A parameterized view on matroid optimization problems

Matroid theory gives us powerful techniques for understanding combinatorial optimization problems and for designing polynomial-time algorithms. However, several natural matroid problems, such as 3-matroid intersection, are NP-hard. Here we investigate these problems from the parameterized complexity...

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Bibliographic Details
Published in:Theoretical computer science Vol. 410; no. 44; pp. 4471 - 4479
Main Author: Marx, Dániel
Format: Journal Article Conference Proceeding
Language:English
Published: Oxford Elsevier B.V 17.10.2009
Elsevier
Subjects:
Applied sciences
Calculus of variations and optimal control
Combinatorial optmization
Computer science; control theory; systems
Convex and discrete geometry
Exact sciences and technology
Fixed-parameter tractability
Geometry
Mathematical analysis
Mathematics
Matroids
Miscellaneous
Numerical analysis
Numerical analysis. Scientific computation
Numerical methods in mathematical programming, optimization and calculus of variations
Sciences and techniques of general use
Theoretical computing
Matroids
Fixed-parameter tractability
Combinatorial optmization
Intersection
Polynomial
Partition
Computer theory
Combinatorial problem
Matroid
Optimization method
Combinatorial optimization
Complexity
Polynomial time
Network
Independent set
ISSN:0304-3975, 1879-2294
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Summary:Matroid theory gives us powerful techniques for understanding combinatorial optimization problems and for designing polynomial-time algorithms. However, several natural matroid problems, such as 3-matroid intersection, are NP-hard. Here we investigate these problems from the parameterized complexity point of view: instead of the trivial n O ( k ) time brute force algorithm for finding a k -element solution, we try to give algorithms with uniformly polynomial (i.e., f ( k ) ⋅ n O ( 1 ) ) running time. The main result is that if the ground set of a represented linear matroid is partitioned into blocks of size ℓ , then we can determine in randomized time f ( k , ℓ ) ⋅ n O ( 1 ) whether there is an independent set that is the union of k blocks. As a consequence, algorithms with similar running time are obtained for other problems such as finding a k -element set in the intersection of ℓ matroids, or finding k terminals in a network such that each of them can be connected simultaneously to the source by ℓ disjoint paths.
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2009.07.027