Nonconvexity of the Set of Hypergraph Degree Sequences

Liu, Ricky Ini

doi:10.37236/2719

Cited by 6 publications

(6 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We prove the following theorem solving a problem raised 30 years ago by Colbourn, Kocay and Stinson [3]. The proof is partially inspired by an argument from [17].…”

Section: The Complexity Of Deciding Hypergraph Degree Sequencesmentioning

confidence: 98%

“…and the vertices of D n 2 were characterized in [13] as precisely the degree sequences of threshold graphs. More recently, the polytopes D n k for k ≥ 3 were studied in [12,17,19], but neither a complete inequality description nor a complete characterization of vertices is known.…”

Section: Convex Functions and Degree Sequence Polytopesmentioning

confidence: 99%

“…, m} n → R which is not necessarily separable as considered above. For this we discuss the degree sequence polytopes studied in [12,13,17,19,21] and references therein, introduce and study degree sequence polytopes of hypergraphs with prescribed number of edges, and show that for k = 2 their vertices correspond to suitable threshold graphs [18].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimization over Degree Sequences

Deza¹,

Levin²,

Meesum³

et al. 2018

SIAM J. Discrete Math.

View full text Add to dashboard Cite

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.

show abstract

“…We prove the following theorem solving a problem raised 30 years ago by Colbourn, Kocay and Stinson [3]. The proof is partially inspired by an argument from [17].…”

Section: The Complexity Of Deciding Hypergraph Degree Sequencesmentioning

confidence: 98%

Section: Convex Functions and Degree Sequence Polytopesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Optimization over Degree Sequences

Deza¹,

Levin²,

Meesum³

et al. 2018

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…However, Problem 4.3 has not been solved for most models listed in this review. For example, the polytope of hypergraph degree sequences has been studied in the literature and some partial results are known: [Liu13] shows that the set of hypergraph degree sequences for uniform hypergraphs is non-convex, and thus Erdös-Gallai-type theorems do not hold. [MS02] also offers a very nice review of the problem, and shows that vertices of the polytope are known by Theorem 2.5.…”

Section: Polytopes Play An Important Role In Ergmsmentioning

confidence: 99%

A survey of discrete methods in (algebraic) statistics for networks

Petrović¹

2017

Algebraic and Geometric Methods in Discrete Mathematics

View full text Add to dashboard Cite

Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view of a particular family of statistical models for networks called exponential random graph models. The problems and underlying constructions are also related to well-known concepts in commutative algebra and graph-theoretic concepts in computer science. We outline a few lines of recent work that highlight the natural connection between these fields and unify them into some open problems. While these problems are often relevant in discrete mathematics in their own right, the emphasis here is on statistical relevance with the hope that these lines of research do not remain disjoint. Suggested specific open problems and general research questions should advance algebraic statistics theory as well as applied statistical tools for rigorous statistical analysis of networks.

show abstract

“…Structures, properties, and several related results were also obtained for 𝐷𝐷 𝑚𝑚 (𝑛𝑛). Convex hull of degree sequences of 𝑘𝑘-uniform hypergraphs was investigated in [4], [15][16][17]. In [16], it is verified computationally that the set of degree sequences for 𝑘𝑘 -uniform hypergraphs is the intersection of a lattice and a convex polytope for 𝑘𝑘 = 3 and 𝑛𝑛 ≤ 8.…”

Section: Introductionmentioning

confidence: 99%

On Nonconvexity of the Set of Hypergraphic Sequences

Sahakyan

2023

Proceedings of Computer Science and Information Technologies 2023 Conference

View full text Add to dashboard Cite

In this paper, we prove that 𝑫𝑫 𝒎𝒎 (𝒏𝒏), the set of hypergraphic sequences of all simple hypergraphs ([𝒏𝒏], 𝑬𝑬), where [𝒏𝒏] = {𝟏𝟏, 𝟐𝟐, ⋯ , 𝒏𝒏}, and |𝑬𝑬| = 𝒎𝒎; being a subset of 𝒏𝒏dimensional 𝒎𝒎 + 𝟏𝟏 -valued grid 𝜩𝜩 𝒎𝒎+𝟏𝟏 𝒏𝒏 , is not a convex set in 𝜩𝜩 𝒎𝒎+𝟏𝟏 𝒏𝒏 ; also, we characterize the smallest convex set containing 𝑫𝑫 𝒎𝒎 (𝒏𝒏).

show abstract

Nonconvexity of the Set of Hypergraph Degree Sequences

Cited by 6 publications

References 5 publications

Optimization over Degree Sequences

Optimization over Degree Sequences

A survey of discrete methods in (algebraic) statistics for networks

On Nonconvexity of the Set of Hypergraphic Sequences

Contact Info

Product

Resources

About