An Efficient Exact Algorithm for Constraint Bipartite Vertex Cover

The constraint bipartite vertex cover problem (CBVC for short) is as follows: given a bipartite graph G with n vertices and two positive integers k₁, k₂, is there a vertex cover taking at most k₁ vertices from one and at most k₂ vertices from the other vertex set of G? CBVC is NP-complete. It formalizes the spare allocation problem for reconfigurable arrays, an important problem from VLSI manufacturing. We provide a nontrivial so-called fixed parameter algorithm for CBVC, running in O(1.3999^{k₁ + k₂} + (k₁ + k₂)n) time. Our algorithm is efficient and practical for small values of k₁ and k₂, as occurring in applications. The analysis of the search tree is based on a novel bonus point system: after the processing of the search tree (which takes time exponential in k), a polynomial-time final analysis follows. Parts of the computation that would be normally done within the search-tree phase can be postponed; nevertheless, knowledge about the size of those parts can be used to reduce the length of the search paths (and hence the depth of the search tree as a whole) by a sort of bonus points.     

An efficient representation of Benes networks and its applications

The most popular bounded-degree derivative network of the hypercube is the butterfly network. The Benes network consists of back-to-back butterflies. There exist a number of topological representations that are used to describe butterfly—like architectures. We identify a new topological representation of butterfly and Benes networks.The minimum metric dimension problem is to find a minimum set of vertices of a graph G(V,E) such that for every pair of vertices u and v of G, there exists a vertex w with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. It is NP-hard in the general sense. We show that it remains NP-hard for bipartite graphs. The algorithmic complexity status of this NP-hard problem is not known for butterfly and Benes networks, which are subclasses of bipartite graphs. By using the proposed new representations, we solve the minimum metric dimension problem for butterfly and Benes networks. The minimum metric dimension problem is important in areas such as robot navigation in space applications.     

Exact algorithms and applications for Tree-like Weighted Set Cover

We introduce an NP-complete special case of the Weighted Set Cover problem and show its fixed-parameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant applications. More precisely, in this practically relevant variant we require that the given collection C of subsets of a base set S should be “tree-like”. That is, the subsets in C can be organized in a tree T such that every subset one-to-one corresponds to a tree node and, for each element s of S, the nodes corresponding to the subsets containing s induce a subtree of T. This is equivalent to the problem of finding a minimum edge cover in an edge-weighted acyclic hypergraph. Our main result is an algorithm running in O(3^kmn) time where k denotes the maximum subset size, n:=|S|, and m:=|C|. The algorithm also implies a fixed-parameter tractability result for the NP-complete Multicut in Trees problem, complementing previous approximation results. Our results find applications in computational biology in phylogenomics and for saving memory in tree decomposition based graph algorithms.     

An adaptive insertion algorithm for the single-vehicle dial-a-ride problem with narrow time windows

The dial-a-ride problem (DARP) is a widely studied theoretical challenge related to dispatching vehicles in demand-responsive transport services, in which customers contact a vehicle operator requesting to be carried from specified origins to specified destinations. An important subproblem arising in dynamic dial-a-ride services can be identified as the single-vehicle DARP, in which the goal is to determine the optimal route for a single vehicle with respect to a generalized objective function. The main result of this work is an adaptive insertion algorithm capable of producing optimal solutions for a time constrained version of this problem, which was first studied by Psaraftis in the early 1980s. The complexity of the algorithm is analyzed and evaluated by means of computational experiments, implying that a significant advantage of the proposed method can be identified as the possibility of controlling computational work smoothly, making the algorithm applicable to any problem size.     

An efficient evolutionary algorithm for the ring star problem

This paper addresses the ring star problem (RSP). The goal is to locate a cycle through a subset of nodes of a network aiming to minimize the sum of the cost of installing facilities on the nodes on the cycle, the cost of connecting them and the cost of assigning the nodes not on the cycle to their closest node on the cycle. A fast and efficient evolutionary algorithm is developed which is based on a new formulation of the RSP as a bilevel programming problem with one leader and two independent followers. The leader decides which nodes to include in the ring, one follower decides about the connections of the cycle and the other follower decides about the assignment of the nodes not on the cycle. The bilevel approach leads to a new form of chromosome encoding in which genes are associated to values of the upper level variables. The quality of each chromosome is evaluated by its fitness, by means of the objective function of the RSP. Hence, in order to compute the value of the lower level variables, two optimization problems are solved for each chromosome. The computational results show the efficiency of the algorithm in terms of the quality of the solutions yielded and the computing time. A study to select the best configuration of the algorithm is presented. The algorithm is tested on a set of benchmark problems providing very accurate solutions within short computing times. Moreover, for one of the problems a new best solution is found.     

The complexity of the weight problem for permutation and matrix groups

Given a metric d on a permutation group G, the corresponding weight problem is to decide whether there exists an element π∈G such that d(π,e)=k, for some given value k. Here we show that this problem is NP-complete for many well-known metrics. An analogous problem in matrix groups, eigenvalue-free problem, and two related problems in permutation groups, the maximum and minimum weight problems, are also investigated in this paper.     

Improved worst-case complexity for the MIN 3-SET COVERING problem

We consider MIN SET COVERING when the subsets are constrained to have maximum cardinality 3. We propose an exact algorithm whose worst-case complexity is bounded above by O^*(1.3957^m), where m is the number of sets in the instance. This result improves upon the previously known bound of O^*(1.4391^m).     

