{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11585","name":"Generalized Budgeted Submodular Set Function Maximization","abstract":"In this paper we consider a generalization of the well-known budgeted maximum coverage problem. We are given a ground set of elements and a set of bins. The goal is to find a subset of elements along with an associated set of bins, such that the overall cost is at most a given budget, and the profit is maximized. Each bin has its own cost and the cost of each element depends on its associated bin. The profit is measured by a monotone submodular function over the elements.\nWe first present an algorithm that guarantees an approximation factor of 1\/2(1-1\/e^alpha), where alpha <= 1 is the approximation factor of an algorithm for a sub-problem. We give two polynomial-time algorithms to solve this sub-problem. The first one gives us alpha=1- epsilon if the costs satisfies a specific condition, which is fulfilled in several relevant cases, including the unitary costs case and the problem of maximizing a monotone submodular function under a knapsack constraint. The second one guarantees alpha=1-1\/e-epsilon for the general case. The gap between our approximation guarantees and the known inapproximability bounds is 1\/2.\nWe extend our algorithm to a bi-criterion approximation algorithm in which we are allowed to spend an extra budget up to a factor beta >= 1 to guarantee a 1\/2(1-1\/e^(alpha beta))-approximation. If we set beta=1\/(alpha)ln (1\/(2 epsilon)), the algorithm achieves an approximation factor of 1\/2-epsilon, for any arbitrarily small epsilon>0.","keywords":["Submodular set function","Approximation algorithms","Budgeted Maximum Coverage"],"author":[{"@type":"Person","name":"Cellinese, Francesco","givenName":"Francesco","familyName":"Cellinese","affiliation":"Gran Sasso Science Institute, L'Aquila, Italy"},{"@type":"Person","name":"D'Angelo, Gianlorenzo","givenName":"Gianlorenzo","familyName":"D'Angelo","affiliation":"Gran Sasso Science Institute, L'Aquila, Italy"},{"@type":"Person","name":"Monaco, Gianpiero","givenName":"Gianpiero","familyName":"Monaco","affiliation":"University of L\u2019Aquila, L'Aquila, Italy"},{"@type":"Person","name":"Velaj, Yllka","givenName":"Yllka","familyName":"Velaj","affiliation":"University of Chieti-Pescara, Pescara, Italy"}],"position":31,"pageStart":"31:1","pageEnd":"31:14","dateCreated":"2018-08-27","datePublished":"2018-08-27","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Cellinese, Francesco","givenName":"Francesco","familyName":"Cellinese","affiliation":"Gran Sasso Science Institute, L'Aquila, Italy"},{"@type":"Person","name":"D'Angelo, Gianlorenzo","givenName":"Gianlorenzo","familyName":"D'Angelo","affiliation":"Gran Sasso Science Institute, L'Aquila, Italy"},{"@type":"Person","name":"Monaco, Gianpiero","givenName":"Gianpiero","familyName":"Monaco","affiliation":"University of L\u2019Aquila, L'Aquila, Italy"},{"@type":"Person","name":"Velaj, Yllka","givenName":"Yllka","familyName":"Velaj","affiliation":"University of Chieti-Pescara, Pescara, Italy"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.MFCS.2018.31","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/dx.doi.org\/10.1016\/j.tcs.2018.01.017","isPartOf":{"@type":"PublicationVolume","@id":"#volume6320","volumeNumber":117,"name":"43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)","dateCreated":"2018-08-27","datePublished":"2018-08-27","editor":[{"@type":"Person","name":"Potapov, Igor","givenName":"Igor","familyName":"Potapov"},{"@type":"Person","name":"Spirakis, Paul","givenName":"Paul","familyName":"Spirakis"},{"@type":"Person","name":"Worrell, James","givenName":"James","familyName":"Worrell"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11585","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6320"}}}