Problem For Hypergraphs Research Articles

Acquaintance Time of Random Graphs Near Connectivity Threshold

Benjamini, Shinkar, and Tsur stated the following conjecture on the acquaintance time: asymptotically almost surely ${\mathcal A \mathcal C}(G) \le p^{-1} \log^{O(1)} n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently, Kinnersley, Mitsche, and Prałat made a major step toward this conjecture by showing that asymptotically almost surely ${\mathcal A \mathcal C}(G) = O(\log n / p)$, provided that $G$ has a Hamiltonian cycle. In this paper, we finish the task by showing that the conjecture holds in the strongest possible sense, that is, it holds right at the time the random graph process creates a connected graph. Moreover, we generalize and investigate the problem for random hypergraphs.

Hypergraphs with Zero Chromatic Threshold

Let F be an r-uniform hypergraph. The chromatic threshold of the family of F-free, r-uniform hypergraphs is the infimum of all non-negative reals c such that the subfamily of F-free, r-uniform hypergraphs H with minimum degree at least $$c \left( {\begin{array}{c}|V(H)| r-1\end{array}}\right) $$c|V(H)|r-1 has bounded chromatic number. The study of chromatic thresholds of various graphs has a long history, beginning with the early work of Erd?s and Simonovits. One interesting problem, first proposed by ?uczak and Thomass? and then solved by Allen, B?ttcher, Griffiths, Kohayakawa and Morris, is the characterization of graphs having zero chromatic threshold, in particular the fact that there are graphs with non-zero Turan density that have zero chromatic threshold. Here, we make progress on this problem for r-uniform hypergraphs, showing that a large class of hypergraphs have zero chromatic threshold in addition to exhibiting a family of constructions showing another large class of hypergraphs have non-zero chromatic threshold. Our construction is based on a particular product of the Bollobas---Erd?s graphs defined earlier by the authors.

Dynamic Shortest Path Algorithms for Hypergraphs

A hypergraph is a set $V$ of vertices and a set of nonempty subsets of $V$ , called hyperedges. Unlike graphs, hypergraphs can capture higher-order interactions in social and communication networks that go beyond a simple union of pairwise relationships. In this paper, we consider the shortest path problem in hypergraphs. We develop two algorithms for finding and maintaining the shortest hyperpaths in a dynamic network with both weight and topological changes. These two algorithms are the first to address the fully dynamic shortest path problem in a general hypergraph. They complement each other by partitioning the application space based on the nature of the change dynamics and the type of the hypergraph. We analyze the time complexity of the proposed algorithms and perform simulation experiments for random geometric hypergraphs, energy efficient routing in multichannel multiradio networks, and the Enron email data set. The experiment with the Enron email data set illustrates the application of the proposed algorithms in social networks for identifying the most important actor and the latent social relationship based on the closeness centrality metric.

Strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs

We consider multiobjective and parametric versions of the global minimum cut problem in undirected graphs and bounded-rank hypergraphs with multiple edge cost functions. For a fixed number of edge cost functions, we show that the total number of supported non-dominated (SND) cuts is bounded by a polynomial in the numbers of nodes and edges, i.e., is strongly polynomial. This bound also applies to the combinatorial facet complexity of the problem, i.e., the maximum number of facets (linear pieces) of the parametric curve for the parametrized (linear combination) objective, over the set of all parameter vectors such that the parametrized edge costs are nonnegative and the parametrized cut costs are positive. We sharpen this bound in the case of two objectives (the bicriteria problem), for which we also derive a strongly polynomial upper bound on the total number of non-dominated (Pareto optimal) cuts. In particular, the bicriteria global minimum cut problem in an n-node graph admits $$O(n^3 \log n)$$O(n3logn) SND cuts and $$O(n^5 \log n)$$O(n5logn) Pareto optimal cuts. These results significantly improve on earlier graph cut results by Mulmuley (SIAM J Comput 28(4):1460---1509, 1999) and Armon and Zwick (Algorithmica 46(1):15---26, 2006). They also imply that the parametric curve and all SND cuts, and, for the bicriteria problems, all Pareto optimal cuts, can be computed in strongly polynomial time when the number of objectives is fixed.

