On the Equivalence between the Primal-Dual Schema and the Local Ratio Technique

Bar-Yehuda, Reuven; Rawitz, Dror

doi:10.1137/050625382

Cited by 50 publications

(34 citation statements)

References 32 publications

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“…The local ratio algorithm by Bar-Noy et al [6], when interpreted as a primaldual algorithm [7], uses an LP relaxation that includes knapsack-cover inequalities. Thus, the 4-approximation algorithm of this paper can be considered a generalization of the algorithm by Bar-Noy et al [6].…”

Section: Previous Resultsmentioning

confidence: 99%

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A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems

Cheung¹,

Mestre²,

Shmoys³

et al. 2017

SIAM J. Discrete Math.

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We consider the following single-machine scheduling problem, which is often denoted 1|| f j : we are given n jobs to be scheduled on a single machine, where each job j has an integral processing time p j , and there is a nondecreasing, nonnegative cost function f j (C j ) that specifies the cost of finishing j at time C j ; the objective is to minimize n j=1 f j (C j ). Bansal & Pruhs recently gave the first constant approximation algorithm with a performance guarantee of 16. We improve on this result by giving a primal-dual pseudo-polynomial-time algorithm based on the recently introduced knapsack-cover inequalities. The algorithm finds a schedule of cost at most four times the constructed dual solution. Although we show that this bound is tight for our algorithm, we leave open the question of whether the integrality gap of the LP is less than 4. Finally, we show how the technique can be adapted to yield, for any ǫ > 0, a (4+ǫ)-approximation algorithm for this problem.

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Section: Previous Resultsmentioning

confidence: 99%

“…It is well known that primal-dual algorithms have an equivalent local-ratio counterpart [7]. For completeness, we also give the local-ratio version of our algorithm and its analysis.…”

Section: Introductionmentioning

confidence: 99%

A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems

Cheung¹,

Mestre²,

Shmoys³

et al. 2017

SIAM J. Discrete Math.

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“…For example, [9] shows a form of equivalence between the primal-dual method and the local-ratio method, but that result only considers problems with solution space {0, 1} n (i.e., 0/1-variables). Also, the standard intuitive interpretation of local-ratio -that the algorithm reduces the coefficients in the cost vector c -works only for 0/1-variables.…”

Section: Relation To Local-ratio Methodsmentioning

confidence: 99%

“…The local-ratio method has most commonly been applied to problems with variables taking values in {0, 1} and with linear objective function c · x (see [7,4,9,5]; for one exception, see [8]). For example, [9] shows a form of equivalence between the primal-dual method and the local-ratio method, but that result only considers problems with solution space {0, 1} n (i.e., 0/1-variables).…”

Section: Relation To Local-ratio Methodsmentioning

confidence: 99%

Greedy ${\ensuremath{\Delta}}$ -Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

Koufogiannakis

Young

2009

Automata, Languages and Programming

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This paper describes a simple greedy ∆-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most ∆ variables of the problem. (A simple example is VERTEX COVER, with ∆ = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.

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“…The time complexity of this algorithm was later improved in [4], where a d-approximation algorithm for Generalized Vertex Cover in d-hypergraphs was given as well. Hassin and Levin [12] studied a problem that extends Generalized Vertex Cover to one in which one pays a penalty for not covering an edge and a smaller penalty for covering an edge only by one of its endpoints.…”

Section: Introduction Given a Graph G = (V E) With Vertex Weights mentioning

confidence: 99%

An Extension of the Nemhauser–Trotter Theorem to Generalized Vertex Cover with Applications

Bar-Yehuda¹,

Hermelin²,

Rawitz³

2010

SIAM J. Discrete Math.

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Abstract. The Nemhauser-Trotter theorem provides an algorithm which is frequently used as a subroutine in approximation algorithms for the classical Vertex Cover problem. In this paper we present an extension of this theorem so it fits a more general variant of Vertex Cover, namely, the Generalized Vertex Cover problem, where edges are allowed not to be covered at a certain predetermined penalty. We show that many applications of the original Nemhauser-Trotter theorem can be applied using our extension to Generalized Vertex Cover. These applications include a (2 − 2/d)-approximation algorithm for graphs of bounded degree d, a polynomial-time approximation scheme (PTAS) for planar graphs, a (2 − lg lg n/2 lg n)-approximation algorithm for general graphs, and a 2k kernel for the parameterized Generalized Vertex Cover problem. Key words.approximation algorithms, generalized vertex cover, local ratio technique, Nemhauser-Trotter theorem AMS subject classifications. 68W25, 05C85, 68W40, 90C27 DOI. 10.1137/090773313 1. Introduction. Given a graph G = (V, E) with vertex weights, the classical Vertex Cover problem asks to find a minimum weight subset of vertices S ⊆ V that covers all edges in G, i.e., a subset S with S ∩ e = ∅ for all e ∈ E. The Vertex Cover problem is one of the most well studied problems in theoretical computer science and discrete mathematics in general, dating back to König's classical early 1930s result [20] and probably even prior to that. In 1972, Karp listed the decision version of Vertex Cover in his famous list of initial 21 NP-complete problems [18].One of the most well known results about Vertex Cover is the half-integrality of the LP-relaxation of the standard integer programming formulation of Vertex Cover (see, e.g., [22]). This result directly implies a 2-approximation algorithm for vertex cover (as observed by Hochbaum [14]). In 1975, only three years after the publication of Karp's famous NP-complete list, Nemhauser and Trotter published their seminal paper [23] in which they present a reduction that reduces the problem of finding a vertex cover in an arbitrary graph G to that of finding a vertex cover in a subgraph of G whose total weight is not much more than the weight of any of its vertex covers. This reduction is based on the half-integrality of Vertex Cover, and it adds additional structure to the Vertex Cover problem in general. Indeed, after applying the Nemhauser-Trotter reduction, one can use the total weight of the graph as a yardstick for analyzing approximate solutions rather than use the weight of the optimal solution of which there is rarely any knowledge. Below is a precise statement of the Nemhauser-Trotter theorem:

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On the Equivalence between the Primal-Dual Schema and the Local Ratio Technique

Cited by 50 publications

References 32 publications

A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems

A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems

Greedy ${\ensuremath{\Delta}}$ -Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

An Extension of the Nemhauser–Trotter Theorem to Generalized Vertex Cover with Applications

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