The Nash rationing problem

This paper studies the problem of allocating utility losses among n agents with cardinal non-comparable utility functions. This problem is referred to as the Nash rationing problem, as it can be regarded as the translation of the Nash bargaining problem to a rationing scenario. We show that there is no single-valued solution satisfying the obvious reformulation of Nash’s axioms, nor a multivalued solution satisfying a certain extension of these axioms. However, there is a multivalued solution that is characterised by an appropriate extension of the axioms. We thus call this mapping the Nash rationing solution. It associates with each rationing problem the set of points that maximises a weighted sum of utilities, in which weights are chosen so that all agents’ weighted losses are equal.We are grateful to Carmen Herrero, Paola Manzini, Karl Schlag, William Thomson, Fernando Vega-Redondo and two careful referees for useful comments. Financial support form the Spanish Ministerio de Ciencia y Tecnología, under project SEJ2004-08011ECON, and the Generalitat Valenciana, are gratefuly acknowledged.     

A fundamental problem in linear inequalities with applications to the travelling salesman problem

We consider a system ofm linearly independent equality constraints inn nonnegative variables:Ax = b, x 0. The fundamental problem that we discuss is the following: suppose we are given a set ofr linearly independent column vectors ofA, known asthe special column vectors. The problem is to develop an efficient algorithm to determine whether there exists a feasible basis which contains all the special column vectors as basic column vectors and to find such a basis if one exists. Such an algorithm has several applications in the area of mathematical programming. As an illustration, we show that the famous travelling salesman problem can be solved efficiently using this algorithm. Recent published work indicates that this algorithm has applications in integer linear programming. An algorithm for this problem using a set covering approach is described.This research has been partially supported by the ISDOS research project and the National Science Foundation under Grant GK-27872 with the University of Michigan.     

A cutting plane algorithm for the capacitated facility location problem

The Capacitated Facility Location Problem (CFLP) is to locate a set of facilities with capacity constraints, to satisfy at the minimum cost the order-demands of a set of clients. A multi-source version of the problem is considered in which each client can be served by more than one facility. In this paper we present a reformulation of the CFLP based on Mixed Dicut Inequalities, a family of minimum knapsack inequalities of a mixed type, containing both binary and continuous (flow) variables. By aggregating flow variables, any Mixed Dicut Inequality turns into a binary minimum knapsack inequality with a single continuous variable. We will refer to the convex hull of the feasible solutions of this minimum knapsack problem as the Mixed Dicut polytope. We observe that the Mixed Dicut polytope is a rich source of valid inequalities for the CFLP: basic families of valid CFLP inequalities, like Variable Upper Bounds, Cover, Flow Cover and Effective Capacity Inequalities, are valid for the Mixed Dicut polytope. Furthermore we observe that new families of valid inequalities for the CFLP can be derived by the lifting procedures studied for the minimum knapsack problem with a single continuous variable. To deal with large-scale instances, we have developed a Branch-and-Cut-and-Price algorithm, where the separation algorithm consists of the complete enumeration of the facets of the Mixed Dicut polytope for a set of candidate Mixed Dicut Inequalities. We observe that our procedure returns inequalities that dominate most of the known classes of inequalities presented in the literature. We report on computational experience with instances up to 1000 facilities and 1000 clients to validate the approach.     

The inverse problem for certain tree parameters

Let p be a graph parameter that assigns a positive integer value to every graph. The inverse problem for p asks for a graph within a prescribed class (here, we will only be concerned with trees), given the value of p. In this context, it is of interest to know whether such a graph can be found for all or at least almost all integer values of p. We will provide a very general setting for this type of problem over the set of all trees, describe some simple examples and finally consider the interesting parameter “number of subtrees”, where the problem can be reduced to some number-theoretic considerations. Specifically, we will prove that every positive integer, with only 34 exceptions, is the number of subtrees of some tree.     

Extended formulations for the A-cut problem

LetG=(V, E) be an undirected graph andA⊆V. We consider the problem of finding a minimum cost set of edges whose deletion separates every pair of nodes inA. We consider two extended formulations using both node and edge variables. An edge variable formulation has previously been considered for this problem (Chopra and Rao (1991), Cunningham (1991)). We show that the LP-relaxations of the extended formulations are stronger than the LP-relaxation of the edge variable formulation (even with an extra class of valid inequalities added). This is interesting because, while the LP-relaxations of the extended formulations can be solved in polynomial time, the LP-relaxation of the edge variable formulation cannot. We also give a class of valid inequalities for one of the extended formulations. Computational results using the extended formulations are performed.     

