Hyperedges Of Size Research Articles

PCPs via the low-degree long code and hardness for constrained hypergraph coloring

We develop new techniques to incorporate the recently proposed “short code” (a low-degree version of the long code) into the construction and analysis of PCPs in the classical “label-cover + Fourier Analysis” framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs and certain coloring-type problems. In particular, we show a hardness for a variant of hypergraph coloring (with hyperedges of size 6), with a gap between 2 and $$\exp ({2^{\Omega (\sqrt {\log \log N} )}})$$ number of colors where N is the number of vertices. This is the first hardness result to go beyond the O(logN) barrier for a coloring-type problem. Our hardness bound is a doubly exponential improvement over the previously known O(log logN)-coloring hardness for 2-colorable hypergraphs, and an exponential improvement over the $${(\log N)^{\Omega (1)}}$$ -coloring hardness for O(1)-colorable hypergraphs. Stated in terms of “covering complexity,” we show that for 6-ary Boolean CSPs, it is hard to decide if a given instance is perfectly satisfiable or if it requires more than $${2^{\Omega (\sqrt {\log \log N} )}}$$ assignments for covering all of the constraints. While our methods do not yield a result for conventional hypergraph coloring due to some technical reasons, we also prove hardness of $${(\log N)^{\Omega (1)}}$$ -coloring 2-colorable 8-uniform hypergraphs (this result relies just on the long code). A key algebraic result driving our analysis concerns a very low-soundness error testing method for Reed-Muller codes. We prove that if a function $$\beta :{\text{F}}_2^m \to {{\text{F}}_2}$$ is $${2^{\Omega (d)}}$$ far in absolute distance from polynomials of degree m − d, then the probability that deg(β g ) ≤ m − 3d/4 for a random degree d/4 polynomial g is doubly exponentially small in d.

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 $$\delta $$?.

Cross-linked structure of network evolution.

We study the temporal co-variation of network co-evolution via the cross-link structure of networks, for which we take advantage of the formalism of hypergraphs to map cross-link structures back to network nodes. We investigate two sets of temporal network data in detail. In a network of coupled nonlinear oscillators, hyperedges that consist of network edges with temporally co-varying weights uncover the driving co-evolution patterns of edge weight dynamics both within and between oscillator communities. In the human brain, networks that represent temporal changes in brain activity during learning exhibit early co-evolution that then settles down with practice. Subsequent decreases in hyperedge size are consistent with emergence of an autonomous subgraph whose dynamics no longer depends on other parts of the network. Our results on real and synthetic networks give a poignant demonstration of the ability of cross-link structure to uncover unexpected co-evolution attributes in both real and synthetic dynamical systems. This, in turn, illustrates the utility of analyzing cross-links for investigating the structure of temporal networks.

Cover-Decomposition and Polychromatic Numbers

A coloring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each color; the polychromatic number is the maximum number of colors in such a coloring. Its dual, the cover-decomposition number, is the maximum number of disjoint hyperedge-covers. In geometric hypergraphs, there is extensive work on lower-bounding these numbers in terms of their trivial upper bounds (minimum hyperedge size and degree); our goal here is to broaden the study beyond geometric settings. We obtain algorithms yielding near-tight bounds for three families of hypergraphs: bounded hyperedge size, paths in trees, and bounded Vapnik--Chervonenkis (VC)-dimension. This reveals that discrepancy theory and iterated linear program relaxation are useful for cover-decomposition. Finally, we discuss the generalization of cover-decomposition to sensor cover.

Friendship 3-hypergraphs

A friendship 3-hypergraph is a 3-hypergraph in which for any 3 distinct vertices u, v and w, there exists a unique fourth vertex x such that uvx, uwx, vwx are 3-hyperedges. Sós constructed friendship 3-hypergraphs using Steiner triple systems. Hartke and Vandenbussche showed that any friendship 3-hypergraph can be partitioned into K43’s. (A K43 is the set of four hyperedges of size three that can be formed from a set of 4 elements.) These K43’s form a set of 4-tuples which we call a friendship design. We define a geometric friendship design to be a resolvable friendship design that can be embedded into an affine geometry. Refining the problem from friendship designs to geometric friendship designs allows us to state some structure results about these geometric friendship designs and decrease the search space when searching for geometric friendship designs. Hartke and Vandenbussche discovered 5 new examples of friendship designs which happen to be geometric friendship designs. We show the 3 non-isomorphic geometric designs on 16 vertices are the only such non-isomorphic geometric designs on 16 vertices. We also improve the known lower and upper bounds on the number of hyperedges in any friendship 3-hypergraph. Finally, we show that no friendship 3-hypergraph exists on 11 or 12 points.

Isomorphism and canonization of tournaments and hypertournaments

We give a polynomial-time oracle algorithm for Tournament Canonization that accesses oracles for Tournament Isomorphism and Rigid-Tournament Canonization. Extending the Babai–Luks Tournament Canonization algorithm (Babai and Luks (1983) [4]), we give an n O ( k 2 + log n ) algorithm for canonization and isomorphism testing of k-hypertournaments, where n is the number of vertices and k is the size of hyperedges.

The union of minimal hitting sets: Parameterized combinatorial bounds and counting

A k-hitting set in a hypergraph is a set of at most k vertices that intersects all hyperedges. We study the union of all inclusion-minimal k-hitting sets in hypergraphs of rank r (where the rank is the maximum size of hyperedges). We show that this union is relevant for certain combinatorial inference problems and give worst-case bounds on its size, depending on r and k. For r=2 our result is tight, and for each r⩾3 we have an asymptotically optimal bound and make progress regarding the constant factor. The exact worst-case size for r⩾3 remains an open problem. We also propose an algorithm for counting all k-hitting sets in hypergraphs of rank r. Its asymptotic runtime matches the best one known for the much more special problem of finding one k-hitting set. The results are used for efficient counting of k-hitting sets that contain any particular vertex.

Hypergraph connectivity augmentation

The hypergraph augmentation problem is to augment a hypergraph by hyperedges to meet prescribed local connectivity requirements. We provide here a minimax theorem for this problem. The result is derived from the degree constrained version of the problem by a standard method. We shall construct the required hypergraph for the latter problem by a greedy type algorithm. A similar minimax result will be given for the problem of augmenting a hypergraph by weighted edges (hyperedges of size two with weights) to meet prescribed local connectivity requirements. Moreover, a special case of an earlier result of Schrijver on supermodular colourings shall be derived from our theorem.

Extremal uncrowded hypergraphs

Let G be a ( k + 1)-graph (a hypergraph with each hyperedge of size k + 1) with n vertices and average degreee t. Assume k ⪡ t ⪡ n. If G is uncrowded (contains no cycle of size 2, 3, or 4) then there exists and independent set of size c k( n t )( ln t) 1 k .