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

Bar-Yehuda, Reuven; Rawitz, Dror

doi:10.1007/3-540-44666-4_7

Cited by 13 publications

(14 citation statements)

References 22 publications

(22 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This algorithm can equivalently be viewed as a primal dual algorithm applied to a linear program with KC inequalities [8]. Combining this result with Theorem 2.1 implies a polynomialtime 16-approximation algorithm for GSP in the case of identical release times.…”

Section: Lemma 26 If There Is a Feasible Solution S To The R2c Instmentioning

confidence: 86%

The Geometry of Scheduling

Bansal¹,

Pruhs²

2014

SIAM J. Comput.

View full text Add to dashboard Cite

We consider the following general scheduling problem. The input consists of n jobs, each with an arbitrary release time, size, and monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as total weighted flow time, total weighted tardiness, and sum of flow time squared. We give an O(log log P ) approximation for this problem, where P is the ratio of the maximum to minimum job size. We also give an O(1) approximation in the special case of identical release times. These results are obtained by reducing the scheduling problem to a geometric capacitated set cover problem in two dimensions. Introduction.We consider the following general offline scheduling problem. General scheduling problem (GSP). The input consists of a collection of n jobs, and for each job j there is a positive integer release time r j , a positive integer size p j and a nondecreasing cost (or weight) function w j (t) ≥ 0 specifying a nonnegative cost for each time t > r j . (We will specify later how these weight functions are represented.) A feasible solution is a preemptive schedule, which assigns to each job j time slots [t, t + 1] (not necessarily consecutive and satisfying t ≥ r j ), during which j is run. A job is completed once it has been run for p j units of time. If job j completes at time t, then a cost of w j (t) is incurred for that job. The objective is to minimize the total cost,

show abstract

Section: Lemma 26 If There Is a Feasible Solution S To The R2c Instmentioning

confidence: 86%

The Geometry of Scheduling

Bansal¹,

Pruhs²

2014

SIAM J. Comput.

View full text Add to dashboard Cite

show abstract

“…This tail-recursive definition follows local-ratio analyses[6]. The more standard primal-dual approach-setting the packing variable for a covering constraint when a step for that constraint is done-doesn't work.…”

mentioning

confidence: 99%

Distributed algorithms for covering, packing and maximum weighted matching

Koufogiannakis

Young²

2011

Distrib. Comput.

View full text Add to dashboard Cite

This paper gives poly-logarithmic-round, distributed δ-approximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodular-cost Covering). The approximation ratio δ is the maximum number of variables in any constraint. Special cases include Covering Mixed Integer Linear Programs (CMIP), and Weighted Vertex Cover (with δ = 2). Via duality, the paper also gives poly-logarithmicround, distributed δ-approximation algorithms for Fractional Packing linear programs (where δ is the maximum number of constraints in which any variable occurs), and for Max Weighted c-Matching in hypergraphs (where δ is the maximum size of any of the hyperedges; for graphs δ = 2). The paper also gives parallel (RNC) 2-approximation algorithms for CMIP with two variables per constraint and Weighted Vertex Cover. The algorithms are randomized. All of the approximation ratios exactly match those of comparable centralized algorithms. 1 Background and resultsMany distributed systems are composed of components that can only communicate locally, yet need to achieve a global (system-wide) goal involving many components. are widely studied [12,38,39,42,49,54]. It is of specific interest to see which fundamental combinatorial optimization problems admit efficient distributed algorithms that achieve approximation guarantees that are as good as those of the best centralized algorithms. Research in this spirit includes works on Set Cover (Dominating Set) [26,40,41], capacitated dominating set [37], Capacitated Vertex Cover [16,17], and many other problems. This paper presents distributed approximation algorithms for some fundamental covering and packing problems.The algorithms use the standard synchronous communication model: in each round, nodes can exchange a constant number of messages with neighbors and perform local computation [54]. There is no restriction on message size or local computation. The algorithms are efficient-they finish in a number of rounds that is poly-logarithmic in the network size [42]. Covering problemsConsider optimization problems of the following form: given a non-decreasing, continuous, and submodular 2 cost function c : R n + → R + , and a set of constraints C where each constraint S ∈ C is closed upwards, 3 1 Preliminary versions appeared in [34,35].

show abstract

“…Thus, an important contribution of this paper is in providing a natural generalization of the framework presented by Goemans and Williamson, and later by Bar-Yehuda. The algorithm we present in this paper for the PCGSF problem is not the local ratio version of the primal-dual 3-approximation algorithm from [13] and cannot be obtained from it using the equivalence between the primal-dual schema and the local ratio technique [5].…”

Section: Our Resultsmentioning

confidence: 99%

Elementary approximation algorithms for prize collecting Steiner tree problems

Gutner¹

2008

Information Processing Letters

View full text Add to dashboard Cite

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

Cited by 13 publications

References 22 publications

The Geometry of Scheduling

The Geometry of Scheduling

Distributed algorithms for covering, packing and maximum weighted matching

Elementary approximation algorithms for prize collecting Steiner tree problems

Contact Info

Product

Resources

About