On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G−I, there is a vertex w∈I such that {u,v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NP.     

The counting complexity of a simple scheduling problem

Let T be a set of tasks. Each task has a non-negative processing time and a deadline. The problem of determining whether or not there is a schedule of the tasks in T such that a single machine can finish processing each of them before its deadline is polynomially solvable. We prove that counting the number of schedules satisfying this condition is #P-complete.     

Exploring the constrained maximum edge-weight connected graph problem

Given an edge weighted graph, the maximum edge-weight connected graph （MECG） is a connected subgraph with a given number of edges and the maximal weight sum. Here we study a special case, i.e. the Constrained Maximum Edge-Weight Connected Graph problem （CMECG）, which is an MECG whose candidate subgraphs must include a given set of k edges, then also called the k-CMECG. We formulate the k-CMECG into an integer linear programming model based on the network flow problem. The k-CMECG is proved to be NP-hard. For the special case 1-CMECG, we propose an exact algorithm and a heuristic algorithm respectively. We also propose a heuristic algorithm for the k-CMECG problem. Some simulations have been done to analyze the quality of these algorithms. Moreover, we show that the algorithm for 1-CMECG problem can lead to the solution of the general MECG problem.     

An unfeasible matching problem

LetG be a bipartite graph with natural edge weights, and letW be a function from the set of vertices ofG into natural numbers. AW-matching ofG is a subset of the set of edges ofG such that for each vertexv the total weight of edges in the subset incident tov does not exceedW(v). Letm be a natural number. We show that the problem of deciding whether there is aW-matching inG whose total weight is not less thanm is NP-complete even ifG is bipartite and its edge weights as well as theW(v)-constraints are constantly bounded.     

A branch-and-cut method for the obnoxious p-median problem

The obnoxious p-median (OpM) problem is the repulsive counterpart of the ore known attractive p-median problem. Given a set I of cities and a set J of possible locations for obnoxious plants, a p-cardinality subset Q of J is sought, such that the sum of the distances between each city of I and the nearest obnoxious site in Q is maximised. We formulate (OpM) as a {0,1} linear programming problem and propose three families of valid inequalities whose separation problem is polynomial. We describe a branch-and-cut approach based on these inequalities and apply it to a set of instances found in the location literature. The computational results presented show the effectiveness of these inequalities for (OpM). The work of the first author has been partially supported by the Coordinated Project C.A.M.P.O. and that of the third author by a short mobility grant, both of the Italian National Research Council.     

Borsuk's problem

The Borsuk number of a bounded set F is the smallest natural number k such that F can be represented as a union of k sets, the diameter of each of which is less than diam F. In this paper we solve the problem of finding the Borsuk number of any bounded set in an arbitrary two-dimensional normed space (the solution is given in terms of the enlargement of a set to a figure of constant width). We indicate spaces for which the solution of Borsuk's problem has the same form as in the Euclidean plane.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 621–631, November, 1977.     

A maximumb-matching problem arising from median location models with applications to the roommates problem

We consider maximumb-matching problems where the nodes of the graph represent points in a metric space, and the weight of an edge is the distance between the respective pair of points. We show that if the space is either the rectilinear plane, or the metric space induced by a tree network, then theb-matching problem is the dual of the (single) median location problem with respect to the given set of points. This result does not hold for the Euclidean plane. However, we show that in this case theb-matching problem is the dual of a median location problem with respect to the given set of points, in some extended metric space. We then extend this latter result to any geodesic metric in the plane. The above results imply that the respective fractionalb-matching problems have integer optimal solutions. We use these duality results to prove the nonemptiness of the core of a cooperative game defined on the roommate problem corresponding to the above matching model. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

The superstar packing problem

Hell and Kirkpatrick proved that in an undirected graph, a maximum size packing by a set of non-singleton stars can be found in polynomial time if this star-set is of the form {S ₁, S ₂, ..., S _k } for some k∈ℤ₊ (S _i is the star with i leaves), and it is NP-hard otherwise. This may raise the question whether it is possible to enlarge a set of stars not of the form {S ₁, S ₂, ..., S _k } by other non-star graphs to get a polynomially solvable graph packing problem. This paper shows such families of depth 2 trees. We show two approaches to this problem, a polynomial alternating forest algorithm, which implies a Berge-Tutte type min-max theorem, and a reduction to the degree constrained subgraph problem of Lovász. Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No. 504438.     

