Frieda Granot

On Some Network Flow Games

We analyze three subclasses of cooperative games arising from network optimization problems in which the resources, such as arcs or nodes in the network, are controlled by individuals who have conflicting objectives. The first subclass of cooperative games is induced by network optimization problems over directed augmented trees. We show that for this subclass of games the kernel coincides with the nucleolus, and that the nucleolus can be characterized as the unique revenue allocation vector in which every pair of arc owners who are adjacent in the tree are located symmetrically with respect to their bargaining range. We further give a linear characterization of the core of this subclass of games, which is then used to provide a more explicit representation of the nucleolus and to construct a strongly polynomial algorithm for generating it. The second subclass of cooperative games is induced by maximum flow problems in simple undirected networks. We provide a useful parametric representation of the core of this subclass of games, which is used to characterize the nucleolus and the intersection of the core and the kernel. Explicitly, we show that the intersection of the core and the kernel consists of all revenue allocations in the core which assign equal payoffs to any pair of unseparated arc owners. We further demonstrate that among the core vectors, the nucleolus is the unique revenue allocation vector in which the smallest allocations are maximized in a lexicographical sense. The third cooperative game that we study is the assignment game, introduced by Shapley and Shubik 1972. This game is induced by the assignment problem which can be cast as a network optimization problem. We investigate the relationship between the kernel and the core of the assignment game, and provide a necessary and sufficient condition for the core to be contained in the kernel. We further show that, in general, the intersection of the kernel and the core is not a convex set. We also exhibit that under certain conditions the nucleolus has a simple characterization as the unique vector in the core in which the smallest revenue allocations are maximized in a lexicographical sense. Finally, we consider the horse market example of Bohm-Bawerk 1923, for which it is shown that the core is contained in the kernel and that the nucleolus is the midpoint of the core.

Multi-terminal maximum flows in node-capacitated networks

Abstract This paper presents a variable-dimension homotopy algorithm that exploits dominant market structure for computing spatial market equilibria. The algorithm is implemented on the problems underlying network. The computational results indicate that the algorithm performs most of its work in lower dimensions and can process large problems effectively.

Substitutes, Complements and Ripples in Network Flows

We study the qualitative variation of minimum-cost network-flows and their associated costs with arc parameters. Demands are given at each node, flow is conserved, each arcs parameter lies in a lattice, and the flow cost is real or infinity-valued. Our main results are roughly as follows. If the flow cost is arc-additive, the problem can be decomposed into independent problems on each biconnected component of the graph. If also the flow cost is convex in the flow, the magnitude of the change in the optimal flow in arc a resulting from changing arc b s parameter diminishes the less biconnected a is to b . Arc a is less-biconnected-to b than is arc d if every simple cycle containing a and b also contains d . This relation is a quasi order with two distinct arcs being equivalent if and only if deleting them disconnects the graph. Hie Hasse diagram of the partial order of the induced equivalence classes is a tree with all arcs directed towards the class containing b . Two arcs are complements (resp., substitutes ) if every simple cycle containing both orients them in the same (resp., opposite) way. For example, two arcs are conformal , i.e., complements or substitutes, if they are either incident, or lie on a common face of a planar graph, or (as Dirac [Dirac, G. A. 1952. A property of 4-chromatic graphs and some remarks on critical graphs. J. London Math. Soc. 27 85--92.] and Duffin [Duffin, R. J. 1965. Topology of series-parallel networks. J. Math. Anal. Appl. 10 303--318.] have shown) lie in a series-parallel graph. If also each arc cost is (lattice) subadditive, then the optimal flow in arc a is nondecreasing (resp., nonincreasing) in the parameter of each complement (resp., substitute) of a . If moreover each parameter lies in a chain, then the minimum cost is superadditive in the parameters of a set of substitutes. Suppose in addition that the arc parameters are real and the arc costs are doubly subadditive. Then the absolute difference of the optimal flows in an arc corresponding to two monotonically-step-connected parameter vectors does not exceed the sum of the absolute differences of their elements. If further the arc costs are affine between successive integers and the difference between the two parameter vectors is a unit vector, then the difference between corresponding integer optimal flows is a unit simple circulation. This fact leads to a parametric algorithm for finding optimal flows. Finally, suppose instead that the flow cost is a sum of a subadditive flow cost for arcs in a set S of complements and a flow cost for arcs not in S that is arc-additive and convex in the flows therein. Then the optimal flow in each arc in S is nondecreasing in the parameters of those arcs, and the minimum cost is subadditive therein. Moreover, the optimal flow in each arc a not in S is nondecreasing (resp., nonincreasing) in the parameters of arcs in S if a is a complement (resp., substitute) of every arc in S .

Computational complexity of a cost allocation approach to a fixed cost spanning forest problem

We present a computational analysis of a game theoretic approach to a cost allocation problem arising from a graph optimization problem, referred to as the fixed cost spanning forest FCSF problem. The customers in the FCSF problem, represented by nodes in a graph G, are in need of service that can be produced at some facilities yet to be constructed. The cost allocation problem is concerned with the fair distribution of the cost of providing the service among customers. We formulate this cost allocation problem as a cooperative game, referred to as the FCSF game. In general, the core of a FCSF game may be empty. However, for the case when G is a tree, it is shown that the core is not empty. Moreover, we prove that in this case core points can be generated in strongly polynomial time. We further provide a nonredundant characterization of the core of the FCSF game defined over a tree in the special case when all nodes are communities. This is shown to lead, in some instances, to a strongly polynomial algorithm for computing the nucleolus.

Some proximity and sensitivity results in quadratic integer programming

AbstractWe show that for any optimal solution

Non-linear integer programming: Sensitivity analysis for branch and bound

\bar z

An Algorithm for Solving Interval Linear Programming Problems

for a given separable quadratic integer programming problem there exist an optimal solution