Minimal spanning tree subject to a side constraint

Aggarwal, Vijay; Aneja, Y. P.; Nair, K. P. K.

doi:10.1016/0305-0548(82)90026-0

Cited by 81 publications

(58 citation statements)

References 5 publications

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“…All the above problems are polynomial-time solvable (see, e.g., [23]) in their unbudgeted version (k = 0), but become NP-hard [1,6] even for a single budget constraint (k = 1). For the case of one budget (k = 1), PTASs are known for spanning tree [35] (see also [21]), 1 The assumption that k is a constant is crucial in this paper, since many of the presented algorithms will have a running time that is exponential in k, but polynomial for constant k. 2 We recall that E is a finite ground set and I ⊆ 2 E is a nonempty family of subsets of E (independent sets) which have to satisfy the following two conditions: (i) I ∈ I, J ⊆ I ⇒ J ∈ I and (ii) I, J ∈ I, |I| > |J| ⇒ ∃z ∈ I \ J : J ∪ {z} ∈ I.…”

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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Section: Introductionmentioning

confidence: 99%

New approaches to multi-objective optimization

et al. 2013

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“…All the problems above are polynomial-time solvable (see, e.g., [24]) in their unbudgeted version (k = 0), but become NP-hard [1,3,7] even for a single budget constraint (k = 1). For the case of one budget (k = 1), polynomial-time approximation schemes (PTASs) are known for spanning tree [21] (see also [11]), shortest path [25] (see also [10,14]), and matching [3].…”

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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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“…As the minimization problem can trivially be transformed into a maximization variant, we ignore this difference in the following. Aggarwal et al [3] were the first describing this problem and called it MST problem subject to a side constraint. They proved its N Phardness and proposed a branch-and-bound approach for solving it.…”

Section: Previous Workmentioning

confidence: 99%

A Lagrangian Decomposition/Evolutionary Algorithm Hybrid for the Knapsack Constrained Maximum Spanning Tree Problem

Pirkwieser

Raidl

Puchinger

2008

Recent Advances in Evolutionary Computation for Combinatorial Optimization

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Abstract. We present a Lagrangian decomposition approach for the Knapsack Constrained Maximum Spanning Tree problem yielding upper bounds as well as heuristic solutions. This method is further combined with an evolutionary algorithm to a sequential hybrid approach. Thorough experimental investigations, including a comparison to a previously suggested simpler Lagrangian relaxation based method, document the advantages of our approach. Most of the upper bounds derived by Lagrangian decomposition are optimal, and when additionally applying local search (LS) and combining it with the evolutionary algorithm, large and supposedly hard instances can be either solved to provable optimality or with a very small remaining gap in reasonable time.

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Minimal spanning tree subject to a side constraint

Cited by 81 publications

References 5 publications

New approaches to multi-objective optimization

New approaches to multi-objective optimization

Approximation Schemes for Multi-Budgeted Independence Systems

A Lagrangian Decomposition/Evolutionary Algorithm Hybrid for the Knapsack Constrained Maximum Spanning Tree Problem

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