The Price of Collusion in Series-Parallel Networks

We consider dynamic equilibria for flows over time under the fluid queuing model. In this model, queues on the links of a network take care of flow propagation. Flow enters the network at a single source and leaves at a single sink. In a dynamic equilibrium, every infinitesimally small flow particle reaches the sink as early as possible given the pattern of the rest of the flow. Although this model has been examined for many decades, progress has been relatively recent. In particular, the derivatives of dynamic equilibria have been characterized as thin flows with resetting, which allows for more structural results. Our two main results are based on the formulation of thin flows with resetting as a linear complementarity problem and its analysis. We present a constructive proof of existence for dynamic equilibria if the inflow rate is right-monotone. The complexity of computing thin flows with resetting, which occurs as a subproblem in this method, is still open. We settle it for the class of two-terminal, series-parallel networks by giving a recursive algorithm that solves the problem for all flow values simultaneously in polynomial time.

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The Recognition of Double Euler Trails in Series-Parallel Networks

Journal of Algorithms ◽

10.1006/jagm.1998.0942 ◽

1998 ◽

Vol 28 (2) ◽

pp. 216-257 ◽

Cited By ~ 2

Author(s):

Tung-Yang Ho ◽

Ting-Yi Sung ◽

Lih-Hsing Hsu ◽

Chang-Hsiung Tsai ◽

Jeng-Yan Hwang

Keyword(s):

Parallel Networks ◽

In Series

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Minimum-Cost Node-Disjoint Steiner Trees in Series-Parallel Networks

VLSI Design ◽

10.1155/1996/43738 ◽

1996 ◽

Vol 4 (1) ◽

pp. 53-57

Author(s):

Sunil Chopra ◽

Kalyan T. Talluri

Keyword(s):

Steiner Tree ◽

Minimum Cost ◽

Steiner Trees ◽

Np Hard ◽

Routing Problem ◽

Vlsi Layout ◽

Disjoint Family ◽

Undirected Network ◽

Parallel Networks ◽

In Series

The routing problem in VLSI-layout can be modeled as a problem of packing node-disjoint Steiner trees in a graph. The problem is as follows: Given an undirected network G = (V, E) and a net list Ψ {Ni, i = 1,..., r} , a family ΓG={TNi = (VNi , ENi), i = 1,..., r} is a node-disjoint family of Steiner trees spanning Ψ if TNi , is a Steiner tree spanning Ni for i = 1, ..., r and VNi ∩ VNj = for i ≠ j. The edge-disjoint version of this problem is known to be NP-hard for t. series-parallel graphs (see Rlchey and Parker [5]). In this paper we give a O(n5) algorithm for finding a minimum-cost node-disjoint family of Steiner trees in series-parallel networks. Our algorithm can be extended to k-trees and is polynomial for fixed k.

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