Minimizing Organizational User Requirement while Meeting Security Constraints

Large systems are complex and typically need automatic configuration to be managed effectively. In any organization, numerous tasks have to be carried out by employees. However, due to security needs, it is not feasible to directly assign any existing task to the first available employee. In order to meet many additional security requirements, constraints such as separation of duty, cardinality and binding have to be taken into consideration. Meeting these requirements imposes extra burden on organizations, which, however, is unavoidable in order to ensure security. While a trivial way of ensuring security is to assign each user to a single task, business organizations would typically like to minimize their costs and keep staffing requirements to a minimum. To meet these contradictory goals, we define the problem of Cardinality Constrained-Mutually Exclusive Task Minimum User Problem (CMUP), which aims to find the minimum users that can carry out a set of tasks while satisfying the given security constraints. We show that the CMUP problem is equivalent to a constrained version of the weak chromatic number problem in hypergraphs, which is NP-hard. We, therefore, propose a greedy solution. Our experimental evaluation shows that the proposed algorithm is both efficient and effective.

Streaming Algorithms for Independent Sets in Sparse Hypergraphs

We give the first treatment of the classic independent set problem in graphs and hypergraphs in the streaming setting. The objective is to find space-efficient algorithms that output independent sets that are "combinatorially optimal", that is, with size guarantee in terms of the degree sequence alone. Our main result is a randomized algorithm that achieves this using space in bits that is linear in the number of vertices. We use this to examine assumptions about the streaming model, and advocate the study of output-efficient algorithms that measure space usage relative to the size of the output solution. In that sense, our main algorithm uses space linear in the output size. We also examine algorithms that use little or no space in addition to the bits storing the output. Our algorithms fall also into an online streaming model, where output-changes can go only in one direction. In particular a feasible solution must be maintained at all times, and items that are removed from the solution can never reenter. We obtain tight bounds on deterministic algorithms for independent sets in graphs in that model.

MaxMinMax problem and sparse equations over finite fields

Asymptotical complexity of polynomial equation systems over finite field $$F_q$$Fq is studied. Let $${\mathcal {X}}=\{X_1,\ldots ,X_m\},\, |\bigcup _{i=1}^{m}X_i|\le n$$X={X1,?,Xm},|?i=1mXi|≤n be a fixed family of variable sets and the polynomials $$f_i(X_i)$$fi(Xi) are taken independently and uniformly at random from the set of all polynomials of degree $${\le }q-1$$≤q-1 in each of the variables in $$X_i$$Xi. In particular, it is proved if $$|X_i|\le 3, m=n$$|Xi|≤3,m=n, then the average complexity of finding all solutions in $$F_q$$Fq to $$f_i(X_i)=0\, (1\le i\le m)$$fi(Xi)=0(1≤i≤m) is at most $$ q^{\frac{n}{5.7883}+O(\log n)}$$qn5.7883+O(logn) for arbitrary $${\mathcal {X}}$$X and $$q$$q. The proof is based on a detailed analysis of MaxMinMax problem, a novel problem for hypergraphs.

Myhill–Nerode Methods for Hypergraphs

We give an analog of the Myhill-Nerode methods from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems: * We provide an algorithm for testing whether a hypergraph has cutwidth at most k that runs in linear time for constant k. In terms of parameterized complexity theory, the problem is fixed-parameter linear parameterized by k. * We show that it is not expressible in monadic second-order logic whether a hypergraph has bounded (fractional, generalized) hypertree width. The proof leads us to conjecture that, in terms of parameterized complexity theory, these problems are W[1]-hard parameterized by the incidence treewidth (the treewidth of the incidence graph). Thus, in the form of the Myhill-Nerode theorem for hypergraphs, we obtain a method to derive linear-time algorithms and to obtain indicators for intractability for hypergraph problems parameterized by incidence treewidth. In an appendix, we point out an error and a fix to the proof of the Myhill-Nerode theorem for graphs in Downey and Fellow's book on parameterized complexity.

