under the assumption that refuting random in- clients. stances of 3SAT is hard on average (hardness of R3SAT as The difficult aspect of this problem (and what distin- in [17]). The inapproximability can be further strengthened guishes it from maximum coverage—see Section 2.4) is in- to 1/(ε log n) for some ε > 0, under a plausible hardness terference between base stations. In the simplest form, there hypothesis about a problem called Balanced Bipartite Inde- is a limit k on the number of base stations that a mobile pendent Set; see Hypothesis B.1. Section 3 proves all of client can reasonably hear without conflict between the sig- these results. nals; any client within range of more than k base stations Our hardness results have other implications regarding cannot communicate because of interference and thus is not the hardness of some well-studied problems. In particular, serviced. More generally, a mobile client’s reception is bet- for the problem of unlimited-supply single-minded (envy- ter when it is within range of fewer base stations, and our free) pricing, a recent result [24] proves an Ω(1/ log n) goal is to maximize total reception. To capture these de- approximation, but no inapproximability result better than sires, the instance specifies the satisfaction si of a client APX-hardness is known. As we show in Section 2.2, our within range of exactly i base stations, such that s0 = 0 and hardness results for the unique coverage problem imply the s1 ≥ s2 ≥ s3 ≥ · · · ≥ 0.
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