Random pseudo-polynomial algorithms for exact matroid problems

Camerini, P.; Galbiati, Giulia; Maffioli, Francesco

doi:10.1016/0196-6774(92)90018-8

Cited by 60 publications

(74 citation statements)

References 11 publications

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“…Similarly, the Monte-Carlo PPT-algorithm for exact matching in [17] gives a PRAS for k-budgeted matching. The Monte-Carlo PPT-algorithms for exact matroid intersection independent set in [5], which works in the special case of representable matroids 5 , implies a PRAS for the corresponding budgeted problem.…”

Section: Theorem 2 (Feasibilization)mentioning

confidence: 99%

Approximation Schemes for Multi-Budgeted Independence Systems

Grandoni

Zenklusen

2010

Algorithms – ESA 2010

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Abstract.A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Some classical optimization problems, such as spanning tree and forest, shortest path, (perfect) matching, independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems efficient deterministic and randomized approximation schemes are known. For two or more budgets, typically only multi-criteria approximation schemes are available, which return slightly infeasible solutions. Not much is known however for strict budget constraints: filling this gap is the main goal of this paper. It is not hard to see that the above-mentioned problems whose solution sets do not correspond to independence systems are inapproximable already for two budget constraints. For the remaining problems, we present approximation schemes for a constant number k of budget constraints using a variety of techniques: i) we present a simple and powerful mechanism to transform multi-criteria approximation schemes into pure approximation schemes. This leads to deterministic and randomized approximation schemes for various of the above-mentioned problems; ii) we show that points in low-dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for k-budgeted matroid independent set; iii) we present a deterministic approximation scheme for 2-budgeted matching. The backbone of this result is a purely topological property of curves in R 2 .

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Section: Theorem 2 (Feasibilization)mentioning

confidence: 99%

Approximation Schemes for Multi-Budgeted Independence Systems

Grandoni

Zenklusen

2010

Algorithms – ESA 2010

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“…Similarly, the MonteCarlo PPT-algorithm for exact matching in [30] gives a PRAS for k-budgeted matching. The Monte-Carlo PPT-algorithms for exact matroid intersection independent set in [9], which works in the special case of representable matroids 3 , implies a PRAS for the corresponding k-budgeted problem.…”

Section: Introductionmentioning

confidence: 99%

New approaches to multi-objective optimization

et al. 2013

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A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Many classical optimization problems, such as maximum spanning tree and forest, shortest path, maximum weight (perfect) matching, maximum weight independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems efficient deterministic and randomized approximation schemes are known. Not much is known however about the case of two or more budgets: filling this gap, at least partially, is the main goal of this paper. In more detail, we obtain the following main results:• Using iterative rounding for the first time in multi-objective optimization, we obtain multi-criteria PTASs (which slightly violate the budget constraints) for spanning tree, matroid basis, and bipartite matching with k = O(1) budget constraints.• We present a simple mechanism to transform multi-criteria approximation schemes into pure approximation schemes for problems whose feasible solutions define an independence system. This gives improved algorithms for several problems. In particular, this mechanism can be applied to the above bipartite matching algorithm, hence obtaining a pure PTAS.• We show that points in low-dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for k-budgeted matroid independent set. • We present a deterministic approximation scheme for k-budgeted matching (in general graphs), where k = O(1). Interestingly, to show that our procedure works, we rely on a non-constructive result by Stromquist and Woodall, which is based on the Ham Sandwich Theorem.

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“…Hence, in contrast with the longest path problem, the /C-WALK problem remains NP-complete even when restricted to DAGs. It is known that the unary version of some intractable (unless P = NP) exact problems defined on graphs can be solved by polynomial time algorithms (and even NC algorithms) [2,3,4, 10], namely they are pseudo-polynomial problems [6]. In the present case we can be more précise since we can prove that the unary &-WALK problem is logspace equivalent to the NL-complete directed GRAPH REACHABILITY problem.…”

Section: : the K-walk Problem Is In Npmentioning

confidence: 64%