The inverse optimal value problem

This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost vectors, determine a cost vector such that the corresponding optimal objective value of the linear program is closest to the desired value. The above problem, referred here as the inverse optimal value problem, is significantly different from standard inverse optimization problems that involve determining a cost vector for a linear program such that a pre-specified solution vector is optimal. In this paper, we show that the inverse optimal value problem is NP-hard in general. We identify conditions under which the problem reduces to a concave maximization or a concave minimization problem. We provide sufficient conditions under which the associated concave minimization problem and, correspondingly, the inverse optimal value problem is polynomially solvable. For the case when the set of feasible cost vectors is polyhedral, we describe an algorithm for the inverse optimal value problem based on solving linear and bilinear programming problems. Some preliminary computational experience is reported.Mathematics Subject Classification (1999):49N45, 90C05, 90C25, 90C26, 90C31, 90C60Acknowledgement This research has been supported in part by the National Science Foundation under CAREER Award DMII-0133943. The authors thank two anonymous reviewers for valuable comments.     

A new algorithm for the quasi-assignment problem

The quasi-assignment problem can be used to solve the bus scheduling problem, the tourist guide problem, and the minimum number of chains in a partially ordered set. A successive shortest path algorithm for the assignment problem is extended to the quasiassignment problem. The algorithm is a variation of the primal-dual algorithm, and its computational complexity isO(n ³).The research for this paper was partly supported by the Chinese National Science Foundation.     

Global classical solution of quasi-stationary Stefan free boundary problem

We consider an one-phase quasi-stationary Stefan problem (Hele–Shaw problem) in multidimensional case. Under some reasonable conditions we prove that the problem has a classical solution globally in time. The method can be used in two-phase problem as well. We also discuss asymptotic behavior of solution as t→+∞. The method developed here can be extended to a general class of free boundary problems.     

Solvability of the vector problem by the linear criteria convolution algorithm

Conditions are found under which a multicriteria problem with a finite set of vector estimates is solvable by means of the linear criteria convolution (LCC) algorithm, that is, any Pareto optimum for the problem can be obtained as an optimum solution to a one-criterion problem with an aggregated criterion defined as an LCC. Also, an algorithm is suggested that is polynomial in dimension and reduces any problem with minimax and minimin criteria to an equivalent vector problem with the same Pareto set solvable by the LCC algorithm. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 502–509, October, 1997. Translated by V. N. Dubrovsky     

An asymptotically exact algorithm for the high-multiplicity bin packing problem

The bin packing problem consists of finding the minimum number of bins, of given capacity D, required to pack a set of objects, each having a certain weight. We consider the high-multiplicity version of the problem, in which there are only C different weight values. We show that when C=2 the problem can be solved in time O( log D). For the general case, we give an algorithm which provides a solution requiring at most C−2 bins more than the optimal solution, i.e., an algorithm that is asymptotically exact. For fixed C, the complexity of the algorithm is O(poly( log D)), where poly(·) is a polynomial function not depending on C.     

The classical decision problem and partial functions

We consider the decision problem for sets of sentences of first-order logic when instead of interpreting function symbols as total functions over the universe of a model (henceforth referred to as the usual interpretation) we interpret them as partial functions.We consider only standard classes, which are certain sets of prenex sentences specified by restrictions on the prefix and on the numbers ofk-place predicate and function symbols for eachk1. Standard classes are introduced in [1] and it is proved there that the decision problem for any set of prenex sentences specified by such restrictions reduces to that for the standard classes.We solve the decision problem completely for standard classes with at least one function symbol and both with and without equality.This problem was suggested to me by my supervisor, Professor Yuri Gurevich who was confident that the results would be very similar to those for the usual interpretation and could be achieved by similar techniques.     

On finitely convergent iterative methods for the convex feasibility problem

Iterative algorithms for the Convex Feasibility Problem can be modified so that at iterationk the original convex sets are perturbed with a parameter ε_k which tends to zero ask increases. We establish conditions on such algorithms which guarantee existence of a sequence of perturbation parameters which make them finitely convergent when applied to a convex feasibility problem whose feasible set has non empty interior.