Superstrings with multiplicities

A superstring of a set of words P = s1, · · · , sp is a string that contains each word of P as substring. Given P, the well known Shortest Linear Superstring problem (SLS), asks for a shortest superstring of P. In a variant of SLS, called Multi-SLS, each word si comes with an integer m(i), its multiplicity, that sets a constraint on its number of occurrences, and the goal is to find a shortest superstring that contains at least m(i) occurrences of si. Multi-SLS generalizes SLS and is obviously as hard to solve, but it has been studied only in special cases (with words of length 2 or with a fixed number of words). The approximability of Multi-SLS in the general case remains open. Here, we study the approximability of Multi-SLS and that of the companion problem Multi-SCCS, which asks for a shortest cyclic cover instead of shortest superstring. First, we investigate the approximation of a greedy algorithm for maximizing the compression offered by a superstring or by a cyclic cover: the approximation ratio is 1/2 for Multi-SLS and 1 for Multi-SCCS. Then, we exhibit a linear time approximation algorithm, Concat-Greedy, and show it achieves a ratio of 4 regarding the superstring length. This demonstrates that for both measures Multi-SLS belongs to the class of APX problems. © 2018 Yoshifumi Sakai; licensed under Creative Commons License CC-BY.Peer reviewe

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

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Approximation of Greedy Algorithms for Max-ATSP, Maximal Compression, Maximal Cycle Cover, and Shortest Cyclic Cover of Strings

International audienceCovering a directed graph by a Hamiltonian path or a set of words by a superstring belong to well studied optimisation problems that prove difficult to approx-imate. Indeed, the Maximum Asymmetric Travelling Salesman Problem (Max-ATSP), which asks for a Hamiltonian path of maximum weight covering a digraph, and the Shortest Superstring Problem (SSP), which, for a finite language P := {s 1 , . . . , s p }, searches for a string of minimal length having each input word as a substring, are both Max-SNP hard. Finding a short superstring requires to choose a permutation of words and the associated overlaps to minimise the superstring length or to maximise the compression of P . Hence, a strong relation exists between Max-ATSP and SSP since solving Max-ATSP on the Overlap Graph for P gives a shortest superstring. Numerous works have designed algorithms that improve the approximation ratio but are increasingly complex. Often, these rely on solving the pendant problems where the cover is made of cycles instead of single path (Max-CC and SCCS). Finally, the greedy algorithm remains an attractive solution for its simplicity and ease of implementation. Its approximation ratios have been obtained by different approaches. In a seminal but complex proof, Tarhio and Ukkonen showed that it achieves 1/2 compression ratio for Max-CC. Here, using the full power of subset systems, we provide a unified approach for proving simply the approximation ratio of a greedy algorithm for these four prob-lems. Especially, our proof for Maximal Compression shows that the Monge property suffices to derive the 1/2 tight bound

Examination scheduling using the ant system.

This work is concerned with heuristic approaches to examination timetabling. It is demonstrated that a relatively new evolutionary method, the Ant System, can be the basis of a successful two-phase solution method. The first phase exploits ant feedback in order both to produce large volumes of feasible timetables and to optimise secondary objectives. The second phase acts as a repair facility where solution quality is improved further while maintaining feasibility. This is accomplished without increasing computational effort to unrealistic levels. The work builds on an existing implementation for the graph colouring problem, the natural model for examination scheduling. It is demonstrated that by adjusting the graph model to allow the accommodation of several side constraints as well incorporating enhancement techniques within the algorithm itself, the Ant System algorithm becomes very effective at producing feasible timetables. The enhancements include a diversification function, new reward functions and trail replenishment tactics. It is observed that the achievement of second-order objectives can be enhanced through a variety of means. A modified elitist strategy (ERF) significantly improves the performance of the Ant System due to the extra emphasis on second-order feedback. It is also shown that through the incorporation of the ERF, trail limits and, in particular, 19th century evolutionary theory the area of the solution space explored by the ants during the infancy of the search can be reduced. In addition, a good level of exploration is maintained as the search matures. This balance between exploration and exploitation is the main determinant of solution quality. The use of a repair facility, as is common practice with evolutionary algorithms, encourages fitter solutions. The interaction between Lamarckian evolution and searching in an extended neighbourhood through the graph theoretic concept of Kempe chains leads to better overall solutions