Connected Colourings of Complete Graphs and Hypergraphs

Gallai's colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all three colours. What happens for more colours: if we $$k$$k-colour the edges of the complete graph, with each colour class connected, how many of the $$\left( {\begin{array}{c}k 3\end{array}}\right) $$k3 triples of colours must appear as triangles? In this note we show that the `obvious' conjecture, namely that there are always at least $$\left( {\begin{array}{c}k-1 2\end{array}}\right) $$k-12 triples, is not correct. We determine the minimum asymptotically. This answers a question of Johnson. We also give some results about the analogous problem for hypergraphs, and we make a conjecture that we believe is the `right' generalisation of Gallai's theorem to hypergraphs.

Randomized Approximation for the Set Multicover Problem in Hypergraphs

Let $$b\in {\mathbb {N}}_{\ge 1}$$b?N?1 and let $${\mathcal {H}}=(V,{\mathcal {E}})$$H=(V,E) be a hypergraph with maximum vertex degree $${\varDelta }$$Δ and maximum edge size $$l$$l. A set $$b$$b-multicover in $${\mathcal {H}}$$H is a set of edges $$C \subseteq {\mathcal {E}}$$C⊆E such that every vertex in $$V$$V belongs to at least $$b$$b edges in $$C$$C. $${\textsc {set }}\, b\text {-}{\textsc {multicover}}$$SETb-MULTICOVER is the problem of finding a set $$b$$b-multicover of minimum cardinality, and for $$b=1$$b=1 it is the fundamental set cover problem. Peleg et al. (Algorithmica 18(1):44---66, 1997) gave a randomized algorithm achieving an approximation ratio of $$\delta \cdot \big (1-\big (\frac{c}{n}\big )^\frac{1}{\delta }\big )$$?·(1-(cn)1?), where $$\delta := {\varDelta }- b + 1$$?:=Δ-b+1 and $$c>0$$c>0 is a constant. As this ratio depends on the instance size $$n$$n and tends to $$\delta $$? as $$n$$n tends to $$\infty $$?, it remained an open problem whether an approximation ratio of $$\delta \alpha $$?? with a constant$$\alpha < 1$$?<1 can be proved. In fact, the authors conjectured that for any fixed $${\varDelta }$$Δ and $$b$$b, the problem is not approximable within a ratio smaller than $$\delta $$?, unless $${\mathcal {P}}={\mathcal {NP}}$$P=NP. We present a randomized algorithm of hybrid type for $${\textsc {set }}\, b\text {-}{\textsc {multicover}}$$SETb-MULTICOVER, $$b \ge 2$$b?2, combining LP-based randomized rounding with greedy repairing, and achieve an approximation ratio of $$\delta \cdot \left( 1 - \frac{11({\varDelta }- b)}{72l} \right) $$?·1-11(Δ-b)72l for hypergraphs with maximum edge size $$l \in {\mathcal {O}}\left( \max \big \{(nb)^\frac{1}{5},n^\frac{1}{4}\big \}\right) $$l?Omax{(nb)15,n14}. In particular, for all hypergraphs where $$l$$l is constant, we get an $$\alpha \delta $$??-ratio with constant $$\alpha < 1$$?<1. Hence the above stated conjecture does not hold for hypergraphs with constant $$l$$l and we have identified the boundedness of the maximum hyperedge size as a relevant parameter responsible for approximations below $$\delta $$?.

An approximation algorithm for the partial vertex cover problem in hypergraphs

Let $$\mathcal {H}=(V,\mathcal {E})$$H=(V,E) be a hypergraph with set of vertices $$V, n:=|V|$$V,n:=|V| and set of (hyper-)edges $$\mathcal {E}, m:=|\mathcal {E}|$$E,m:=|E|. Let $$l$$l be the maxim...

