Faster Parametric Shortest Path and Minimum Balance Algorithms

The parametric shortest path problem is to find the shortest paths in graph where the edge costs are of the form w_ij+lambda where each w_ij is constant and lambda is a parameter that varies. The problem is to find shortest path trees for every possible value of lambda. The minimum-balance problem is to find a ``weighting'' of the vertices so that adjusting the edge costs by the vertex weights yields a graph in which, for every cut, the minimum weight of any edge crossing the cut in one direction equals the minimum weight of any edge crossing the cut in the other direction. The paper presents fast algorithms for both problems. The algorithms run in O(nm+n^2 log n) time. The paper also describes empirical studies of the algorithms on random graphs, suggesting that the expected time for finding a minimum-mean cycle (an important special case of both problems) is O(n log(n) + m)

Polynomial cases of the tarification problem

We consider the problem of determining a set of optimal tariffs for an agent in a network, who owns a subset of the arcs of the network, and who wishes to maximize his revenues on this subset from a set of clients that make use of the network.The general variant of this problem is NP-hard, already with a single client. This paper introduces several new polynomially solvable special cases. An important case is the following.For multiple clients, if the number of tariff arcs is bounded from above, we can solve the problem by a polynomial number of linear programs (each of which is of polynomial size). Furthermore, we show that the parametric tarification problem and the single arc fixed charge tarification problem can be solved in polynomial time.Economics ;

Setting Parameters by Example

We introduce a class of "inverse parametric optimization" problems, in which one is given both a parametric optimization problem and a desired optimal solution; the task is to determine parameter values that lead to the given solution. We describe algorithms for solving such problems for minimum spanning trees, shortest paths, and other "optimal subgraph" problems, and discuss applications in multicast routing, vehicle path planning, resource allocation, and board game programming.Comment: 13 pages, 3 figures. To be presented at 40th IEEE Symp. Foundations of Computer Science (FOCS '99

Pricing Network Edges to Cross a River.

We consider a Stackelberg pricing problem in directed networks:Tariffs (prices) have to be defined by an operator, the leader, for a subset of the arcs. Clients, the followers, choose paths to route their demand through the network selfishly and independently of each other, on the basis of minimal total cost. The problem is to find tariffs such as to maximize the operator''s revenue. We consider the case where each client takes at most one tariff arc to route the demand.The problem, which we refer to as the river tarification problem, is a special case of problems studied previously in the literature.We prove that the problem is strongly NP-hard.Moreover, we show that the polynomially solvable case of uniform tarification yields an m--approximation algorithm, and this is tight. We suggest a new type of analysis that allows to improve the result to \bigO{\log m}, whenever the input data is polynomially bounded. We furthermore derive an \bigO{m^{1-\varepsilon}}--inapproximability result for problems where the operator must serve all clients, and we discuss some polynomial special cases. Finally, a computational study with instances from France Telecom suggests that uniform pricing performs better in practice than theory would suggest.operations research and management science;

Pricing bridges to cross a river.

We consider a Stackelberg pricing problem in directed, uncapacitated networks. Tariffs have to be defined by an operator, the leader, for a subset of m arcs, the tariff arcs. Costs of all other arcs are assumed to be given. There are n clients, the followers, that route their demand independent of each other on paths with minimal total cost. The problem is to find tariffs that maximize the operator's revenue. Motivated by problems in telecommunication networks, we consider a restricted version of this problem, assuming that each client utilizes at most one of the operator's tariff arcs. The problem is equivalent to pricing bridges that clients can use in order to cross a river. We prove that this problem is APX-hard. Moreover, we show that uniform pricing yields both an m–approximation, and a (1 + lnD)–approximation. Here, D is upper bounded by the total demand of all clients. We furthermore discuss some polynomially solvable special cases, and present a short computational study with instances from France Télécom. In addition, we consider the problem under the additional restriction that the operator must serve all clients. We prove that this problem does not admit approximation algorithms with any reasonable performance guarantee, unless NP = ZPP, and we prove the existence of an n–approximation algorithm.Pricing; Networks; Tariffs; Costs; Cost; Demand; Problems; Order; Yield; Studies; Approximation; Algorithms; Performance;

Strictly Balancing Matrices in Polynomial Time Using Osborne\u27s Iteration

Osborne\u27s iteration is a method for balancing n x n matrices which is widely used in linear algebra packages, as balancing preserves eigenvalues and stabilizes their numeral computation. The iteration can be implemented in any norm over R^n, but it is normally used in the L_2 norm. The choice of norm not only affects the desired balance condition, but also defines the iterated balancing step itself. In this paper we focus on Osborne\u27s iteration in any L_p norm, where p < infty. We design a specific implementation of Osborne\u27s iteration in any L_p norm that converges to a strictly epsilon-balanced matrix in O~(epsilon^{-2}n^{9} K) iterations, where K measures, roughly, the number of bits required to represent the entries of the input matrix. This is the first result that proves a variant of Osborne\u27s iteration in the L_2 norm (or any L_p norm, p < infty) strictly balances matrices in polynomial time. This is a substantial improvement over our recent result (in SODA 2017) that showed weak balancing in L_p norms. Previously, Schulman and Sinclair (STOC 2015) showed strict balancing of another variant of Osborne\u27s iteration in the L_infty norm. Their result does not imply any bounds on strict balancing in other norms