Worst-case performance of approximation algorithms for tool management problems

Approximability of the Ground State Problem for Certain Ising Spin Glasses

Improving the performance of stochastic local search for maximum vertex weight clique problem using programming by optimization

Article Approximability and nonapproximability results for minimizing total flow time on a single machine Share on Authors: Hans Kellerer Institut für Statistik, Ökonometrie und Operations Research, Universität Graz, A-8010 Graz, Austria Institut für Statistik, Ökonometrie und Operations Research, Universität Graz, A-8010 Graz, AustriaView Profile , Thomas Tautenhahn Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg, D-39016 Magdeburg, Germany Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg, D-39016 Magdeburg, GermanyView Profile , Gerhard J. Woeginger Eindhoven University of Technology, Department of Mathematics and Computing Science, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands Eindhoven University of Technology, Department of Mathematics and Computing Science, P.O. Box 513, NL-5600 MB Eindhoven, The NetherlandsView Profile Authors Info & Claims STOC '96: Proceedings of the twenty-eighth annual ACM symposium on Theory of ComputingJuly 1996 Pages 418–426https://doi.org/10.1145/237814.237989Online:01 July 1996Publication History 54citation413DownloadsMetricsTotal Citations54Total Downloads413Last 12 Months3Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

In this paper, we give a comprehensive overview of the development of clique algorithms and their use for drug design based on the search for cliques in protein product graphs. The maximum clique problem is a computational problem of finding largest subsets of vertices in a graph that are all pairwise adjacent. A related problem is the maximum weight clique problem and the highest weight k-clique problem, which both extend the algorithm to weighted graphs. The review covers our developed algorithms, starting with our improved branch-and-bound algorithm for finding maximum cliques in undirected graphs from 2007 up to the recent developments of algorithms for weighted graphs in 2024. We show the application of these algorithms to early stages of drug discovery, in particular to protein binding site detection based on protein similarity search in large protein databases and to protein-ligand molecular docking.

Optimal parallel time bounds for the maximum clique problem on intervals

In communication networks such as social networks, wireless networks and biology networks, it is of importance to find all cliques which can help understand the network topology. The clique structures can also be utilized in facilitating message forwarding in wireless networks. For instance, using a set of cliques of maximal and disjoint, one of the nodes in each clique can be elected to forward messages, by which the duplicate message transmissions can be efficiently reduced. Further, it is well known that the maximum clique problem (MCP) is closely related to the maximum independent set and the vertex cover problems. In recent years, the fundamental problem of finding maximal cliques or maximum cliques has attracted lots of attentions. However, few of these works focus on distributed solutions in wireless networks. In this paper, we pay our attention to this missing corner of research. Specifically, we first give a distributed algorithm which can compute all maximal cliques in a wireless network represented by a graph. The algorithm takes O(n) time and uses O(mn) messages, where n is the number of nodes and m the number of edges. Then, with the proposed algorithms MCP (maximum clique problem) and UMCP (unique MCP), we show that a unique maximum clique can be selected from all maximal cliques in O(n) rounds and using O(mn) messages. To the best of our knowledge, our algorithms are the first deterministic distributed solutions for MCP in wireless networks.

Scheduling jobs with equal-processing-time on parallel machines with non-identical capacities to minimize makespan

Given a multiset X={x1 ,. . .,xn} of real numbers, the floating-point set summation problem asks for Sn=x1 + . . . + xn. Let $E^*_n$ denote the minimum worst-case error over all possible orderings of evaluating Sn. We prove that if X has both positive and negative numbers, it is NP-hard to compute Sn with the worst-case error equal to $E^*_n$. We then give the first known polynomial-time approximation algorithm that has a provably small error for arbitrary X. Our algorithm incurs a worst-case error at most $2(\lceil\log(n-1)\rceil+1)E^*_n$. (All logarithms log in this paper are base 2.) After X is sorted, it runs in O(n) time. For the case where X is either all positive or all negative, we give another approximation algorithm with a worst-case error at most $\lceil\log\log n\rceil E^*_n$. Even for unsorted X, this algorithm runs in O(n) time. Previously, the best linear-time approximation algorithm had a worst-case error at most $\lceil\log n\rceil E^*_n$, while $E^*_n$ was known to be attainable in O(n log n) time using Huffman coding.