Minimum number of affine simplices of given dimension

In this paper we formulate and solve extremal problems in the Euclidean space Rd and further in hypergraphs, originating from problems in stoichiometry and elementary linear algebra. The notion of affine simplex is the bridge between the original problems and the presented extremal theorem on set systems. As a sample corollary, it follows that if no triple is collinear in a set S of n points in R3, then S contains at least n4−cn3 affine simplices for some constant c. A function related to Sperner’s Theorem and its well-known extension to reciprocal sums is also considered and its relation to Turán’s hypergraph problems is discussed.

Ramsey theory for discrete structures

This monograph covers some of the most important developments in Ramsey theory from its beginnings in the early 20th century via its many breakthroughs to recent important developments in the early 21st century. The book first presents a detailed discussion of the roots of Ramsey theory before offering a thorough discussion of the role of parameter sets. It presents several examples of structures that can be interpreted in terms of parameter sets and features the most fundamental Ramsey-type results for parameter sets: Hales-Jewett's theorem and Graham-Rothschilds Ramsey theorem as well as their canonical versions and several applications. Next, the book steps back to the most basic structure, to sets. It reviews classic results as well as recent progress on Ramsey numbers and the asymptotic behavior of classical Ramsey functions. In addition, it presents product versions of Ramsey's theorem, a combinatorial proof of the incompleteness of Peano arithmetic, provides a digression to discrepancy theory and examines extensions of Ramsey's theorem to larger cardinals. The next part of the book features an in-depth treatment of the Ramsey problem for graphs and hypergraphs. It gives an account on the existence of sparse and restricted Ramsey theorem's using sophisticated constructions as well as probabilistic methods. Among others it contains a proof of the induced Graham-Rothschild theorem and the random Ramsey theorem. The book closes with a chapter on one of the recent highlights of Ramsey theory: a combinatorial proof of the density Hales-Jewett theorem. This book provides graduate students as well as advanced researchers with a solid introduction and reference to the field.

Monochromatic Loose-Cycle Partitions in Hypergraphs

In this paper we study the monochromatic loose-cycle partition problem for non-complete hypergraphs. Our main result is that in any $r$-coloring of a $k$-uniform hypergraph with independence number $\alpha$ there is a partition of the vertex set into monochromatic loose cycles such that their number depends only on $r$, $k$ and $\alpha$. We also give an extension of the following result of Pósa to hypergraphs: the vertex set of every graph $G$ can be partitioned into at most $\alpha(G)$ cycles, edges and vertices.

A Ramsey-type result for geometric [formula omitted]-hypergraphs

Let n≥ℓ≥2 and q≥2. We consider the minimum N such that whenever we have N points in the plane in general position and the ℓ-subsets of these points are colored with q colors, there is a subset S of n points all of whose ℓ-subsets have the same color and furthermore S is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hypergraphs and the Erdős–Szekeres theorem on convex configurations in the plane. For the special case ℓ=2, we establish a single exponential bound on the minimum N, such that every complete N-vertex geometric graph whose edges are colored with q colors, yields a monochromatic convex geometric graph on n vertices.For fixed ℓ≥2 and q≥4, our results determine the correct exponential tower growth rate for N as a function of n, similar to the usual hypergraph Ramsey problem, even though we require our monochromatic set to be in convex position. Our results also apply to the case of ℓ=3 and q=2 by using a geometric variation of the stepping up lemma of Erdős and Hajnal. This is in contrast to the fact that the upper and lower bounds for the usual 3-uniform hypergraph Ramsey problem for two colors differ by one exponential in the tower.

On the Enumeration of Minimal Dominating Sets and Related Notions

A dominating set $D$ in a graph is a subset of its vertex set such that each vertex is either in $D$ or has a neighbor in $D$. In this paper, we are interested in the enumeration of (inclusionwise) minimal dominating sets in graphs, called the Dom-Enum problem. It is well known that this problem can be polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the problem of enumerating all minimal transversals in a hypergraph. First, we show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum problem. As a consequence there exists an output-polynomial time algorithm for the Trans-Enum problem if and only if there exists one for the Dom-Enum problem. Second, we study the Dom-Enum problem in some graph classes. We give an output-polynomial time algorithm for the Dom-Enum problem in split graphs and introduce the completion of a graph to obtain an output-polynomial time algorithm for the Dom-Enum problem in $P_6$-free chordal graphs, a proper superclass of split graphs. Finally...