An efficient distributed algorithm for canonical labeling on directed split-stars

We consider the problem of canonical labeling in anonymous directed split-stars. This paper proposes a distributed algorithm for finding the vertex sets with specified leading symbols in directed split-stars and which has a linear message and constant time complexity. The algorithm runs on an asynchronous timing model without shared memory. In addition, our algorithm generalizes the previous distributed algorithms on directed split-stars that we know.     

二阶数乘问题的一个最优算法

研究一个有趣的组合优化问题——二阶数乘问题.问题描述如下：给定n≥2个正整数a_1,a_2,…,a_n,设π为{1,2,…,n}的一个置换,表示该问题的一个解,试图找到一个置换π以至∑_（i=1）~n a_（π_i）a_（π_（i＋1））最小,在这里π_（n＋1）=π_1.给出了一个算法复杂度为O（n log n）的最优算法.     

An algorithm for minimax

An algorithm is proposed for computing an unconstrained minimax, based on differential equations with suitable stabilization terms. Methods for accelerating the convergence are discussed. For computing a constrained minimax, the augmented Lagrangian algorithm of Powell, Hestenes and Rockafellar is generalized to minimax, assuming the unconstrained minimax algorithm as a subroutine. An estimate of the convergence rate is obtained.     

An infeasible-interior-point predictor-corrector algorithm for theP *-geometric LCP

AP _*-geometric linear complementarity problem (P _*GP) as a generalization of the monotone geometric linear complementarity problem is introduced. In particular, it contains the monotone standard linear complementarity problem and the horizontal linear complementarity problem. Linear and quadratic programming problems can be expressed in a “natural” way (i.e., without any change of variables) asP _*GP. It is shown that the algorithm of Mizunoet al. [6] can be extended to solve theP _*GP. The extended algorithm is globally convergent and its computational complexity depends on the quality of the starting points. The algorithm is quadratically convergent for problems having a strictly complementary solution. The work of F. A. Potra was supported in part by NSF Grant DMS 9305760     

A splitting type algorithm for multi-valued complementarity problems

We consider a generalized complementarity problem whose cost mapping is multi-valued and is the sum of upper Z and antitone mappings. We suggest a simple splitting type algorithm which utilizes an extended Jacobi iteration. Its convergence is proved under mild assumptions. Preliminary results of numerical experiments confirm efficiency of the algorithm presented. This work was supported by RFBR–NNSF Grant No. 07-01-92101.     

An efficient genetic algorithm for the traveling salesman problem with precedence constraints

The traveling salesman problem with precedence constraints (TSPPC) is one of the most difficult combinatorial optimization problems. In this paper, an efficient genetic algorithm (GA) to solve the TSPPC is presented. The key concept of the proposed GA is a topological sort (TS), which is defined as an ordering of vertices in a directed graph. Also, a new crossover operation is developed for the proposed GA. The results of numerical experiments show that the proposed GA produces an optimal solution and shows superior performance compared to the traditional algorithms.     

An ant colony algorithm for the pos/neg weighted p-median problem

Let a graph G = (V, E) with vertex set V and edge set E be given. The classical graph version of the p-median problem asks for a subset of cardinality p, so that the (weighted) sum of the minimum distances from X to all other vertices in V is minimized. We consider the semi-obnoxious case, where every vertex has either a positive or a negative weight. This gives rise to two different objective functions, namely the weighted sum of the minimum distances from X to the vertices in V\X and, differently, the sum over the minimum weighted distances from X to V\X. In this paper an Ant Colony algorithm with a tabu restriction is designed for both problems. Computational results show its superiority with respect to a previously investigated variable neighborhood search and a tabu search heuristic.This research has partially been supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung.     

An efficient algorithm for the three-guard problem

Given a simple polygon P with two vertices u and v, the three-guard problem asks whether three guards can move from u to v such that the first and third guards are separately on two boundary chains of P from u to v and the second guard is always kept to be visible from two other guards inside P. It is a generalization of the well-known two-guard problem, in which two guards move on the boundary chains from u to v and are always kept to be mutually visible. In this paper, we introduce the concept of link-2-ray shots, which can be considered as ray shots under the notion of link-2-visibility. Then, we show a one-to-one correspondence between the structure of the restrictions placed on the motion of two guards and the one placed on the motion of three guards, and generalize the solution for the two-guard problem to that for the three-guard problem. We can decide whether there exists a solution for the three-guard problem in O(nlogn) time, and if so generate a walk in O(nlogn+m) time, where n denotes the number of vertices of P and the size of the optimal walk. This improves upon the previous time bounds O(n²) and O(n²logn), respectively. Moreover, our results can be used to solve other more sophisticated geometric problems.     

Solving more linear complementarity problems with Murty's bard-type algorithm

The linear complementarity problem (M|q) is to findw andz inR ⁿ such thatw–Mz=q,w0,z0,w ^t z=0, givenM inR ^n×n andq in . Murty's Bard-type algorithm for solving LCP is modeled as a digraph.Murty's original convergence proof considered allq inR ⁿ andM inR ^n×n , aP-matrix. We show how to solve more LCP's by restricting the set ofq vectors and enlarging the class ofM matrices beyondP-matrices. The effect is that the graph contains an embedded graph of the type considered by Stickney and Watson wheneverM is a matrix containing a principal submatrix which is aP-matrix. Examples are presented which show what can happen when the hypotheses are further weakened.