{"id":157282,"date":"2008-06-01T00:00:00","date_gmt":"2008-06-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/degree-bounded-matroids-and-submodular-flows\/"},"modified":"2018-10-16T21:51:59","modified_gmt":"2018-10-17T04:51:59","slug":"degree-bounded-matroids-and-submodular-flows","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/degree-bounded-matroids-and-submodular-flows\/","title":{"rendered":"Degree Bounded Matroids and Submodular Flows"},"content":{"rendered":"

We consider two related problems, the Minimum Bounded Degree Matroid Basis<\/span> problem and the Minimum Bounded Degree Submodular Flow<\/span> problem. The first problem is a generalization of the Minimum Bounded Degree Spanning Tree<\/span> problem: we are given a matroid and a hypergraph on its ground set with lower and upper bounds f<\/em>(e<\/em>)\u2009\u2264\u2009g<\/em>(e<\/em>) for each hyperedge e<\/em>. The task is to find a minimum cost basis which contains at least f<\/em>(e<\/em>) and at most g<\/em>(e<\/em>) elements from each hyperedge e<\/em>. In the second problem we have a submodular flow problem, a lower bound f<\/em>(v<\/em>) and an upper bound g<\/em>(v<\/em>) for each node v<\/em>, and the task is to find a minimum cost 0-1 submodular flow with the additional constraint that the sum of the incoming and outgoing flow at each node v<\/em> is between f<\/em>(v<\/em>) and g<\/em>(v<\/em>). Both of these problems are NP-hard (even the feasibility problems are NP-complete), but we show that they can be approximated in the following sense. Let opt<\/span> be the value of the optimal solution. For the first problem we give an algorithm that finds a basis B<\/em> of cost no more than opt<\/span> such that f<\/em>(e<\/em>)\u2009\u2212\u20092\u0394<\/em>\u2009+\u20091\u2009\u2264\u2009|B<\/em>\u2009\u2229\u2009e<\/em>|\u2009\u2264\u2009g<\/em>(e<\/em>)\u2009+\u20092\u0394<\/em>\u2212\u20091 for every hyperedge e<\/em>, where \u0394<\/em> is the maximum degree of the hypergraph. If there are only upper bounds (or only lower bounds), then the violation can be decreased to \u0394<\/em>\u2212\u20091. For the second problem we can find a 0-1 submodular flow of cost at most opt<\/span> where the sum of the incoming and outgoing flow at each node v<\/em> is between f<\/em>(v<\/em>)\u2009\u2212\u20091 and g<\/em>(v<\/em>) \u2009+\u20091. These results can be applied to obtain approximation algorithms for different combinatorial optimization problems with degree constraints, including the Minimum Crossing Spanning Tree<\/span> problem, the Minimum Bounded Degree Spanning Tree Union<\/span> problem, the Minimum Bounded Degree Directed Cut Cover<\/span> problem, and the Minimum Bounded Degree Graph Orientation<\/span> problem.<\/p>\n","protected":false},"excerpt":{"rendered":"

We consider two related problems, the Minimum Bounded Degree Matroid Basis problem and the Minimum Bounded Degree Submodular Flow problem. The first problem is a generalization of the Minimum Bounded Degree Spanning Tree problem: we are given a matroid and a hypergraph on its ground set with lower and upper bounds f(e)\u2009\u2264\u2009g(e) for each hyperedge […]<\/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","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"Integer Programming and Combinatorial Optimization, 13th International Conference, IPCO 2008, Bertinoro, Italy, May 26-28, 2008, Proceedings","msr_editors":"","msr_how_published":"","msr_isbn":"978-3-540-68886-0","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"259\u2013272","msr_page_range_start":"259","msr_page_range_end":"272","msr_series":"Lecture Notes in Computer 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