Weighted stability number of graphs and weighted satisfiability: The two facets of pseudo-Boolean optimization

We exhibit links between pseudo-Boolean optimization, graph theory and logic. We show the equivalence of maximizing a pseudo-Boolean function and finding a maximum weight stable set; symmetrically minimizing a pseudo-Boolean function is shown to be equivalent to solving a weighted satisfiability problem.     

Robust maximum weighted independent-set problems on interval graphs

We study the maximum weighted independent-set problem on interval graphs with uncertainty on the vertex weights. We use the absolute robustness criterion and the min–max regret criterion to evaluate solutions. For a discrete scenario set, we find that the problem is NP-hard for each of the robustness criteria; we also provide pseudo-polynomial time algorithms when there is a constant number of scenarios and show that the problem is strongly NP-hard when the set of scenarios is unbounded. When the scenario set is a Cartesian product, we prove that the problem is equivalent to a maximum weighted independent-set problem on the same interval graph but without uncertainty for the first objective function and that the scenario set can be reduced for the second objective function.     

最小最大树划分的近似算法与最小和树划分的精确算法

本文研究把连通赋权图的点集划分成p个子集,要求每个点子集的导出子图都连通,并且使得所得到的p个子图的最小支撑树中权重最大者的权重达到最小(最小最大树划分问题),或者使得所得到的p个子图的最小支撑树权重之和达到最小(最小和树划分问题)．文中给出了最小最大树划分问题的强NP困难性证明,并给出了一个多项式时间算法,该算法是最小最大树划分问题的竞争比为p的近似算法,同时是最小和树划分问题的精确算法．     

An algorithm and upper bounds for the weighted maximal planar graph problem

In this paper, we investigate the weighted maximal planar graph (WMPG) problem. Given a complete, edge-weighted, simple graph, the WMPG problem involves finding a subgraph with the highest sum of edge weights that is maximal planar, namely, it can be embedded in the plane without any of its edges intersecting, and no additional edge can be added to the subgraph without violating its planarity. We present a new integer linear programming (ILP) model for this problem. We then develop a cutting-plane algorithm to solve the WMPG problem based on the proposed ILP model. This algorithm enables the problem to be solved more efficiently than previously reported algorithms. New upper bounds are also provided, which are useful in evaluating the quality of heuristic solutions or in generating initial solutions for meta-heuristics. Computational results are reported for a set of 417 test instances of size varying from 6 to 100 nodes including 105 instances from the literature and 312 randomly generated instances. The computational results indicate that instances with up to 24 nodes can be solved optimally in reasonable computational time and the new upper bounds for larger instances significantly improve existing upper bounds.     

Distributed Online Frequency Assignment in Cellular Networks

A cellular network is generally modeled as a subgraph of the triangular lattice. The distributed online frequency assignment problem can be abstracted as a multicoloring problem on a weighted graph, where the weight vector associated with the vertices models the number of calls to be served at the vertices and is assumed to change over time. In this paper, we develop a framework for studying distributed online frequency assignment in cellular networks. We present the first distributed online algorithms for this problem with proven bounds on their competitive ratios. We show a series of algorithms that use at each vertex information about increasingly larger neighborhoods of the vertex, and that achieve better competitive ratios. In contrast, we show lower bounds on the competitive ratios of some natural classes of online algorithms.     

Multiple-Load Truss Topology and Sizing Optimization: Some Properties of Minimax Compliance

This paper considers the mathematical properties of discrete or discretized mechanical structures under multiple loadings which are optimal w.r.t. maximal stiffness. We state a topology and/or sizing problem of maximum stiffness design in terms of element volumes and displacements. Multiple loads are handled by minimizing the maximum of compliance of all load cases, i.e., minimizing the maximal sum of displacements along an applied force. Generally, the problem considered may contain constraints on the design variables. This optimization problem is first reformulated in terms of only design variables. Elastic equilibrium is hidden in potential energy terms. It is shown that this transformed objective function is convex and continuous, including infinite values. We deduce that maximum stiffness structures are dependent continuously on the bounds of the element volumes as parameters. Consequently, solutions to sizing problems with small positive lower bounds on the design variables can be considered as good approximations of solutions to topology problems with zero lower bounds. This justifies heuristic approaches such as the well-known stress-rationing method for solving truss topology problems.     

Certain generalization of the maximum traveling salesman problem

The problem is considered of finding in a complete undirected weighted graph a connected spanning subgraph with the given degrees of the vertices having maximum total weight of the edges. An approximate polynomial algorithm is presented for this problem. The algorithm is analyzed, and some upper bounds of its error are established in the general case as well as in the metric and Euclidean cases.     

