Choose Outsiders First: a mean 2-approximation random algorithm for covering problems

A high number of discrete optimization problems, including Vertex Cover, Set Cover or Feedback Vertex Set, can be unified into the class of covering problems. Several of them were shown to be inapproximable by deterministic algorithms. This article proposes a new random approach, called Choose Outsiders First, which consists in selecting randomly ele- ments which are excluded from the cover. We show that this approach leads to random outputs which mean size is at most twice the optimal solution.Comment: 8 pages The paper has been withdrawn due to an error in the proo

Approximating Weighted Duo-Preservation in Comparative Genomics

Motivated by comparative genomics, Chen et al. [9] introduced the Maximum Duo-preservation String Mapping (MDSM) problem in which we are given two strings $s_1$ and $s_2$ from the same alphabet and the goal is to find a mapping $\pi$ between them so as to maximize the number of duos preserved. A duo is any two consecutive characters in a string and it is preserved in the mapping if its two consecutive characters in $s_1$ are mapped to same two consecutive characters in $s_2$. The MDSM problem is known to be NP-hard and there are approximation algorithms for this problem [3, 5, 13], but all of them consider only the "unweighted" version of the problem in the sense that a duo from $s_1$ is preserved by mapping to any same duo in $s_2$ regardless of their positions in the respective strings. However, it is well-desired in comparative genomics to find mappings that consider preserving duos that are "closer" to each other under some distance measure [19]. In this paper, we introduce a generalized version of the problem, called the Maximum-Weight Duo-preservation String Mapping (MWDSM) problem that captures both duos-preservation and duos-distance measures in the sense that mapping a duo from $s_1$ to each preserved duo in $s_2$ has a weight, indicating the "closeness" of the two duos. The objective of the MWDSM problem is to find a mapping so as to maximize the total weight of preserved duos. In this paper, we give a polynomial-time 6-approximation algorithm for this problem.Comment: Appeared in proceedings of the 23rd International Computing and Combinatorics Conference (COCOON 2017

Bridging gap between standard and differential polynomial approximation: The case of bin-packing

AbstractThe purpose of this paper is mainly to prove the following theorem: for every polynomial time algorithm running in time T(n) and guaranteeing standard-approximation ratio ϱ for bin-packing, there exists an algorithm running in time O(nT(n)) and achieving differential-approximation ratio 2 − ϱ for BP. This theorem has two main impacts. The first one is “operational”, deriving a polynomial time differential-approximation schema for bin-packing. The second one is structural, establishing a kind of reduction (to our knowledge not existing until now) between standard approximation and differential one

Lagrangian Relaxation and Partial Cover

Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover.Comment: 20 pages, extended abstract appeared in STACS 200