Problem For Hypergraphs Research Articles

Partial recovery and weak consistency in the non-uniform hypergraph stochastic block model

Abstract We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $\gamma$ fraction of the vertices classified correctly, where $\gamma \in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati ((2017) Ann. Stat.45(1) 289–315.) for non-uniform HSBMs. Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularised adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularisation of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

The growth rate of multicolor Ramsey numbers of 3-graphs

The q-color Ramsey number of a k-uniform hypergraph G, denoted r(G; q), is the minimum integer N such that any coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of G. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior of r(G; q) for fixed G and q tending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height of r(G; q) as a function of q. More precisely, given a hypergraph G, we determine when r(G; q) behaves polynomially, exponentially or double exponentially in q. This answers a question of Axenovich, Gyárfás, Liu and Mubayi.

Set-Coloring Ramsey Numbers via Codes

For positive integers 𝑛, 𝑟, 𝑠 with 𝑟 > 𝑠, the set-coloring Ramsey number 𝑅(𝑛; 𝑟, 𝑠) is the minimum 𝑁 such that if every edge of the complete graph 𝐾𝑁 receives a set of 𝑠 colors from a palette of 𝑟 colors, then there is guaranteed to be a monochromatic clique on 𝑛 vertices, that is, a subset of 𝑛 vertices where all of the edges between them receive a common color. In particular, the case 𝑠 = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on 𝑅(𝑛; 𝑟, 𝑠) which imply that 𝑅(𝑛; 𝑟, 𝑠) = 2Θ(𝑛𝑟) if 𝑠/𝑟 is bounded away from 0 and 1. The upper bound extends an old result of Erdős and Szemerédi, who treated the case 𝑠 = 𝑟 − 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.

Wickets in 3-uniform hypergraphs

In this note, we consider a Turán-type problem in hypergraphs. What is the maximum number of edges if we forbid a subgraph? Let Hn(3) be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, called wicket, is formed by three rows and two columns of a 3×3 point matrix. We describe two linear hypergraphs that if we forbid either of them in Hn(3), then the hypergraph is sparse, i.e. the number of its edges is o(n2). Since both contain a wicket, it implies a conjecture of Gyárfás and Sárközy that wicket-free hypergraphs are sparse.

Polynomial reconstruction problem for hypergraphs

We show that, in general, the characteristic polynomial of a hypergraph is not determined by its “polynomial deck”, the multiset of characteristic polynomials of its vertex-deleted subgraphs, thus settling the “polynomial reconstruction problem” for hypergraphs in the negative. The proof proceeds by showing that a construction due to Kocay of an infinite family of pairs of 3-uniform hypergraphs which are non-isomorphic but share the same hypergraph deck, in fact, have different characteristic polynomials. The question remains unresolved for ordinary graphs.

Large cliques or cocliques in hypergraphs with forbidden order-size pairs

AbstractThe well-known Erdős-Hajnal conjecture states that for any graph $F$ , there exists $\epsilon \gt 0$ such that every $n$ -vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^{\epsilon }$ . We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$ , then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$ , for every $S \subseteq \{0,1,2,3,4\}$ , we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$ . In most cases the bounds are essentially tight. We also determine, for all $S$ , whether the growth rate is polynomial or polylogarithmic. Some open problems remain.

Using edge cuts to find Euler tours and Euler families in hypergraphs

An Euler tour in a hypergraph is a closed walk that traverses each edge of the hypergraph exactly once, while an Euler family is a family of closed walks that jointly traverse each edge exactly once and cannot be concatenated. In this paper, we show how the problem of existence of an Euler tour (family) in a hypergraph H can be reduced to the analogous problem in some smaller hypergraphs that are derived from H using an edge cut of H. In the process, new techniques of edge cut assignments and collapsed hypergraphs are introduced. Moreover, we describe algorithms based on these characterizations that determine whether or not a hypergraph admits an Euler tour (family), and can also construct an Euler tour (family) if it exists.

The critical node detection problem in hypergraphs using weighted node degree centrality.

Network analysis is an indispensable part of today's academic field. Among the different types of networks, the more complex hypergraphs can provide an excellent challenge and new angles for analysis. This study proposes a variant of the critical node detection problem for hypergraphs using weighted node degree centrality as a form of importance metric. An analysis is done on both generated synthetic networks and real-world derived data on the topic of United States House and Senate committees, using a newly designed algorithm. The numerical results show that the combination of the critical node detection on hypergraphs with the weighted node degree centrality provides promising results and the topic is worth exploring further.

