The unimodular intersection problem

Kaibel, Volker; Onn, Shmuel; Sarrabezolles, Pauline

doi:10.1016/j.orl.2015.09.005

Cited by 8 publications

(12 citation statements)

References 5 publications

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“…Then x ∼ y so x is also in the left hand side of (5). Moreover, by an algorithmic version of [7] of this decomposition theorem, such a y can be found in polynomial time, solving problem (4) over S. By the equality in (5), to solve problem (3) over S with a given profit c, we can maximize c over the right hand side of (5), and this can be done in polynomial time by linear programming since the defining matrix of this set is [A, ..., A], which is totally unimodular since A is. Lemmas 2.1 and 2.2 now imply that problems (2) and (1) over S can also be solved in polynomial time.…”

Section: Remarksmentioning

confidence: 99%

“…Lemma 2.1 [7] The Lexicographic Combinatorial Optimization problem (1) can be reduced in polynomial time to the Shifted Combinatorial Optimization problem (2).…”

Section: Shifted Combinatorial Optimizationmentioning

confidence: 99%

“…In Section 4 we discuss matroid intersections, partially solve the shifted problem over the intersection of two strongly base orderable matroids (which include gammoids) in Theorem 4.3, and leave open the complexity of the problem for the intersection of two arbitrary matroids. We conclude in Section 5 with some final remarks about the polynomial time solvability of the shifted and lexicographic problems over totally unimodular systems from [7], and show that these problems are NP-hard over matchings already for n = 2 and cubic graphs.…”

mentioning

confidence: 99%

“…The complexity of this problem depends on the presentation of the set S. In [7] it was shown that it is polynomial time solvable for S = {z ∈ {0, 1} d : Az = b} for any totally unimodular A and any integer b. Here we solve the problem for matroids.…”

mentioning

confidence: 99%

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Shifted matroid optimization

Levin

Onn

2016

Operations Research Letters

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Section: Remarksmentioning

confidence: 99%

“…Lemma 2.1 [7] The Lexicographic Combinatorial Optimization problem (1) can be reduced in polynomial time to the Shifted Combinatorial Optimization problem (2).…”

Section: Shifted Combinatorial Optimizationmentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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Shifted matroid optimization

Levin

Onn

2016

Operations Research Letters

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“…• when S is presented by totally unimodular inequalities, in particular when S is the set of source-sink dipaths in a digraph or matchings in a bipartite graph [11];…”

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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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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The unimodular intersection problem

Abstract: We show that finding minimally intersecting n paths from s to t in a directed graph or n perfect matchings in a bipartite graph can be done in polynomial time. This holds more generally for unimodular set systems.

Cited by 8 publications

References 5 publications

Shifted matroid optimization

Shifted matroid optimization

Optimization over Degree Sequences

Parameterized Shifted Combinatorial Optimization

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