Hypergraph Decomposition Research Articles

Octahedral designs

AbstractAn oriented octahedral design of order v, or OCT(v), is a decomposition of all oriented triples on v points into oriented octahedra. Hanani [H. Hanani, Decomposition of hypergraphs into octahedra, Second International Conference on Combinatorial Mathematics (New York, 1978), Annals of the New York Academy of Sciences, 319, New York Academy of Science, New York, 1979, pp. 260–264.] settled the existence of these designs in the unoriented case. We show that an OCT(v) exists if and only if v≡1, 2, 6 (mod 8) (the admissible numbers), and moreover the constructed OCT(v) are unsplit, i.e. their octahedra cannot be paired into mirror images. We show that an OCT(v) with a subdesign OCT(U) exists if and only if v and u are admissible and v≥u+4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:319–327, 2010

Generalized hypertree decompositions: NP-hardness and tractable variants

The generalized hypertree width GHW( H ) of a hypergraph H is a measure of its cyclicity. Classes of conjunctive queries or constraint satisfaction problems whose associated hypergraphs have bounded GHW are known to be solvable in polynomial time. However, it has been an open problem for several years if for a fixed constant k and input hypergraph H it can be determined in polynomial time whether GHW( H ) ≤ k . Here, this problem is settled by proving that even for k = 3 the problem is already NP-hard. On the way to this result, another long standing open problem, originally raised by Goodman and Shmueli [1984] in the context of join optimization is solved. It is proven that determining whether a hypergraph H admits a tree projection with respect to a hypergraph G is NP-complete. Our intractability results on generalized hypertree width motivate further research on more restrictive tractable hypergraph decomposition methods that approximate generalized hypertree decomposition (GHD). We show that each such method is dominated by a tractable decomposition method definable through a function that associates a set of partial edges to a hypergraph. By using one particular such function, we define the new Component Hypertree Decomposition method, which is tractable and strictly more general than other approximations to GHD published so far.

On decompositions of complete hypergraphs

We study the minimum number of complete r-partite r-uniform hypergraphs needed to partition the edges of the complete r-uniform hypergraph on n vertices and we improve previous results of Alon.

A backtracking-based algorithm for hypertree decomposition

Hypertree decompositions of hypergraphs are a generalization of tree decompositions of graphs. The corresponding hypertree-width is a measure for the acyclicity and therefore an indicator for the tractability of the associated computation problem. Several NP-hard decision and computation problems are known to be tractable on instances whose structure is represented by hypergraphs of bounded hypertree-width. Roughly speaking, the smaller the hypertree-width, the faster the computation problem can be solved. In this paper, we present the new backtracking-based algorithm det- k -decomp for computing hypertree decompositions of small width. Our benchmark evaluations have shown that det- k -decomp significantly outperforms opt- k -decomp, the only exact hypertree decomposition algorithm so far. Even compared to the best heuristic algorithm, we obtained competitive results as long as the hypergraphs are sufficiently simple.

Decompositions of the 3-uniform hypergraphs K v (3) into hypergraphs of a certain type

In this paper, we investigate a generalization of graph decomposition, called hypergraph decomposition. We show that a decomposition of a 3-uniform hypergraph K v (3) into a special kind of hypergraph K 4 (3) − e exists if and only if v ≡ 0, 1, 2 (mod 9) and v ⩾ 9.

Fractional decompositions of dense hypergraphs

A seminal result of Rodl (the Rodl nibble) asserts that the edges of the complete r-uniform hypergraph K$_{n}^{r}$ can be packed, almost completely, with copies of K$_{k}^{r}$, where k is fixed. This result is considered one of the most fruitful applications of the probabilistic method. It was not known whether the same result also holds in a dense hypergraph setting. In this paper we prove that it does. We prove that for every r-uniform hypergraph H0, there exists a constant α=α(H0) r > 2. We prove that there exists a constant α=α(k,r) < 1 such that every r-uniform hypergraph with n (sufficiently large) vertices in which every (r–1)-set is contained in at least αn edges has a fractional K$_{k}^{r}$-decomposition. We then apply a recent result of Rodl, Schacht, Siggers and Tokushige to obtain our integral packing result. The proof makes extensive use of probabilistic arguments and additional combinatorial ideas.

Planar Branch Decompositions II: The Cycle Method

This is the second of two papers dealing with the relationship of branchwidth and planar graphs. Branchwidth and branch decompositions, introduced by Robertson and Seymour, have been shown to be beneficial for both proving theoretical results on graphs and solving NP-hard problems modeled on graphs. The first practical implementation of an algorithm of Seymour and Thomas for computing optimal branch decompositions of planar hypergraphs is presented. This algorithm encompasses another algorithm of Seymour and Thomas for computing the branchwidth of any planar hypergraph, whose implementation is discussed in the first paper. The implementation also includes the addition of a heuristic to decrease the run times of the algorithm. This method, called the cycle method, is an improvement on the algorithm by using a “divide-and-conquer” approach.

