{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7590","name":"Improved Approximation Algorithm for Steiner k-Forest with Nearly Uniform Weights","abstract":"In the Steiner k-Forest problem we are given an edge weighted graph, a collection D of node pairs, and an integer k \\leq |D|. The goal is to find a minimum cost subgraph that connects at least k pairs. The best known ratio for this problem is min{O(sqrt{n}),O(sqrt{k})} [Gupta et al., 2008]. In [Gupta et al., 2008] it is also shown that ratio rho for Steiner k-Forest implies ratio O(rho log^2 n) for the Dial-a-Ride problem: given an edge weighted graph and a set of items with a source and a destination each, find a minimum length tour to move each object from its source to destination, but carrying at most k objects at a time. The only other algorithm known for Dial-a-Ride, besides the one resulting from [Gupta et al., 2008], has ratio O(sqrt{n}) [Charikar and Raghavachari, 1998]. We obtain ratio n^{0.448} for Steiner k-Forest and Dial-a-Ride with unit weights, breaking the O(sqrt{n}) ratio barrier for this natural special case. We also show that if the maximum weight of an edge is O(n^{epsilon}), then one can achieve ratio O(n^{(1+epsilon) 0.448}), which is less than sqrt{n} if epsilon is small enough. To prove our main result we consider the following generalization of the Minimum k-Edge Subgraph (Mk-ES) problem, which we call Min-Cost l-Edge-Profit Subgraph (MCl-EPS): Given a graph G=(V,E) with edge-profits p={p_e: e in E} and node-costs c={c_v: v in V}, and a lower profit bound l, find a minimum node-cost subgraph of G of edge profit at least l. The Mk-ES problem is a special case of MCl-EPS with unit node costs and unit edge profits. The currently best known ratio for Mk-ES is n^{3-2*sqrt{2} + epsilon} (note that 3-2*sqrt{2} < 0.1716). We extend this ratio to MCl-EPS for arbitrary node weights and edge profits that are polynomial in n, which may be of independent interest.","keywords":"k-Steiner Forest; Uniform weights; Densest k-Subgraph; Approximation algorithms","author":[{"@type":"Person","name":"Dinitz, Michael","givenName":"Michael","familyName":"Dinitz"},{"@type":"Person","name":"Kortsarz, Guy","givenName":"Guy","familyName":"Kortsarz"},{"@type":"Person","name":"Nutov, Zeev","givenName":"Zeev","familyName":"Nutov"}],"position":9,"pageStart":115,"pageEnd":127,"dateCreated":"2014-09-04","datePublished":"2014-09-04","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dinitz, Michael","givenName":"Michael","familyName":"Dinitz"},{"@type":"Person","name":"Kortsarz, Guy","givenName":"Guy","familyName":"Kortsarz"},{"@type":"Person","name":"Nutov, Zeev","givenName":"Zeev","familyName":"Nutov"}],"copyrightYear":"2014","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2014.115","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6231","volumeNumber":28,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2014)","dateCreated":"2014-09-04","datePublished":"2014-09-04","editor":[{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"Rolim, Jos\u00e9","givenName":"Jos\u00e9","familyName":"Rolim"},{"@type":"Person","name":"Devanur, Nikhil R.","givenName":"Nikhil R.","familyName":"Devanur"},{"@type":"Person","name":"Moore, Cristopher","givenName":"Cristopher","familyName":"Moore"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7590","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":"#volume6231"}}}