Almost optimal query algorithm for hitting set using a subset query

In this paper, we focus on Hitting-Set, a fundamental problem in combinatorial optimization, through the lens of sublinear time algorithms. Given access to the hypergraph through a subset query oracle in the query model, we give sublinear time algorithms for Hitting-Set with almost tight parameterized query complexity. In parameterized query complexity, we estimate the number of queries to the oracle based on the parameter k, the size of the Hitting-Set. The subset query oracle we use in this paper is called Generalized d-partite Independent Set query oracle (GPIS) and it was introduced by Bishnu et al. (ISAAC'18). GPIS is a generalization to hypergraphs of the Bipartite Independent Set query oracle (BIS) introduced by Beame et al. (ITCS'18 and TALG'20) for estimating the number of edges in graphs. Since its introduction GPIS query oracle has been used for estimating the number of hyperedges independently by Dell et al. (SODA'20 and SICOMP'22) and Bhattacharya et al. (STACS'22), and for estimating the number of triangles in a graph by Bhattacharya et al. (ISAAC'19 and TOCS'21). Formally, GPIS is defined as follows: GPISoracle for a d-uniform hypergraphHtakes as input d pairwise disjoint non-empty subsetsA1,…,Adof vertices inHand answers whether there is a hyperedge inHthat intersects each setAi, wherei∈{1,2,…,d}. For d=2, the GPIS oracle is nothing but BIS oracle.We show that d-Hitting-Set, the hitting set problem for d-uniform hypergraphs, can be solved using O˜d(kdlog⁡n)GPIS queries. Additionally, we also showed that d-Decision-Hitting-Set, the decision version of d-Hitting-Set can be solved with O˜d(min⁡{kdlog⁡n, k2d2})GPIS queries. We complement these parameterized upper bounds with an almost matching parameterized lower bound that states that any algorithm that solves d-Decision-Hitting-Set requires Ω((k+dd))GPIS queries.

Minimum Cut and Minimum k -Cut in Hypergraphs via Branching Contractions

On hypergraphs with m hyperedges and n vertices, where p denotes the total size of the hyperedges, we provide the following results: We give an algorithm that runs in \(\widetilde{O}(mn^{2k-2})\) time for finding a minimum k -cut in hypergraphs of arbitrary rank. This algorithm betters the previous best running time for the minimum k -cut problem, for k > 2. We give an algorithm that runs in \(\widetilde{O}(n^{\max \lbrace r,2k-2\rbrace })\) time for finding a minimum k -cut in hypergraphs of constant rank r . This algorithm betters the previous best running times for both the minimum cut and minimum k -cut problems for dense hypergraphs. Both of our algorithms are Monte Carlo, i.e., they return a minimum k -cut (or minimum cut) with high probability. These algorithms are obtained as instantiations of a generic branching randomized contraction technique on hypergraphs, which extends the celebrated work of Karger and Stein on recursive contractions in graphs. Our techniques and results also extend to the problems of minimum hedge-cut and minimum hedge- k -cut on hedgegraphs, which generalize hypergraphs.

On optimization problems in acyclic hypergraphs

Binary polynomial optimization (BPO) is the problem of maximizing a polynomial function on the Boolean domain. This problem can be formulated by using a hypergraph, and various properties of the input hypergraph have been investigated from the viewpoint of polynomial-time solvability. In this paper, we especially focus on β-acyclic hypergraphs. For BPO over β-acyclic hypergraphs, Del Pia and Di Gregorio proposed a polynomial-time algorithm. We prove that the algorithm proposed by Del Pia and Di Gregorio can be extended to a more general optimization problem in β-acyclic hypergraphs.

Path planning for a tracked robot traversing uneven terrains based on tip‐over stability

AbstractIn the real‐world environment, the path planning method of tracked robot is widely studied when driving on uneven terrain. How to solve the problem that the traditional path planning algorithm cannot adapt to uneven terrain because of the constraints of obstacle avoidance and path length is a challenge for tracked robots. In this paper, a stability‐based path planning framework for tracked robot is proposed to reduce the risk of rollover when the tracked robot passes through uneven terrain. First, a virtual plane method is proposed to estimate the attitude of tracked robot. Second, on this basis, a dynamic high‐stability path evaluation algorithm for tracked robot based on force angle stability margin (FASM) is proposed, which transforms the stability‐based path planning problem into a hypergraph problem. Moreover, considering that the optimization problem is strongly nonlinear and nonconvex, a hybrid algorithm of covariance matrix adaptive evolution strategy (CMAES) and Levenberg–Marquardt (LM) is designed under the framework of generalized graph optimization (G2O) to improve the solution efficiency. Finally, simulation and experiments show that the stability‐based path planning framework can effectively plan the high‐quality path, and the maximum stability of the tracked robot is 0.9156 when the robot crosses uneven terrain using optimal path 2.

