On the Complexity of Timetable and Multicommodity Flow Problems

Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primal-dual schema and the local ratio technique. Recently, primal-dual algorithms were devised by first constructing a local ratio algorithm and then transforming it into a primal-dual algorithm. This was done in the case of the 2-approximation algorithms for the feedback vertex set problem and in the case of the first primal-dual algorithms for maximization problems. Subsequently, the nature of the connection between the two paradigms was posed as an open question by Williamson [Math. Program., 91 (2002), pp. 447-478]. In this paper we answer this question by showing that the two paradigms are equivalent.

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“…First, we can check whether ϕ is satisfiable by using the algorithm from [21]. Thus, we may assume that ϕ is satisfiable.…”

Section: Minimum Weight 2-satisfiabilitymentioning

confidence: 99%

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

Bar-Yehuda

Rawitz

2001

Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques

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“…We use the term k-CNF to mean any of a family of CNF expressions all of whose clauses have width k. It is well known that 2-CNF expressions are solved in polynomial time [38]. Therefore, we will only be interested in generating k-CNF expressions where k ≥ 3.…”

Section: Cnf Expressionsmentioning

confidence: 99%

Probabilistic Analysis of Satisfiability Algorithms

Franco¹,

Crama²,

Hammer³

2010

Boolean Models and Methods in Mathematics, Computer Science, and Engineering

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“…Later, Garey and Johnson [10] observed that the modification of a construction by Even et al [7] shows that the problem is NP-hard even if the capacity of every arc is 1. Srinathan et al [18] furthered this, showing that the problem remains NP-hard even if all capacities are 1 and all arcs in a homologous set originate from the same node.…”

Section: Problem Definitionmentioning

confidence: 99%

“…Sahni [16] proved that the maximum flow version of this problem is NP-hard with a reduction from NonTautology. Later, Even, Itai, and Shamir [7] showed via a reduction from Satisfiability that the problem remains NP-hard even if the capacity of each arc is 1. Srinathan et al [18] showed by a reduction from Exact Cover by 3-Sets that this problem also remains NP-hard if we further require that all arcs in a set R i must originate from the same node.…”

Section: Introductionmentioning

confidence: 99%