On Generalizations of Network Design Problems with Degree Bounds

Bansal, Nikhil; Khandekar, Rohit; Könemann, Jochen; Nagarajan, Viswanath; Peis, Britta

doi:10.1007/978-3-642-13036-6_9

Cited by 13 publications

(26 citation statements)

References 28 publications

(33 reference statements)

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“…Recently Bansal et.al. [1] show how to extend the iterative relaxation method to obtain new or improved bicriteria approximation algorithms for minimum crossing spanning tree, crossing matroid intersection, and crossing lattice polyhedra.…”

Section: Discussionmentioning

confidence: 99%

Degree bounded matroids and submodular flows

2012

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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)≤g(e) for each hyperedge e. The task is to find a minimum cost basis which contains at least f(e) and at most g(e) elements from each hyperedge e. In the second problem we have a submodular flow problem, a lower bound f(v) and an upper bound g(v) for each node v, 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 is between f(v) and g(v). 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 be the value of the optimal solution. For the first problem we give an algorithm that finds a basis B of cost no more than opt such that f(e)-2Δ+1≤|B∩e|≤g(e)+2Δ-1 for every hyperedge e, where Δ is the maximum degree of the hypergraph. If there are only upper bounds (or only lower bounds), then the violation can be decreased to Δ-1. For the second problem we can find a 0-1 submodular flow of cost at most opt where the sum of the incoming and outgoing flow at each node v is between f(v)-1 and g(v)+1. These results can be applied to obtain approximation algorithms for several combinatorial optimization problems with degree constraints, including the Minimum Crossing Spanning Tree problem, the Minimum Bounded Degree Spanning Tree Union problem, the Minimum Bounded Degree Directed Cut Cover problem, and the Minimum Bounded Degree Graph Orientation problem. © 2012 János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg

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Section: Discussionmentioning

confidence: 99%

Degree bounded matroids and submodular flows

2012

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“…We remark that, whereas [21] obtains an O(1)-additive violation of the matroid constraints for matroidal degree-bounded MST problem, such a polytime additive guarantee is not possible for gmdst unless P = NP. This follows from the same replication idea used in Appendix A to rule out small additive violations for (1).…”

Section: An Algorithm Based On Iterative Refinement and Relaxationmentioning

confidence: 99%

“…Perhaps the two notions that first come to mind are additive and multiplicative violation of the rank constraints. Whereas additive violations are common in the study of degree-bounded MST problems, which can be cast as special cases of (1), it turns out that such a guarantee is impossible to obtain (in polytime) for (1). More precisely, we show in Appendix A (via a replication idea) that, even for k = 2, if we could find in polytime a basis B of M 0 satisfying |B| ≤ r i (B) + α for i = 1, 2 for α = O(|N | 1− ) for any > 0, then we could efficiently find a basis of M 0 that is independent in M 1 , M 2 ; the latter problem is easily seen to be NP-hard via a reduction from Hamiltonian path.…”

Section: Introductionmentioning

confidence: 99%

“…Also, interesting variations have been suggested that do not just drop constraints but may relax constraints by replacing a constraint by a weaker family (see work by Bansal etal. [1]). In contrast, our method does not just relax constraints, but also strengthens the constraint family in some iterations, so as to simplify it and enable one to drop constraints later on.…”

Section: Introductionmentioning

confidence: 99%

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Approximate multi-matroid intersection via iterative refinement

et al. 2020

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We introduce a new iterative rounding technique to round a point in a matroid polytope subject to further matroid constraints. This technique returns an independent set in one matroid with limited violations of the other ones. On top of the classical steps of iterative relaxation approaches, we iteratively refine/split involved matroid constraints to obtain a more restrictive constraint system, that is amenable to iterative relaxation techniques. Hence, throughout the iterations, we both tighten constraints and later relax them by dropping constraints under certain conditions. Due to the refinement step, we can deal with considerably more general constraint classes than existing iterative relaxation/rounding methods, which typically round on one matroid polytope with additional simple cardinality constraints that do not overlap too much. We show how our rounding method, combined with an application of a matroid intersection algorithm, yields the first 2-approximation for finding a maximum-weight common independent set in 3 matroids. Moreover, our 2-approximation is LP-based, and settles the integrality gap for the natural relaxation of the problem. Prior to our work, no upper bound better than 3 was known for the integrality gap, which followed from the greedy algorithm. We also discuss various other applications of our techniques, including an extension that allows us to handle a mixture of matroid and knapsack constraints.

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“…Spanning tree problems with a somewhat different notion of generalized degree bounds have been considered in [2] and [1]. In these papers, the term "generalized degree bounds" is used as follows: given is a family of sets E 1 , .…”

Section: Related Workmentioning

confidence: 99%

Matroidal Degree-Bounded Minimum Spanning Trees

Zenklusen

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the minimum spanning tree (MST) problem under the restriction that for every vertex v, the edges of the tree that are adjacent to v satisfy a given family of constraints. A famous example thereof is the classical degree-bounded MST problem, where for every vertex v, a simple upper bound on the degree is imposed. Iterative rounding/relaxation algorithms became the tool of choice for degreeconstrained network design problems. A cornerstone for this development was the work of Singh and Lau [18], who showed that for the degree-bounded MST problem, one can find a spanning tree violating each degree bound by at most one unit and with cost at most the cost of an optimal solution that respects the degree bounds.However, current iterative rounding approaches face several limits when dealing with more general degree constraints. In particular, when several constraints are imposed on the edges adjacent to a vertex v, as for example when a partition of the edges adjacent to v is given and only a fixed number of elements can be chosen out of each set of the partition, current approaches might violate each of the constraints by a constant, instead of violating the whole family of constraints by at most a constant number of edges. Furthermore, it is also not clear how previous iterative rounding approaches can be used for degree constraints where some edges are in a super-constant number of constraints.We extend iterative rounding/relaxation approaches both on a conceptual level as well as aspects involving their analysis to address these limitations. Based on these extensions, we present an algorithm for the degree-constrained MST problem where for every vertex v, the edges adjacent to v have to be independent in a given matroid. The algorithm returns a spanning tree of cost at most OPT such that for every vertex v, it suffices to remove at most 8 edges from the spanning tree to satisfy the matroidal degree constraint at v. *

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On Generalizations of Network Design Problems with Degree Bounds

Cited by 13 publications

References 28 publications

Degree bounded matroids and submodular flows

Degree bounded matroids and submodular flows

Approximate multi-matroid intersection via iterative refinement

Matroidal Degree-Bounded Minimum Spanning Trees

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