The r-coloring and maximum stable set problem in hypergraphs with bounded matching number and edge size

Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting ν(G) denote the maximum size of a matching in H, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed r,k,s∈N:•r-Coloring in hypergraphs G with edge size at most k and ν(G)≤s;•r-Precoloring Extension in k-uniform hypergraphs G with ν(G)≤s;•r-Precoloring Extension in hypergraphs G with edge size at most k and ν(G)≤s;•Maximum Stable Set in k-uniform hypergraphs G with ν(G)≤s;•Maximum Weight Stable Set in k-uniform hypergraphs with ν(G)≤s; as well as partial results for r-Coloring in k-uniform hypergraphs ν(G)≤s. We then turn our attention to 2-Coloring in 3-uniform hypergraphs with forbidden induced subhypergraphs, and give a polynomial-time algorithm when restricting the input to hypergraphs excluding a fixed one-edge hypergraph. Finally, we consider linear 3-uniform hypergraphs (in which every two edges share at most one vertex), and show that excluding an induced matching in G implies that ν(G) is bounded by a constant; and that 3-coloring linear 3-uniform hypergraphs G with ν(G)≤532 is NP-hard.

An efficient adaptive degree-based heuristic algorithm for influence maximization in hypergraphs

Influence maximization (IM) has shown wide applicability in immense fields over the past decades. Previous researches on IM mainly focused on the dyadic relationship but lacked the consideration of higher-order relationship between entities, which has been constantly revealed in many real systems. An adaptive degree-based heuristic algorithm, i.e., Hyper Adaptive Degree Pruning (HADP) which aims to iteratively select nodes with low influence overlap as seeds, is proposed in this work to tackle the IM problem in hypergraphs. Furthermore, we extend algorithms from ordinary networks as baselines. Results on 8 empirical hypergraphs show that HADP surpasses the baselines in terms of both effectiveness and efficiency with a maximally 46.02% improvement. Moreover, we test the effectiveness of our algorithm on synthetic hypergraphs generated by different degree heterogeneity. It shows that the improvement of our algorithm effectiveness increases from 2.66% to 14.67% with the increase of degree heterogeneity, which indicates that HADP shows high performance especially in hypergraphs with high heterogeneity, which is ubiquitous in real-world systems.

Covering random graphs with monochromatic trees

AbstractGiven an ‐edge‐colored complete graph , how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well‐known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an ‐edge‐colored random graph . Recently, Bucić, Korándi, and Sudakov established a connection between this problem and a certain Helly‐type local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal, and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the 3‐color case by showing that is a threshold at which point three monochromatic components are needed to cover all vertices of a 3‐edge‐colored random graph, answering a question posed by Kohayakawa, Mendonça, Mota, and Schülke. Our approach also allows us to determine the answer in the general ‐edge colored instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Bucić, Korándi, and Sudakov.

Zeon and Idem-Clifford Formulations of Hypergraph Problems

Zeon algebras have proven to be useful for enumerating structures in graphs, such as paths, trails, cycles, matchings, cliques, and independent sets. In contrast to an ordinary graph, in which each edge connects exactly two vertices, an edge (or, “hyperedge”) can join any number of vertices in a hypergraph. Hypergraphs have been used for problems in biology, chemistry, image processing, wireless networks, and more. In the current work, zeon (“nil-Clifford”) and “idem-Clifford” graph-theoretic methods are generalized to hypergraphs. In particular, zeon and idem-Clifford methods are used to enumerate paths, trails, independent sets, cliques, and matchings in hypergraphs. An approach for finding minimum hypergraph transversals is developed, and zeon formulations of some open hypergraph problems are presented.

A unified approach to hypergraph stability

We present a method which provides a unified framework for most stability theorems that have been proved in graph and hypergraph theory. Our main result reduces stability for a large class of hypergraph problems to the simpler question of checking that a hypergraph H with large minimum degree that omits the forbidden structures is vertex-extendable. This means that if v is a vertex of H and H−v is a subgraph of the extremal configuration(s), then H is also a subgraph of the extremal configuration(s). In many cases vertex-extendability is quite easy to verify.We illustrate our approach by giving new short proofs of hypergraph stability results of Pikhurko, Hefetz-Keevash, Brandt-Irwin-Jiang, Bene Watts-Norin-Yepremyan and others. Since our method always yields minimum degree stability, which is the strongest form of stability, in some of these cases our stability results are stronger than what was known earlier. Along the way, we clarify the different notions of stability that have been previously studied.

Hypergraph Isomorphism for Groups with Restricted Composition Factors

We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group Γ over domain V , and asking whether there is a permutation γ ε Γ that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time ( n + m ) O ((log d ) c ) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for this problem due to Schweitzer and Wiebking (STOC 2019) runs in time n O ( d ) m O (1) . As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K 3, h ( h ≥ 3) as a minor in time n O ((log h ) c ) . In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time n O ((log g ) c ) .

High dimensional Hoffman bound and applications in extremal combinatorics

The n-th tensor power of a graph with vertex set V is the graph on the vertex set V n , where two vertices are connected by an edge if they are connected in each coordinate. One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. In this paper we introduce the problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many prominent open problems in extremal combinatorics, such as the Turán problem for graphs and hypergraphs, can be encoded as special cases of this problem. We generalize the Hoffman bound to hypergraphs, and give several applications.

The Turan density problem for hypergraphs

The Turan density problem for hypergraphs