This chapter is devoted to some algorithm design techniques which became known by the term heuristics. The term heuristic in the area of combinatorial optimization is not unambiguously specified and so it is used with different meanings. A heuristic algorithm in a very general sense is a consistent algorithm for an optimization problem that is based on some transparent (usually simple) strategy (idea) of searching in the set of all feasible solutions, and that does not guarantee finding any optimal solution. In this context people speak about local search heuristics, or a greedy heuristic, even when this heuristic technique results in an approximation algorithm. In a narrow sense a heuristic is a technique providing a consistent algorithm for which nobody is able to prove that it provides feasible solutions of a reasonable quality in a reasonable (for instance, polynomial) time, but the idea of the heuristic seems to promise good behavior for typical instances of the optimization problem considered. Thus, a polynomial-time approximation algorithm cannot be considered as a heuristic in this sense, independently of the simplicity of its design idea. Observe that the description of a heuristic in this narrow sense is a relative term because an algorithm can be considered to be a heuristic one while nobody is able to analyze its behavior. But after proving some reasonable bounds on its complexity and the quality of the produced solutions (even with an error-bounded probability in the randomized case), this algorithm becomes a (randomized) approximation algorithm and is not considered to be a heuristic any more.

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L⊆N. We investigate how well L-cycle covers of minimum weight can be approximated. For undirected graphs, we devise a polynomial-time approximation algorithm that achieves a constant approximation ratio for all sets L. On the other hand, we prove that the problem cannot be approximated within a factor of 2 - Ɛ for certain sets L. For directed graphs, we present a polynomial-time approximation algorithm that achieves an approximation ratio of O(n), where n is the number of vertices. This is asymptotically optimal: We show that the problem cannot be approximated within a factor of o(n). To contrast the results for cycle covers of minimum weight, we show that the problem of computing L-cycle covers of maximum weight can, at least in principle, be approximated arbitrarily well.

The complexity of minimum convex coloring

Given a set ofn points on the plane, the rectilinearm-center problem is to findn rectilinear squares covering all thesen points such that the maximum side length of these squares is minimized. In this paper we prove that there is no polynomial-time algorithm with an error ratio ɛ < 2 for the rectilinearm-center problem unless NP = P. A polynomial-time approximation algorithm with an error ratio of 2 is also proposed.

The maximum clique problem is a classical combinatorial optimization problem which is one of the first problems shown to be NP-complete. Moreover, it does not admit a polynomial-time approximation algorithm unless P=NP. So many heuristic algorithms have been proposed for this problem. In this paper, we introduce an improved ant colony optimization (ACO) for the maximum clique problem. In the proposed algorithm, both a new pheromone updating method and a parameter tuning method are introduced. Simulations are performed on some graphs of the DIMACS clique instances in the second DIMACS challenge. The results show that the improved ant colony optimization can yield satisfactory results.

Graph pattern mining is an important part of the emerging social network science, and the research of the maximum clique problem is one of the most important research branches. In the big data environment, the mass of nodes and complexity of edges in the graph set a higher requirement on the speed and accuracy of the maximum clique (MCP) study. In this paper, we present a parallel graph partitioning framework for solving the maximum clique problem(PPMC) based on Hadoop. Firstly, This paper presents a new graph partition method based on degree sorting(GPD), the subgraph scale is greatly reduced. In order to balance the workload of solving the maximum clique problems. Secondly, the improved ant algorithm proposed by our team is used to solve the maximum clique of each subgraph. Finally, the framework is deployed on Hadoop distributed cloud computing platform and tested by Stanford large-scale datasets. The experimental results show that GPD algorithm can greatly reduce the subgraph scale, the subgraph number, and the complexity of time and space of solving the maximum clique problem of large-scale instances. And the efficiency and scalability of the Hadoop based parallel graph partitioning framework for solving the maximum clique problems has been further proved.

We consider the routing open shop problem being a generalization of the open shop and the metric travelling salesman problems. The jobs are located at nodes of some transportation network, and the machines travel on the network to execute the jobs in the open shop environment. The machines are initially located at the same node (depot) and must return to the depot after completing all the jobs. It is required to find a non-preemptive schedule that minimizes the makespan. The problem is NP-hard even on a two-node network with two machines. We present new polynomial-time approximation algorithms with worst-case performance guarantees.KeywordsRouting open shopapproximation algorithmworst-case analysis