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Approximating the minimum-degree Steiner tree to within one of optimal
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Handbook of combinatorics (vol. 1)
The primal-dual method for approximation algorithms and its application to network design problems
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Approximation algorithms for NP-hard problems
A representation for crossing set families with applications to submodular flow problems
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
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FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
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Approximating minimum bounded degree spanning trees to within one of optimal
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WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
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ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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| Hi-index | 5.23 |
Given a graph H=(V,F) with edge weights {w"e:e@?F}, the weighted degree of a node v in H is @?{w"v"u:vu@?F}. We give bicriteria approximation algorithms for problems that seek to find a minimum cost directed graph that satisfies both intersecting supermodular connectivity requirements and weighted degree constraints. The input to such problems is a directed graph G=(V,E) with edge-costs {c"e:e@?E} and edge-weights {w"e:e@?E}, an intersecting supermodular set-function f on V, and degree bounds {b(v):v@?B@?V}. The goal is to find a minimum cost f-connected subgraph H=(V,F) (namely, at least f(S) edges in F enter every S@?V) of G with weighted degrees @?b(v). Our algorithm computes a solution of cost @?2@?opt, so that the weighted degree of every v@?V is at most: 7b(v) for arbitrary f and 5b(v) for a 0,1-valued f; 2b(v)+4 for arbitrary f and 2b(v)+2 for a 0,1-valued f in the case of unit weights. Another algorithm computes a solution of cost @?3@?opt and weighted degrees @?6b(v). We obtain similar results when there are both indegree and outdegree constraints, and better results when there are indegree constraints only: a (1,4b(v))-approximation algorithm for arbitrary weights and a polynomial time algorithm for unit weights. Similar results are shown for crossing supermodular f. We also consider the problem of packing maximum number k of pairwise edge-disjoint arborescences so that their union satisfies weighted degree constraints, and give an algorithm that computes a solution of value at least @?k/36@?. Finally, for unit weights and without trying to bound the cost, we give an algorithm that computes a subgraph so that the degree of every v@?V is at most b(v)+3, improving over the approximation b(v)+4 of Bansal et al. (2008) [2].