{"id":324326,"date":"2016-11-18T15:27:28","date_gmt":"2016-11-18T23:27:28","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=324326"},"modified":"2018-10-16T20:19:47","modified_gmt":"2018-10-17T03:19:47","slug":"approximability-sparse-integer-programs","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/approximability-sparse-integer-programs\/","title":{"rendered":"Approximability of Sparse Integer Programs"},"content":{"rendered":"<p>The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min\u2009<em class=\"EmphasisTypeItalic \">cx<\/em>:<em class=\"EmphasisTypeItalic \">Ax<\/em>\u2265<em class=\"EmphasisTypeItalic \">b<\/em>,<strong class=\"EmphasisTypeBold \">0<\/strong>\u2264<em class=\"EmphasisTypeItalic \">x<\/em>\u2264<em class=\"EmphasisTypeItalic \">d<\/em>} where <em class=\"EmphasisTypeItalic \">A<\/em> has at most <em class=\"EmphasisTypeItalic \">k<\/em>nonzeroes per row, we give a <em class=\"EmphasisTypeItalic \">k<\/em>-approximation algorithm. (We assume <em class=\"EmphasisTypeItalic \">A<\/em>,<em class=\"EmphasisTypeItalic \">b<\/em>,<em class=\"EmphasisTypeItalic \">c<\/em>,<em class=\"EmphasisTypeItalic \">d<\/em> are nonnegative.) For any <em class=\"EmphasisTypeItalic \">k<\/em>\u22652 and <em class=\"EmphasisTypeItalic \">\u03b5<\/em>>0, if <span class=\"EmphasisFontCategorySansSerif \">P<\/span>\u2260<span class=\"EmphasisFontCategorySansSerif \">NP<\/span> this ratio cannot be improved to <em class=\"EmphasisTypeItalic \">k<\/em>\u22121\u2212<em class=\"EmphasisTypeItalic \">\u03b5<\/em>, and under the unique games conjecture this ratio cannot be improved to <em class=\"EmphasisTypeItalic \">k<\/em>\u2212<em class=\"EmphasisTypeItalic \">\u03b5<\/em>. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max\u2009<em class=\"EmphasisTypeItalic \">cx<\/em>:<em class=\"EmphasisTypeItalic \">Ax<\/em>\u2264<em class=\"EmphasisTypeItalic \">b<\/em>,<strong class=\"EmphasisTypeBold \">0<\/strong>\u2264<em class=\"EmphasisTypeItalic \">x<\/em>\u2264<em class=\"EmphasisTypeItalic \">d<\/em>} where <em class=\"EmphasisTypeItalic \">A<\/em> has at most <em class=\"EmphasisTypeItalic \">k<\/em> nonzeroes per column, we give a (2<em class=\"EmphasisTypeItalic \">k<\/em><sup>2<\/sup>+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when <em class=\"EmphasisTypeItalic \">k<\/em>=2, and for both problems when every <em class=\"EmphasisTypeItalic \">A<\/em><sub><em class=\"EmphasisTypeItalic \">ij<\/em><\/sub> is small compared to <em class=\"EmphasisTypeItalic \">b<\/em><sub><em class=\"EmphasisTypeItalic \">i<\/em><\/sub>. Finally, we demonstrate a 17\/16-inapproximability for covering integer programs with at most two nonzeroes per column.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min\u2009cx:Ax\u2265b,0\u2264x\u2264d} where A has at most knonzeroes per row, we give a k-approximation algorithm. (We assume A,b,c,d are nonnegative.) For any k\u22652 and \u03b5>0, if P\u2260NP this ratio cannot be improved to k\u22121\u2212\u03b5, [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Springer-Verlag New York, Inc. 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