A fast algorithm for the generalized parametric minimum cut problem and applications

Abstract

Abstract. Many combinatorial optimization problems are solved by a sequence of network flow computations on a network whose edge capacities are given as a function of a parameter 2. Recently Gallo et al. I-7] made a major advance in solving such parametric flow problems. They showed that for an important class of networks, called monotone parametric flow networks, a sequence of O(n) flow computations could be solved in the same worst-case time bound as a single flow. However, these results require one of two special assumptions: either that the 2 values are presented in increasing or decreasing order; or that the edge capacity functions are affine functions of 2. In this paper we show how to remove both of these assumptions while obtaining the same running times as in [7]. This observation generalizes and unifies the two major results of 1-7], and allows its ideas to be applied to many new combinatorial problems. Of greatest importance, it allows the efficient application of binary search and successive binary search to a sequence of network flow problems.

Efficient Algorithms for the Optimal-Ratio Region Detection Problems in Discrete Geometry with Applications

“…13,10 To compute an optimal-ratio smooth lower-half region in Γ, as in Section 2.2, we need to compute an optimal smooth lower-half region with a maximized parametric net-cost for each of a sequence of parameters {θ 1 , θ 2 ,…, θ m } generated by the hand probing process (see Section 2.1), where m = O(n). Due to the monotonicity of G st (θ), we can apply Gusfield and Martel's parametric minimum cut algorithm 13 to compute all those O(n) optimal smooth lower-half regions in Γ in the complexity of solving a single maximum flow problem. Note that negative edge costs in the various related minimum cut problems can be made nonnegative using the transformation suggested in Ref.…”

“…We notice that the O(n) calls to the minimum st cut algorithm are on a sequence of weighted directed graphs, which forms a monotone parametric flow network. 13,10 Hence, the optimal-ratio smooth lower-half region in Γ can be detected in the complexity of computing a single maximum flow using Gusfield and Martel's algorithm. 13 Furthermore, for the case of using the normalization function g(R) = | R|, we establish a connection between our convex hull model for the problem and the traditional Newton based approach for the fractional programming problem (see, e.g., Gondran and Minous 12 ).…”

“…13,10 Hence, the optimal-ratio smooth lower-half region in Γ can be detected in the complexity of computing a single maximum flow using Gusfield and Martel's algorithm. 13 Furthermore, for the case of using the normalization function g(R) = | R|, we establish a connection between our convex hull model for the problem and the traditional Newton based approach for the fractional programming problem (see, e.g., Gondran and Minous 12 ). This connection enables us to apply Gallo et al 's simple parametric minimum s-t cut algorithm, 10 yielding an O(n 2 log n) time algorithm for computing an optimal-ratio smooth lower-half region.…”

A New Property and a Faster Algorithm for Baseball Elimination

“…We construct a bipartite network in which feasible integral flows correspond to outcomes of the remaining schedule. The following network flow formulation is due to Schwartz [15]: Gusfield and Martel [10] give an alternate construction. There are nodes corresponding to teams and to remaining games.…”

“…Robinson [14] gave a linear programming based model that finds the maximum lead a team can have at the end of the season. Gusfield and Martel [10] and McCormick [12] determined the elimination number, i.e., the minimum number of remaining games a team must win in order to have any chance of finishing in first place. Their methods use different extensions of the parametric maximum flow techniques of Gallo, Grigoriadis, and Tarjan [5].…”