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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Vydáno v:	Theoretical computer science Ročník 410; číslo 44; s. 4471 - 4479
Hlavní autor:	Marx, Dániel
Médium:	Journal Article Konferenční příspěvek
Jazyk:	angličtina
Vydáno:	Oxford Elsevier B.V 17.10.2009 Elsevier
Témata:	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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Abstract	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.
AbstractList	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 ?, then we can determine in randomized time f(k,?)?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 ? matroids, or finding k terminals in a network such that each of them can be connected simultaneously to the source by ? disjoint paths. 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.
Author	Marx, Dániel
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Title	A parameterized view on matroid optimization problems
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