Shifted matroid optimization

Levin, Asaf; Onn, Shmuel

doi:10.1016/j.orl.2016.05.013

Cited by 11 publications

(18 citation statements)

References 10 publications

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“…We proceed with the (positive) algorithmic statements of the theorem. For part 1, which was also proved in [18], just note that for fixed m, there are O(r m−1 ) feasible solutions in (4), obtained by taking integers 0 ≤ r 1 , . .…”

Section: Sets Given Explicitlymentioning

confidence: 77%

“…The complexity of the shifted combinatorial optimization (SCO) problem depends on c and on the presentation of S, and is typically harder than the corresponding standard combinatorial optimization problem. Say, when S is the set of perfect matchings in a graph, the standard problem is polynomial time solvable, but the shifted problem is NP-hard even for r = 2 and cubic graphs, as the optimal value of the above 2-vulnerability problem is 0 if and only if the graph is 3-edge-colorable [18]. The minimization of 2-vulnerable arcs with S the set of s-t dipaths in a digraph, also called the Minimum shared edges problem, was recently shown to be NP-hard for r variable in [21], polynomial time solvable for fixed r in [1], and fixed-parameter tractable with r as a parameter in [6].…”

Section: Introductionmentioning

confidence: 99%

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Parameterized Shifted Combinatorial Optimization

Gajarský

Hliněný

Koutecký

et al. 2017

Lecture Notes in Computer Science

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Shifted combinatorial optimization is a new nonlinear optimization framework which is a broad extension of standard combinatorial optimization, involving the choice of several feasible solutions at a time. This framework captures well studied and diverse problems ranging from so-called vulnerability problems to sharing and partitioning problems. In particular, every standard combinatorial optimization problem has its shifted counterpart, which is typically much harder. Already with explicitly given input set the shifted problem may be NP-hard. In this article we initiate a study of the parameterized complexity of this framework. First we show that shifting over an explicitly given set with its cardinality as the parameter may be in XP, FPT or P, depending on the objective function. Second, we study the shifted problem over sets definable in MSO logic (which includes, e.g., the well known MSO partitioning problems). Our main results here are that shifted combinatorial optimization over MSO definable sets is in XP with respect to the MSO formula and the treewidth (or more generally clique-width) of the input graph, and is W[1]-hard even under further severe restrictions.

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Section: Sets Given Explicitlymentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Parameterized Shifted Combinatorial Optimization

Gajarský

Hliněný

Koutecký

et al. 2017

Lecture Notes in Computer Science

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“…A fundamental difference is that Megiddo considers fractional flows while we characterize integer-valued flows which are inc-max on the set of source edges. Finally, we mention the 'shifted matroid optimization' problem due to Levin and Onn [27] which seeks for k bases Z 1 , . .…”

Section: Background Problemsmentioning

confidence: 99%

Decreasing Minimization on M-convex Sets: Background and Structures

Frank

Murota

2020

Preprint

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The present work is the first member of a pair of papers concerning decreasingly-minimal (dec-min) elements of a set of integral vectors, where a vector is dec-min if its largest component is as small as possible, within this, the next largest component is as small as possible, and so on. This notion showed up earlier under various names at resource allocation, network flow, matroid, and graph orientation problems. Its fractional counterpart was also seriously investigated.The domain we consider is an M-convex set, that is, the set of integral elements of an integral base-polyhedron. A fundamental difference between the fractional and the discrete case is that a base-polyhedron has always a unique dec-min element, while the set of dec-min elements of an M-convex set admits a rich structure, described here with the help of a 'canonical chain'. As a consequence, we prove that this set arises from a matroid by translating the characteristic vectors of its bases with an integral vector.By relying on these characterizations, we prove that an element is dec-min if and only if the square-sum of its components is minimum, a property resulting in a new type of min-max theorems. The characterizations also give rise, as shown in the companion paper, to a strongly polynomial algorithm and to a proof of a conjecture on dec-min orientations.

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“…• when S is the set of independent sets in a matroid, in particular spanning trees in a graph, or the intersection of two so-called strongly-base-orderable matroids [16];…”

Section: Shifted Combinatorial Optimizationmentioning

confidence: 99%

Optimization over Degree Sequences

Deza¹,

Levin²,

Meesum³

et al. 2018

SIAM J. Discrete Math.

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We introduce and study the problem of optimizing arbitrary functions over degree sequences of hypergraphs and multihypergraphs. We show that over multihypergraphs the problem can be solved in polynomial time. For hypergraphs, we show that deciding if a given sequence is the degree sequence of a 3-hypergraph is NP-complete, thereby solving a 30 year long open problem. This implies that optimization over hypergraphs is hard even for simple concave functions. In contrast, we show that for graphs, if the functions at vertices are the same, then the problem is polynomial time solvable. We also provide positive results for convex optimization over multihypergraphs and graphs and exploit connections to degree sequence polytopes and threshold graphs. We then elaborate on connections to the emerging theory of shifted combinatorial optimization.

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Shifted matroid optimization

Abstract: We show that finding lexicographically minimal n bases in a matroid can be done in polynomial time in the oracle model. This follows from a more general result that the shifted problem over a matroid can be solved in polynomial time as well.

Cited by 11 publications

References 10 publications

Parameterized Shifted Combinatorial Optimization

Parameterized Shifted Combinatorial Optimization

Decreasing Minimization on M-convex Sets: Background and Structures

Optimization over Degree Sequences

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