We explore the complexity and exact computation of a variant of the classical stable marriage problem in which we seek matchings that are not only stable, but are also "fair" in a formal sense. In particular...
详细信息
We explore the complexity and exact computation of a variant of the classical stable marriage problem in which we seek matchings that are not only stable, but are also "fair" in a formal sense. In particular, we study the sex-equal stable marriage problem (SESM), in which, roughly speaking, we wish to find a stable matching with the property that the men's happiness is as close as possible to the women's happiness. This problem is known to be strongly NP-hard (Kato in Jpn. J. Ind. Appl. Math. 10:1-19, 1993). We specifically consider SESM instances in which the preference lists of the men and/or women are bounded in length by a constant. On the negative side, we show that SESM is NP-hard, even if both the men's and women's preference lists are of length at most three, and is not even in the class XP when parameterized by the objective value of the solution. This strengthens the NP-hardness results of Kato (Jpn. J. Ind. Appl. Math. 10:1-19, 1993). On the positive side, we show that our hardness result is "tight" by giving a polynomial-time algorithm for the case in which the preference lists on one side (say the men) are of length at most two, and the lengths of the lists on the other side (the women) are unbounded. Furthermore, we give a low-order exponential-time algorithm for SESM in which the preference lists on one side are of length at most l (and the lengths of the lists on the other side are unbounded). In particular, for every pair of constants and there is an algorithm with running time bounded by . Hence, if I mu is chosen to be a sufficiently small constant, the running time is in O (a
In this paper, we address the line-capacitated minimum Steiner tree problem (the Lc-MStT problem, for short), which is a variant of the (Euclidean) capacitated minimum Steiner tree problem and defined as follows. Give...
详细信息
In this paper, we address the line-capacitated minimum Steiner tree problem (the Lc-MStT problem, for short), which is a variant of the (Euclidean) capacitated minimum Steiner tree problem and defined as follows. Given a set X = (r(1), r(2),...,r(n)} of n terminals in R-2, a demand function d : X -> N and a positive integer C, we are asked to determine the location of a line l and a Steiner tree T-l to interconnect these n terminals in X and at least one point located on this line l such that the total demand of terminals in each maximal subtree (of TO connected to the line l, where the terminals in such maximal subtree are all located at the same side of this line l, does not exceed the bound C. The objective is to minimize total weight Sigma(e is an element of Tl) w(e) of such a Steiner tree T-l among all line-capacitated Steiner trees mentioned-above, where weight w(e) = 0 if two endpoints of that edge e is an element of T-l are located on the line l and otherwise weight w(e) is the Euclidean distance between two endpoints of that edge e is an element of T-l . In addition, when this line l is as an input in R-2 and Sigma(r is an element of X) d(r) <= C holds, we refer to this version as the 1-line-fixed minimum Steiner tree problem (the 1Lf-MStT problem, for short). We obtain three main results. (1) Given a rho(st )-approximation algorithm to solve the Euclidean minimum Steiner tree problem and a rho(1Lf)-approximation algorithm to solve the 1Lf-MStT problem, respectively, we design a (rho(st)rho(1Lf )+2)-approximation algorithm to solve the Lc-MStT problem. (2) Whenever demand of each terminal r is an element of X is less than c/2, we provide a (rho(1Lf) + 2)-approximation algorithm to resolve the Lc-MStT problem. (3) Whenever demand of each terminal r is an element of X is at least c/2, using the Edmonds' algorithm to solve the minimum weight perfect matching as a subroutine, we present an exact algorithm in polynomial time to resolve the Lc-MStT problem.
The MAXIMUM SATISFIABILITY problem (MAXSAT) is a fundamental NP-hard problem which has significant applications in many areas. Based on refined observations, we derive a branching algorithm of running time O*(1.2989(m...
详细信息
The MAXIMUM SATISFIABILITY problem (MAXSAT) is a fundamental NP-hard problem which has significant applications in many areas. Based on refined observations, we derive a branching algorithm of running time O*(1.2989(m)) for the MAXSAT problem, where m denotes the number of clauses in the given CNF formula. Our algorithm considerably improves the previous best result O*(1.3248(m)) published in 2004. For our purpose, we derive improved branching strategies for variables of degrees 3, 4, and 5. The worst case of our branching algorithm is at certain degree-4 variables. To serve the branching rules, we also propose a variety of reduction rules which can be exhaustively applied in polynomial time.
Based on the vehicle distribution optimization problem with time window in postal logistics, this paper intends to apply various operations research optimization algorithms such as exact algorithms, tabu search algori...
详细信息
We consider stochastic problems in which both the objective function and the feasible set are affected by uncertainty. We address these problems using a K-adaptability approach, in which K solutions for a given proble...
