Optimization over Degree Sequences

Deza, Antoine; Levin, Asaf; Meesum, Syed Mohammad; Onn, Shmuel

doi:10.1137/17m1134482

Cited by 28 publications

(31 citation statements)

References 20 publications

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“…Another interesting special case is when H = K n , that is, when the optimization is over any graph G on [n]. This special case was recently shown in [4] to be solvable in polynomial time when all the functions are the same, f 1 = · · · = f n , extending an earlier result of [12].…”

Section: Introductionmentioning

confidence: 83%

Optimization over degree sequences of graphs

Deza

Onn

2021

Discrete Applied Mathematics

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We consider the problem of finding a subgraph of a given graph minimizing the sum of given functions at vertices evaluated at their subgraph degrees. While the problem is NP-hard already for bipartite graphs when the functions are convex on one side and concave on the other, we show that when all functions are convex, the problem can be solved in polynomial time for any graph. We also provide polynomial time solutions for bipartite graphs with one side fixed for arbitrary functions, and for arbitrary graphs when all but a fixed number of functions are either nondecreasing or nonincreasing. We note that the general factor problem and the (l,u)-factor problem over a graph are special cases of our problem, as well as the intriguing exact matching problem. The complexity of the problem remains widely open, particularly for arbitrary functions over complete graphs. Hardness and exact matchingWe begin by showing that the optimization problem over degree sequence is generally hard.Proposition 1.1 Deciding if the optimal value in our problem is zero is NP-complete already:1. when f 1 = · · · = f n = f are identical, with f (0) = f (3) = 0 and f (i) = 1 for i = 0, 3.2. when H = (I, J, E) is bipartite and f i is convex for all i ∈ I and concave for all i ∈ J.

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Section: Introductionmentioning

confidence: 83%

Optimization over degree sequences of graphs

Deza

Onn

2021

Discrete Applied Mathematics

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“…Recently, Deza et al [7] proved that, for any fixed integer k ≥ 3, deciding the k-graphicality of a sequence is an NP-complete problem.…”

Section: Introductionmentioning

confidence: 99%

On null 3-hypergraphs

Frosini

Kocay

Palma

et al. 2021

Discrete Applied Mathematics

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Given a 3-uniform hypergraph H consisting of a set V of vertices, and T ⊆ V 3 triples, a null labelling is an assignment of ±1 to the triples such that each vertex is contained in an equal number of triples labelled +1 and −1. Thus, the signed degree of each vertex is zero. A necessary condition for a null labelling is that the degree of every vertex of H is even. The Null Labelling Problem is to determine whether H has a null labelling. It is proved that this problem is NP-complete. Computer enumerations suggest that most hypergraphs which satisfy the necessary condition do have a null labelling. Some constructions are given which produce hypergraphs satisfying the necessary condition, but which do not have a null labelling. A self complementary 3-hypergraph with this property is also constructed.

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“…And an algorithm to construct a graph with a given degree sequence was given by Havel [10] and Hakimi [9]. On the other hand, the same problem for hypergraphs remained unsolved till 2018, when Deza et al in [6] proved its NP-completeness, even for the simplest case of 3-uniform hypergraphs.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, but still before the result in [6], some necessary conditions had been given for a sequence to be the degree sequence of an h-uniform hypergraph. Such a sequence is called an h-sequence.…”

Section: Introductionmentioning

confidence: 99%

“…The present study focuses on investigating the properties of the 3-sequences that originate from a generalization of the gadget used by Deza et al in [6] for their NP-completeness proof. These are denoted by D n , according to their length n. The relevance of those sequences is mainly due to their uniqueness property, i.e., there exists a unique 3-hypergraph compatible with them, up to isomorphism.…”

Section: Introductionmentioning

confidence: 99%

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Properties of Unique Degree Sequences of 3-Uniform Hypergraphs

Ascolese¹,

Frosini²,

Kocay³

et al. 2021

Lecture Notes in Computer Science

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In 2018 Deza et al. proved the NP-completeness of deciding wether there exists a 3-uniform hypergraph compatible with a given degree sequence. A well known result of Erdös and Gallai (1960) shows that the same problem related to graphs can be solved in polynomial time. So, it becomes relevant to detect classes of uniform hypergraphs that are reconstructible in polynomial time. In particular, our study concerns 3-uniform hypergraphs that are defined in the N P -completeness proof of Deza et al. Those hypergraphs are constructed starting from a nonincreasing sequence s of integers and have very interesting properties. In particular, they are unique, i.e., there do not exist two non isomorphic 3-uniform hypergraphs having the same degree sequence ds. This property makes us conjecture that the reconstruction of these hypergraphs from their degree sequences can be done in polynomial time. So, we first generalize the computation of the ds degree sequences by Deza et al., and we show their uniqueness. We proceed by defining the equivalence classes of the integer sequences determining the same ds and we define a (minimal) representative. Then, we find the asymptotic growth rate of the maximal element of the representatives in terms of the length of the sequence, with the aim of generating and then reconstructing them. Finally, we show an example of a unique 3-uniform hypergraph similar to those defined by Deza et al. that does not admit a generating integer sequence s. The existence of this hypergraph makes us conjecture an extended generating algorithm for the sequences of Deza et al. to include a much wider class of unique 3-uniform hypergraphs. Further studies could also include strategies for the identification and reconstruction of those new sequences and hypergraphs.

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Optimization over Degree Sequences

Cited by 28 publications

References 20 publications

Optimization over degree sequences of graphs

Optimization over degree sequences of graphs

On null 3-hypergraphs

Properties of Unique Degree Sequences of 3-Uniform Hypergraphs

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