Approximation hardness of min-max tree covers

We prove the first inapproximability bounds to study approximation hardness for a min-max k-tree cover problem and its variants. The problem is to find a set of k trees to cover vertices of a given graph with metric edge weights, so as to minimize the maximum total edge weight of any of the k trees. Our technique can also be applied to improve inapproximability bounds for min-max problems that use other covering objectives, such as stars, paths, and tours.     

A branch and bound algorithm for the maximum diversity problem

This article begins with a review of previously proposed integer formulations for the maximum diversity problem (MDP). This problem consists of selecting a subset of elements from a larger set in such a way that the sum of the distances between the chosen elements is maximized. We propose a branch and bound algorithm and develop several upper bounds on the objective function values of partial solutions to the MDP. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the medium-sized instances (with 50 elements) as well as some large-sized instances (with 100 elements). We compare our method with the best previous linear integer formulation solved with the well-known software Cplex. The comparison favors the proposed procedure.     

Gap functions and error bounds for nonsmooth convex vector optimization problem

Abstract

In this article, our main aim is to develop gap functions and error bounds for a (non-smooth) convex vector optimization problem. We show that by focusing on convexity we are able to quite efficiently compute the gap functions and try to gain insight about the structure of set of weak Pareto minimizers by viewing its graph. We will discuss several properties of gap functions and develop error bounds when the data are strongly convex. We also compare our results with some recent results on weak vector variational inequalities with set-valued maps, and also argue as to why we focus on the convex case.     

混合图上最小-最大圈覆盖问题的近似算法

考虑一个混合图上的最小-最大圈覆盖问题。给定一个正整数k和一个混合加权图G=（V,E,A）,这里V表示顶点集,E表示边集,A表示弧集。E中的每条边和A中的每条弧关联一个权重。问题的要求是确定k个环游,使得这k个环游能够经过A中的所有弧。目标是极小化最大环游的权重。该问题是运筹学和计算机科学中一个重要的组合优化问题,它和它的变形在诸如快递配送、垃圾收集、积雪清扫等相关行业具有广泛应用。针对该问题,通过结合二分搜索和环游撕裂的技巧,首次给出了一个近似比为37/5的近似算法。     

Performance Bounds with Curvature for Batched Greedy Optimization

The batched greedy strategy is an approximation algorithm to maximize a set function subject to a matroid constraint. Starting with the empty set, the batched greedy strategy iteratively adds to the current solution set a batch of elements that results in the largest gain in the objective function while satisfying the matroid constraints. In this paper, we develop bounds on the performance of the batched greedy strategy relative to the optimal strategy in terms of a parameter called the total batched curvature. We show that when the objective function is a polymatroid set function, the batched greedy strategy satisfies a harmonic bound for a general matroid constraint and an exponential bound for a uniform matroid constraint, both in terms of the total batched curvature. We also study the behavior of the bounds as functions of the batch size. Specifically, we prove that the harmonic bound for a general matroid is nondecreasing in the batch size and the exponential bound for a uniform matroid is nondecreasing in the batch size under the condition that the batch size divides the rank of the uniform matroid. Finally, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.     

Unrelated machine scheduling with time-window and machine downtime constraints: An application to a naval battle-group problem

This paper deals with an unrelated machine scheduling problem of minimizing the total weighted flow time, subject to time-window job availability and machine downtime side constraints. We present a zero-one integer programming formulation of this problem. The linear programming relaxation of this formulation affords a tight lower bound and often generates an integer optimal solution for the problem. By exploiting the special structures inherent in the formulation, we develop some classes of strong valid inequalities that can be used to tighten the initial formulation, as well as to provide cutting planes in the context of a branch-and-cut procedure. A major computational bottleneck is the solution of the underlying linear programming relaxation because of the extremely high degree of degeneracy inherent in the formulation. In order to overcome this difficulty, we employ a Lagrangian dual formulation to generate lower and upper bounds and to drive the branch-and-bound algorithm. As a practical instance of the unrelated machine scheduling problem, we describe a combinatorial naval defense problem. This problem seeks to schedule a set of illuminators (passive homing devices) in order to strike a given set of targets using surface-to-air missiles in a naval battle-group engagement scenario. We present computational results for this problem using suitable realistic data.     