Spectral Extremal Problems for Hypergraphs

In this paper we consider spectral extremal problems for hypergraphs. We give two general criteria under which such results may be deduced from “strong stability” forms of the corresponding (pure) extremal results. These results hold for the $\alpha$-spectral radius defined using the $\alpha$-norm for any $\alpha>1$; the usual spectral radius is the case $\alpha=2$. Our results imply that any hypergraph Turán problem which has the stability property and whose extremal construction satisfies some rather mild continuity assumptions admits a corresponding spectral result. A particular example is to determine the maximum $\alpha$-spectral radius of any 3-uniform hypergraph on $n$ vertices not containing the Fano plane, when $n$ is sufficiently large. Another is to determine the maximum $\alpha$-spectral radius of any graph on $n$ vertices not containing some fixed color-critical graph, when $n$ is sufficiently large; this generalizes a theorem of Nikiforov who proved stronger results in the case $\alpha=2$. We also obtain an $\alpha$-spectral version of the Erdös--Ko--Rado theorem on $t$-intersecting $k$-uniform hypergraphs.

On the Erdős-Hajnal problem for 3-uniform hypergraphs

On the Erdős-Hajnal problem for 3-uniform hypergraphs

On an extremal hypergraph problem related to combinatorial batch codes

Let n,r,k be positive integers such that 3≤k<n and 2≤r≤k−1. Let m(n,r,k) denote the maximum number of edges an r-uniform hypergraph on n vertices can have under the condition that any collection of i edges spans at least i vertices for all 1≤i≤k. We are interested in the asymptotic nature of m(n,r,k) for fixed r and k as n→∞. This problem is related to the forbidden hypergraph problem introduced by Brown, Erdős, and Sós and very recently discussed in the context of combinatorial batch codes (CBCs). In this short paper we obtain the following results. (i)Using a result due to Erdős we are able to show m(n,r,k)=o(nr) for 7≤k, and 3≤r≤k−1−⌈logk⌉. This result is best possible with respect to the upper bound on r as we subsequently show through explicit construction that for 6≤k, and k−⌈logk⌉≤r≤k−1,m(n,r,k)=Θ(nr).This explicit construction improves on the non-constructive general lower bound obtained by Brown, Erdős, and Sós for the considered parameter values.(ii)For 2-uniform CBCs we obtain the following results. (a)We provide exact value of m(n,2,5) for n≥5.(b)Using a result of Lazebnik et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n,2,k) (Ω(nk+1k−1)) for all k≥8 and infinitely many values of n.(c)We show m(n,2,k)=O(n1+1⌊k4⌋) by using a result due to Bondy and Simonovits, and also show m(n,2,k)=Θ(n32) for k=6,7,8 by using a result of Kővári, Sós, and Turán.

Hypergraph Ramsey numbers: Triangles versus cliques

A celebrated result in Ramsey Theory states that the order of magnitude of the triangle-complete graph Ramsey numbers R(3,t) is t2/logt. In this paper, we consider an analogue of this problem for uniform hypergraphs. A triangle is a hypergraph consisting of edges e,f,g such that |e∩f|=|f∩g|=|g∩e|=1 and e∩f∩g=∅. For all r⩾2, let R(C3,Ktr) be the smallest positive integer n such that in every red–blue coloring of the edges of the complete r-uniform hypergraph Knr, there exists a red triangle or a blue Ktr. We show that there exist constants a,br>0 such that for all t⩾3,at32(logt)34⩽R(C3,Kt3)⩽b3t32 and for r⩾4t32(logt)34+o(1)⩽R(C3,Ktr)⩽brt32. This determines up to a logarithmic factor the order of magnitude of R(C3,Ktr). We conjecture that R(C3,Ktr)=o(t3/2) for all r⩾3. We also study a generalization to hypergraphs of cycle-complete graph Ramsey numbers R(Ck,Kt) and a connection to r3(N), the maximum size of a set of integers in {1,2,…,N} not containing a three-term arithmetic progression.