On Lagrangian relaxation for constrained maximization and reoptimization problems

Kulik, Ariel; Shachnai, Hadas; Tamir, Gal

doi:10.1016/j.dam.2020.10.001

Cited by 2 publications

(11 citation statements)

References 26 publications

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“…The paper includes a d-approximation for the special case of uniform weights MS on (d + 1)-claw free graphs, and a polynomial time approximation scheme (PTAS) for MS on planar graphs. We note that the results of [23] can improve the approximation algorithm for MS on d + 1-claw free graphs (see Section 6). For MS on line graphs (or, the budgeted matching problem) there is a PTAS [6] and also an EPTAS [11].…”

Section: Related Workmentioning

confidence: 99%

“…The proof of Lemma 6 is given in Section 5. The proof utilizes a known approximation algorithm for the unbudgeted version of BM [19,21] which is then transformed into an approximation algorithm for BM using a technique of [25].…”

Section: Representative Setmentioning

confidence: 99%

“…In this section we prove Lemma 6. The proof combines an existing approximation algorithm for the unbudgeted version of BM [19,21] with the Lagrangian relaxation technique of [25].…”

Section: A Polynomial Time 1 2•ℓ -Approximation For Bmmentioning

confidence: 99%

“…The next lemma shows that an ℓ-matchoid is in fact an ℓ-extendible set system. 6 Therefore, using a technique of [25], we have the following. There is an algorithm that, given some ε > 0, returns a solution for the BM instance I of profit at least…”

Section: A Polynomial Time 1 2•ℓ -Approximation For Bmmentioning

confidence: 99%

“…Instead of the above approach, we reduce the MSP problem to a subset selection problem with a budget constraint [23], relying on the Lagrangian relaxation of the MSP problem. This, combined with Theorem 1.1, gives the tight upper bound for MSP and MBS (recall that MSB is a special case of MSP).…”

Section: Introductionmentioning

confidence: 99%

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An EPTAS for Budgeted Matroid Independent Set

Doron-Arad¹,

Kulik²,

Shachnai³

2023

Symposium on Simplicity in Algorithms (SOSA)

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We study the budgeted versions of the well known matching and matroid intersection problems. While both problems admit a polynomial-time approximation scheme (PTAS) [Berger et al. (Math. Programming, 2011), Chekuri, Vondrák and Zenklusen (SODA 2011)], it has been an intriguing open question whether these problems admit a fully PTAS (FPTAS), or even an efficient PTAS (EPTAS).In this paper we answer the second part of this question affirmatively, by presenting an EPTAS for budgeted matching and budgeted matroid intersection. A main component of our scheme is a novel construction of representative sets for desired solutions, whose cardinality depends only on ε, the accuracy parameter. Thus, enumerating over solutions within a representative set leads to an EPTAS. This crucially distinguishes our algorithms from previous approaches, which rely on exhaustive enumeration over the solution set. Our ideas for constructing representative sets may find use in tackling other budgeted optimization problems, and are thus of independent interest. ACM Subject Classification Theory of computationKeywords and phrases budgeted matching, budgeted matroid intersection, efficient polynomial-time approximation scheme. DigitalObject Identifier 10.4230/LIPIcs...23 1 1 For a function f : A → R and a subset of elements C ⊆ A, we define f (C) = e∈C f (e).

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Section: Related Workmentioning

confidence: 99%

Section: Representative Setmentioning

confidence: 99%

“…In this section we prove Lemma 6. The proof combines an existing approximation algorithm for the unbudgeted version of BM [19,21] with the Lagrangian relaxation technique of [25].…”

Section: A Polynomial Time 1 2•ℓ -Approximation For Bmmentioning

confidence: 99%

Section: A Polynomial Time 1 2•ℓ -Approximation For Bmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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