On Approximability of Bounded Degree Instances of Selected Optimization Problems

In order to cope with the approximation hardness of an underlying optimization problem, it is advantageous to consider specific families of instances with properties that can be exploited to obtain efficient approximation algorithms for the restricted version of the problem with improved performance guarantees. In this thesis, we investigate the approximation complexity of selected NP-hard optimization problems restricted to instances with bounded degree, occurrence or weight parameter. Specifically, we consider the family of dense instances, where typically the average degree is bounded from below by some function of the size of the instance. Complementarily, we examine the family of sparse instances, in which the average degree is bounded from above by some fixed constant. We focus on developing new methods for proving explicit approximation hardness results for general as well as for restricted instances. The fist part of the thesis contributes to the systematic investigation of the VERTEX COVER problem in k-hypergraphs and k-partite k-hypergraphs with density and regularity constraints. We design efficient approximation algorithms for the problems with improved performance guarantees as compared to the general case. On the other hand, we prove the optimality of our approximation upper bounds under the Unique Games Conjecture or a variant. In the second part of the thesis, we study mainly the approximation hardness of restricted instances of selected global optimization problems. We establish improved or in some cases the first inapproximability thresholds for the problems considered in this thesis such as the METRIC DIMENSION problem restricted to graphs with maximum degree 3 and the (1,2)-STEINER TREE problem. We introduce a new reductions method for proving explicit approximation lower bounds for problems that are related to the TRAVELING SALESPERSON (TSP) problem. In particular, we prove the best up to now inapproximability thresholds for the general METRIC TSP problem, the ASYMMETRIC TSP problem, the SHORTEST SUPERSTRING problem, the MAXIMUM TSP problem and TSP problems with bounded metrics

Approximate solution of NP optimization problems

AbstractThis paper presents the main results obtained in the field of approximation algorithms in a unified framework. Most of these results have been revisited in order to emphasize two basic tools useful for characterizing approximation classes, that is, combinatorial properties of problems and approximation preserving reducibilities. In particular, after reviewing the most important combinatorial characterizations of the classes PTAS and FPTAS, we concentrate on the class APX and, as a concluding result, we show that this class coincides with the class of optimization problems which are reducible to the maximum satisfiability problem with respect to a polynomial-time approximation preserving reducibility

Combinatorial and Probabilistic Approaches to Motif Recognition

Short substrings of genomic data that are responsible for biological processes, such as gene expression, are referred to as motifs. Motifs with the same function may not entirely match, due to mutation events at a few of the motif positions. Allowing for non-exact occurrences significantly complicates their discovery. Given a number of DNA strings, the motif recognition problem is the task of detecting motif instances in every given sequence without knowledge of the position of the instances or the pattern shared by these substrings. We describe a novel approach to motif recognition, and provide theoretical and experimental results that demonstrate its efficiency and accuracy. Our algorithm, MCL-WMR, builds an edge-weighted graph model of the given motif recognition problem and uses a graph clustering algorithm to quickly determine important subgraphs that need to be searched further for valid motifs. By considering a weighted graph model, we narrow the search dramatically to smaller problems that can be solved with significantly less computation. The Closest String problem is a subproblem of motif recognition, and it is NP-hard. We give a linear-time algorithm for a restricted version of the Closest String problem, and an efficient polynomial-time heuristic that solves the general problem with high probability. We initiate the study of the smoothed complexity of the Closest String problem, which in turn explains our empirical results that demonstrate the great capability of our probabilistic heuristic. Important to this analysis is the introduction of a perturbation model of the Closest String instances within which we provide a probabilistic analysis of our algorithm. The smoothed analysis suggests reasons why a well-known fixed parameter tractable algorithm solves Closest String instances extremely efficiently in practice. Although the Closest String model is robust to the oversampling of strings in the input, it is severely affected by the existence of outliers. We propose a refined model, the Closest String with Outliers problem, to overcome this limitation. A systematic parameterized complexity analysis accompanies the introduction of this problem, providing a surprising insight into the sensitivity of this problem to slightly different parameterizations. Through the application of probabilistic and combinatorial insights into the Closest String problem, we develop sMCL-WMR, a program that is much faster than its predecessor MCL-WMR. We apply and adapt sMCL-WMR and MCL-WMR to analyze the promoter regions of the canola seed-coat. Our results identify important regions of the canola genome that are responsible for specific biological activities. This knowledge may be used in the long-term aim of developing crop varieties with specific biological characteristics, such as being disease-resistant

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

28th Annual Symposium on Combinatorial Pattern Matching : CPM 2017, July 4-6, 2017, Warsaw, Poland

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