详细信息
We consider stochastic problems in which both the objective function and the feasible set are affected by uncertainty. We address these problems using a K-adaptability approach, in which K solutions for a given problem are computed before the uncertainty dissolves and afterwards the best of them can be chosen for the realized scenario. We analyze the complexity of the resulting problem from a theoretical viewpoint, showing that, even in case the deterministic problem can be solved in polynomial time, deciding if a feasible solution exists is NP-hard for discrete probability distributions. Besides that, we prove that an approximation factor for the underlying problem can be carried over to our problem. Finally, we present exact approaches including a branch-and-price algorithm. An extensive computational analysis compares the performances of the proposed algorithms on a large set of randomly generated instances.
We consider the problem of scheduling jobs with equal lengths on uniform parallel batch machines with non-identical capacities where each job can only be processed on a specified subset of machines called its processi...
详细信息
We consider the problem of scheduling jobs with equal lengths on uniform parallel batch machines with non-identical capacities where each job can only be processed on a specified subset of machines called its processing set. For the case of equal release times, we give efficient exact algorithms for various objective functions. For the case of unequal release times, we give efficient exact algorithms for minimizing makespan.
We consider the problem of scheduling production jobs on a single machine with sequence dependent family setup times and individual job deadlines. Given a set of jobs, the goal is to minimize the total time to process...
详细信息
ISBN:
(纸本)9781665493130
We consider the problem of scheduling production jobs on a single machine with sequence dependent family setup times and individual job deadlines. Given a set of jobs, the goal is to minimize the total time to process all jobs while every job meets its deadline. We study algorithms that compute an exact solution to the problem. Motivated by one example use case, we exploit a natural structural observation that occurs in many production settings: the number of product configurations may be significantly smaller than the total number of jobs. We identify an algorithm that is efficient in this setting in terms of performance. We experimentally evaluate its running time and compare it with two other natural approaches of exact job scheduling.
NP-hard problems are infamous for having highly varying and extreme runtimes for different problem instances. This study quantifies the instance hardness of NP-hard protein folding problem instances within the HP-mode...
详细信息
ISBN:
(纸本)9798350310177
NP-hard problems are infamous for having highly varying and extreme runtimes for different problem instances. This study quantifies the instance hardness of NP-hard protein folding problem instances within the HP-model. A custom dataset is generated, consisting of 1000 problem instances of lengths 10, 15, 20, and 25, totalling to 4000 instances. The computational costs for solving a problem instance using a depth-first branch and bound algorithm are measured. The resulting hardness distributions can all be characterized by a probit function regardless of the corresponding protein length. Knowing the instance hardness distribution makes it possible to extrapolate the range of expected runtimes for larger problem instances. As a result, it allows for a prediction before folding a protein.
This survey investigates the field of moderate exponential-time algorithms for NP-hard scheduling problems, i.e., exact algorithms whose worst-case time complexity is moderately exponential with respect to brute force...
详细信息
This survey investigates the field of moderate exponential-time algorithms for NP-hard scheduling problems, i.e., exact algorithms whose worst-case time complexity is moderately exponential with respect to brute force algorithms. Scheduling problems are very challenging problems for which interesting results have emerged in the literature since 2010. We will provide a comprehensive overview of the known results of these problems before detailing three general techniques to derive moderate exponential-time algorithms. These techniques are Sort&Search, Inclusion-Exclusion and Branching. In the last part of this survey, we will focus on side topics such as approximation in moderate exponential time, the design of lower bounds on worst-case time complexities or fixed-parameter tractability. We will also discuss the potential benefits of moderate exponential-time algorithms for efficiently solving in practice scheduling problems.
This paper studies binary quadratic programs in which the objective is defined by the maximisation of a Euclidean distance matrix, subject to a general polyhedral constraint set. This class of nonconcave maximisation ...
详细信息
This paper studies binary quadratic programs in which the objective is defined by the maximisation of a Euclidean distance matrix, subject to a general polyhedral constraint set. This class of nonconcave maximisation problems, which we refer to as the Euclidean Max -Sum problem, includes the capacitated, generalised and maxsum diversity problems as special cases. Due to the nonconcave objective, traditional cutting plane algorithms are not guaranteed to converge globally. In this paper, we introduce two exact cutting plane algorithms to address this limitation. The new algorithms remove the need for a concave reformulation, which is known to significantly slow down convergence. We establish exactness of the new algorithms by examining the concavity of the quadratic objective in a given direction, a concept we refer to as directional concavity . Numerical results show that the algorithms outperform other exact methods for benchmark diversity problems (capacitated, generalised and max -sum), and can easily solve problems of up to three thousand variables.
暂无评论