On the optimum choice of weight functions in a class of variational calculations

Summary We consider the variational solution of a particular class of second order differential equations and show that expansions in terms of Chebychev and a range of ultraspherical polynomials lead to operator matrices that are asymptotically diagonal, and that hence their convergence properties can be completely characterised using a previously developed analysis. For a given class of weight functions bounds are given on the convergence of the coefficients and of the weighted mean square error, in terms of the analyticity properties of the coefficients in the differential equation. These bounds are used to discuss the optimum choice of weight function for such a calculation.     

Single machine scheduling with a general exponential learning effect

In this paper we consider the scheduling problem with a general exponential learning effect and past-sequence-dependent (p-s-d) setup times. By the general exponential learning effect, we mean that the processing time of a job is defined by an exponent function of the total weighted normal processing time of the already processed jobs and its position in a sequence, where the weight is a position-dependent weight. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the δ ? 0th power of completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Edge searching weighted graphs

In traditional edge searching one tries to clean all of the edges in a graph employing the least number of searchers. It is assumed that each edge of the graph initially has a weight equal to one. In this paper we modify the problem and introduce the Weighted Edge Searching Problem by considering graphs with arbitrary positive integer weights assigned to its edges. We give bounds on the weighted search number in terms of related graph parameters including pathwidth. We characterize the graphs for which two searchers are sufficient to clear all edges. We show that for every weighted graph the minimum number of searchers needed for a not-necessarily-monotonic weighted edge search strategy is enough for a monotonic weighted edge search strategy, where each edge is cleaned only once. This result proves the NP-completeness of the problem.     

Random algorithms for convex minimization problems

This paper deals with iterative gradient and subgradient methods with random feasibility steps for solving constrained convex minimization problems, where the constraint set is specified as the intersection of possibly infinitely many constraint sets. Each constraint set is assumed to be given as a level set of a convex but not necessarily differentiable function. The proposed algorithms are applicable to the situation where the whole constraint set of the problem is not known in advance, but it is rather learned in time through observations. Also, the algorithms are of interest for constrained optimization problems where the constraints are known but the number of constraints is either large or not finite. We analyze the proposed algorithm for the case when the objective function is differentiable with Lipschitz gradients and the case when the objective function is not necessarily differentiable. The behavior of the algorithm is investigated both for diminishing and non-diminishing stepsize values. The almost sure convergence to an optimal solution is established for diminishing stepsize. For non-diminishing stepsize, the error bounds are established for the expected distances of the weighted averages of the iterates from the constraint set, as well as for the expected sub-optimality of the function values along the weighted averages.     

Efficient solutions for weight-balanced partitioning problems

We prove polynomial-time solvability of a large class of clustering problems where a weighted set of items has to be partitioned into clusters with respect to some balancing constraints. The data points are weighted with respect to different features and the clusters adhere to given lower and upper bounds on the total weight of their points with respect to each of these features. Further the weight-contribution of a vector to a cluster can depend on the cluster it is assigned to. Our interest in these types of clustering problems is motivated by an application in land consolidation where the ability to perform this kind of balancing is crucial.Our framework maximizes an objective function that is convex in the summed-up utility of the items in each cluster. Despite hardness of convex maximization and many related problems, for fixed dimension and number of clusters, we are able to show that our clustering model is solvable in time polynomial in the number of items if the weight-balancing restrictions are defined using vectors from a fixed, finite domain. We conclude our discussion with a new, efficient model and algorithm for land consolidation.     

The Exact Weighted Independent Set Problem in Perfect Graphs and Related Classes

The exact weighted independent set (EWIS) problem consists in determining whether a given vertex-weighted graph contains an independent set of given weight. This problem is a generalization of two well-known problems, the NP-complete subset sum problem and the strongly NP-hard maximum weight independent set (MWIS) problem. Since the MWIS problem is polynomially solvable for some special graph classes, it is interesting to determine the complexity of this more general EWIS problem for such graph classes.We focus on the class of perfect graphs, which is one of the most general graph classes where the MWIS problem can be solved in polynomial time. It turns out that for certain subclasses of perfect graphs, the EWIS problem is solvable in pseudo-polynomial time, while on some others it remains strongly NP-complete. In particular, we show that the EWIS problem is strongly NP-complete for bipartite graphs of maximum degree three, but solvable in pseudo-polynomial time for cographs, interval graphs and chordal graphs, as well as for some other related graph classes.     

A priori bounds in weighted spaces

In unbounded domains we state some a priori bounds for solutions of the Dirichlet problem for linear second order elliptic differential equations in nondivergence form with discontinuous coefficients in weighted spaces. The weight function is related to the distance function from a fixed subset S of ∂Ω.