Max Flow and Min Cut with bounded-length paths: complexity, algorithms, and approximation

We consider the “flow on paths” versions of Max Flow and Min Cut when we restrict to paths having at most B arcs, and for versions where we allow fractional solutions or require integral solutions. We show that the continuous versions are polynomial even if B is part of the input, but that the integ...

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Bibliographic Details
Published in:Mathematical programming Vol. 124; no. 1-2; pp. 271 - 284
Main Authors: Mahjoub, A. Ridha, McCormick, S. Thomas
Format: Journal Article Conference Proceeding
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.07.2010
Springer
Springer Nature B.V
Subjects:
Algorithms
Applied sciences
Approximation
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Communications networks
Complexity
Complexity theory
Connectivity
Exact sciences and technology
Flows in networks. Combinatorial problems
Full Length Paper
Gaps
Integer programming
Integrals
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Operational research and scientific management
Operational research. Management science
Studies
Theoretical
90B10
90C57
68M10
Algorithm complexity
Minimax method
Primal dual method
Best approximation
Approximation algorithm
Combinatorial optimization
Mathematical programming
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We consider the “flow on paths” versions of Max Flow and Min Cut when we restrict to paths having at most B arcs, and for versions where we allow fractional solutions or require integral solutions. We show that the continuous versions are polynomial even if B is part of the input, but that the integral versions are polynomial only when B  ≤ 3. However, when B  ≤ 3 we show how to solve the problems using ordinary Max Flow/Min Cut. We also give tight bounds on the integrality gaps between the integral and continuous objective values for both problems, and between the continuous objective values for the bounded-length paths version and the version allowing all paths. We give a primal–dual approximation algorithm for both problems whose approximation ratio attains the integrality gap, thereby showing that it is the best possible primal–dual approximation algorithm.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-010-0366-6