Optimization over Degree Sequences

@article{Deza2018OptimizationOD,
  title={Optimization over Degree Sequences},
  author={Antoine Deza and Asaf Levin and Syed Mohammad Meesum and Shmuel Onn},
  journal={SIAM J. Discret. Math.},
  year={2018},
  volume={32},
  pages={2067-2079}
}
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 already for simple concave functions. In contrast, we show that for… 
Optimization over Degree Sequences of Graphs
On Degree Sequence Optimization
  • S. Onn
  • Mathematics, Computer Science
    Oper. Res. Lett.
  • 2020
Reconstruction of hypergraphs from line graphs and degree sequences
In this paper we consider the problem to reconstruct a k-uniform hypergraph from its line graph. In general this problem is hard. We solve this problem when the number of hyperedges containing any
On the Degree Sequence of 3-Uniform Hypergraph: A New Sufficient Condition
The study of the degree sequences of h-uniform hypergraphs, say h-sequences, was a longstanding open problem in the case of \(h>2\), until very recently where its decision version was proved to be
Construction of Simplicial Complexes with Prescribed Degree-Size Sequences
TLDR
It is found that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes, which unveils a fundamental constraint on the degree-size sequences.
On null 3-hypergraphs
Separable and Equitable Hypergraphs
A k -hypergraph is separable if its vertices admit a certain labeling, and is equitable if the edges of the complete k -hypergraph admit a certain labeling. We show that these classes of hypergraphs
On Line Sum Optimization
  • S. Onn
  • Mathematics
    Linear Algebra and its Applications
  • 2021
On the Computational Complexity of Finding Bipartite Graphs with a Small Number of Short Cycles and Large Girth
TLDR
It is proved that for a given set of integers $\alpha, \beta$, and $\gamma$, and degree sequences $\pi$ and $\pi$’, the problem of determining whether there exists a simple bipartite graph with degree sequences $(\pi,\ \pi')$ that has at most $\alpha$ ($\beta$ and $\Gamma$) cycles of length four is NP-complete.
On the Hypercube Subset Partitioning Varieties
TLDR
The NP-hardness of the QDP problem is proved and it is shown that QDP are in a correspondence to the upper homogeneous area elements of the n-cube and to the monotone Boolean functions.
...
...

References

SHOWING 1-10 OF 29 REFERENCES
Nonconvexity of the Set of Hypergraph Degree Sequences
TLDR
It is shown that the set of possible degree sequences for a simple k-uniform hypergraph on n vertices is not the intersection of a lattice and a convex polytope for k 3 and n k+13, and an analogous nonconvexity result is shown.
The polytope of degree sequences of hypergraphs
Extreme degree sequences of simple graphs
On the Swap-Distances of Different Realizations of a Graphical Degree Sequence
TLDR
This paper develops formulae (Gallai-type identities) for the swap-distances of any two realizations of simple undirected or directed degree sequences, which considerably improve the known upper bounds on the Swap-Distances.
Parameterized Shifted Combinatorial Optimization
TLDR
It is shown 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.
Shifted Set Families, Degree Sequences, and Plethysm
TLDR
Higher weight theory is used to explain how shifted k-families provide the ``top part'' of these plethysm expansions of elementary symmetric functions e_m[e_k], along with offering a conjecture about a further relation.
Some NP-complete problems for hypergraph degree sequences
Degree multisets of hypergraphs
  • D. Billington
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 1982
A multiset is a "set" which may have repeated elements. If S is a positive integer then an s-uniform hypergraph is a hypergraph in which every block, or edge, contains exactly s points. A hypergraph
(n,e)-Graphs with maximum sum of squares of degrees*
Among all simple graphs on n vertices and e edges, which ones have the largest sum of squares of the vertex degrees? It is easy to see that they must be threshold graphs, but not every threshold
A PROCEDURE FOR COMPUTING THE K BEST SOLUTIONS TO DISCRETE OPTIMIZATION PROBLEMS AND ITS APPLICATION TO THE SHORTEST PATH PROBLEM
TLDR
It is shown how the K shortest (loopless) paths in an n-node network with positive and negative arcs can be computed with an amount of computation which is O(Kn 3 ).
...
...