On the Hamiltonian cycle Decompositions of Complete 3-Uniform Hypergraphs

Abstract A 3-uniform hypergraph H is a pair (V,ϵ), where V is vertex set, ϵ, is a family of 3-subsets of V. If ϵ consists of all 3-subsets of V, H is a complete 3-uniform hypergraph on n vertices and is denoted by K n(3). If V is the disjoint union of sets (so-called parts) V1 and V2, and ϵ consists of all possible 3-subsets ζ of V1 ∪ V2 such that ζ ⊈ Vi, i = 1, 2, then H is a complete bipartite 3-uniform hypergraph and is denoted by K m,n(3), if ∣V1∣ = m, ∣V2∣ = n. In this paper we show that following results on the decomposition of hypergraph into Hamiltonian cycles. (i) K n,n(3) has a decomposition into Hamiltonian cycles, n ≥ 2. (ii) all complete 3-uniform hypergraphs K 2m(3) with 2m vertices has a decomposition into Hamiltonian cycles, m ≥ 2.

Local spanning trees in graphs and hypergraph decomposition with respect to edge connectivity

Local spanning trees in graphs and hypergraph decomposition with respect to edge connectivity

An Algorithm for the Modular Decomposition of Hypergraphs

We propose an O(n4) algorithm to build the modular decomposition tree of hypergraphs of dimension three and show how this algorithm can be generalized to compute in O(n3k−5) time the decomposition of hypergraphs of any fixed dimension k.

Local spanning trees in graphs and hypergraph decomposition with respect to edge connectivity

Abstract We present a sufficient condition to a subset A of vertices of a graph G for the existence of a system of k edge-disjoint A-spanning trees. As a corollary, we prove that every k − ( k + 1)-edge-connected hypergraph of rank at most k + 1 is decomposable into k connected spanning subhy-pergraphs. AMS subject classification: 05C40, 05C65.

Solution of a delta-system decomposition problem

A decomposition of a hypergraph H into hypergraphs H 1,…, H q is a partition of the edge set ϵ of H into subsets ϵ 1,…, ϵ q such that H i is generated by ϵ i , for i = 1,…, q. An h-uniform hypergraph generated by edges E 1,…, E c is called a deltasystem Δ( p, h, c) if there is a set F such that | F| = p and E j ∪ E j = F ∀ i ≠ j. In this paper we find a necessary and sufficient condition for the existence of a decomposition of the complete 3-uniform hypergraph K n 3 into delta-systems Δ(1, 3, c). This way we settle a conjecture by Mouyart and Sterboul. Moreover, we solve problems of the maximum (respectively minimum) number of delta-systems Δ(1, 3, c) to be packed into K n 3 (resp. to cover K n 3) for “most” pairs of integers n, c. In the remaining cases we estimate the numbers.

On decomposition of hypergraphs into Δ-systems

We show that every uniform hypergraph H of size being a multiple of a positive integer k can be decomposed into k-edge Δ-systems provided that H contains a matching of sufficiently large, but independent of parameters of H, size.

Decomposition of the complete hypergraph into hyperclaws

On etudie la decomposition d'un hypergraphe complet en «hypergriffes». On conjecture une condition necessaire et suffisante d'existence d'une telle decomposition

A note on hypergraph decomposition based on extended minimal sets

We discuss the problem of finding a partition of the nodes of a hypergraph Y with weighted arcs.Extended minimal sets (em-sets) of nodes are defined, intuitively corresponding to subsets of nodes of Y with a high number of inter-connections, and a partition of Y in em-sets is derived by a polynomial time algorithm. To show the nature of the partition, several properties of em-sets are proved. In particular, these sets can totally but not partially overlap. As an example of an application, the role of our partition in the field of VLSI design is briefly discussed.

Decompositions of hypergraphs into hyperstars

In the paper we investigate decompositions of hypergraphs into hyperstars. A hyperstar with center F and size c is every hypergraph ( X, ¢E) such that F ⊆ ∩ ¢E and |¢E|=c. A decomposition of a hypergraph ( X, ¢E) into hypergraphs from a certain class ¢K is a family of hypergraphs {H i = (X, ¢E i):i ϵ I} such that {¢E i:i ϵ I} is a partition of ¢E and each H i is isomorphic to hypergraph in ¢K. In the paper we find necessary and sufficient conditions for existence of a decomposition of a hypergraph into hyperstars with given centers and sizes. This result is then applied to obtain sufficient conditions for existence of a hyperstar decomposition of the hypergraphs P m = (X, ¢P (X) β {0}) and K m n = (X, ¢P n(X)), |X| = m . As a corollary, these results give a partial solution of a problem of Yamamoto and Tazawa [7] related to hyperstar decompositions of K m n .

Decomposition of the complete hypergraph into delta-systems II

On caracterise les valeurs de n pour lesquelles K n 3 peut etre decompose en Δ(1, 3, c) dans les cas C=4, 5, et 6

Decomposition of submodular functions

A decomposition theory for submodular functions is described. Any such function is shown to have a unique decomposition consisting of indecomposable functions and certain highly decomposable functions, and the latter are completely characterized. Applications include decompositions of hypergraphs based on edge and vertex connectivity, the decomposition of matroids based on three-connectivity, the Gomory—Hu decomposition of flow networks, and Fujishige’s decomposition of symmetric submodular functions. Efficient decomposition algorithms are also discussed.

Minimal decompositions of hypergraphs into mutually isomorphic subhypergraphs

Minimal decompositions of hypergraphs into mutually isomorphic subhypergraphs

Decompositions of graphs and hypergraphs into isomorphic factors with a given diameter

Decompositions of graphs and hypergraphs into isomorphic factors with a given diameter