16 research outputs found
Any monotone property of 3-uniform hypergraphs is weakly evasive
© 2014 Elsevier B.V. For a Boolean function f, let D(f) denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine f. In a classic paper, Rivest and Vuillemin [11] show that any non-constant monotone property P:{0,1}(n2)→{0,1} of n-vertex graphs has D(P)=Ω(n2).We extend their result to 3-uniform hypergraphs. In particular, we show that any non-constant monotone property P:{0,1}(n3)→{0,1} of n-vertex 3-uniform hypergraphs has D(P)=Ω(n3).Our proof combines the combinatorial approach of Rivest and Vuillemin with the topological approach of Kahn, Saks, and Sturtevant [6]. Interestingly, our proof makes use of Vinogradov's Theorem (weak Goldbach Conjecture), inspired by its recent use by Babai et al. [1] in the context of the topological approach. Our work leaves the generalization to k-uniform hypergraphs as an intriguing open question
Graph Properties in Node-Query Setting: Effect of Breaking Symmetry
The query complexity of graph properties is well-studied when queries are on the edges. We investigate the same when queries are on the nodes. In this setting a graph G = (V,E) on n vertices and a property P are given. A black-box access to an unknown subset S of V is provided via queries of the form "Does i belong to S?". We are interested in the minimum number of queries needed in the worst case in order to determine whether G[S] - the subgraph of G induced on S - satisfies P.
Our primary motivation to study this model comes from the fact that it allows us to initiate a systematic study of breaking symmetry in the context of query complexity of graph properties. In particular, we focus on the hereditary graph properties - properties that are closed under deletion of vertices as well as edges. The famous Evasiveness Conjecture asserts that even with a minimal symmetry assumption on G, namely that of vertex-transitivity, the query complexity for any hereditary graph property in our setting is the worst possible, i.e., n.
We show that in the absence of any symmetry on G it can fall as low as O(n^{1/(d + 1)}) where d denotes the minimum possible degree of a minimal forbidden sub-graph for P. In particular, every hereditary property benefits at least quadratically. The main question left open is: Can it go exponentially low for some hereditary property? We show that the answer is no for any hereditary property with finitely many forbidden subgraphs by exhibiting a bound of Omega(n^{1/k}) for a constant k depending only on the property. For general ones we rule out the possibility of the query complexity falling down to constant by showing Omega(log(n)*log(log(n))) bound. Interestingly, our lower bound proofs rely on the famous Sunflower Lemma due to Erdos and Rado
On the Sensitivity Complexity of k-Uniform Hypergraph Properties
In this paper we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n^{ceil(k/3)}) for any k >= 3, where n is the number of vertices. Moreover, we can do better when k = 1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n^{ceil(k/3)-1/2}). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Omega(n^{k/2}). We also investigate the sensitivity complexity of other weakly symmetric functions and show that for many classes of transitive-invariant Boolean functions the minimum achievable sensitivity complexity can be O(N^{1/3}), where N is the number of variables. Finally, we give a lower bound for sensitivity of k-uniform hypergraph properties, which implies the sensitivity conjecture of k-uniform hypergraph properties for any constant k
EXPLORING DIFFERENT MODELS OF QUERY COMPLEXITY AND COMMUNICATION COMPLEXITY
Ph.DDOCTOR OF PHILOSOPH
Extremal Problems on the Hypercube
PhDThe hypercube, Qd, is a natural and much studied combinatorial object, and we discuss
various extremal problems related to it.
A subgraph of the hypercube is said to be (Qd; F)-saturated if it contains no copies of
F, but adding any edge forms a copy of F. We write sat(Qd; F) for the saturation number,
that is, the least number of edges a (Qd; F)-saturated graph may have. We prove the
upper bound sat(Qd;Q2) < 10 2d, which strongly disproves a conjecture of Santolupo that
sat(Qd;Q2) =
�� 1 4 + o(1)
d2d��1. We also prove upper bounds on sat(Qd;Qm) for general
m.Given a down-set A and an up-set B in the hypercube, Bollobás and Leader conjectured
a lower bound on the number of edge-disjoint paths between A and B in the directed
hypercube. Using an unusual form of the compression argument, we confirm the conjecture
by reducing the problem to a the case of the undirected hypercube. We also prove an
analogous conjecture for vertex-disjoint paths using the same techniques, and extend both
results to the grid.
Additionally, we deal with subcube intersection graphs, answering a question of Johnson
and Markström of the least r = r(n) for which all graphs on n vertices may be represented as
subcube intersection graph where each subcube has dimension exactly r. We also contribute
to the related area of biclique covers and partitions, and study relationships between various
parameters linked to such covers and partitions.
Finally, we study topological properties of uniformly random simplicial complexes, employing
a characterisation due to Korshunov of almost all down-sets in the hypercube as a
key tool
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Any monotone property of 3-uniform hypergraphs is weakly evasive
For a Boolean function f, let D(f) denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine f. In a classic paper, Rivest and Vuillemin [18] show that any non-constant monotone property P: {0,1} (2n) → {0,1} of n-vertex graphs has D(P) = Ω (n). We extend their result to 3-uniform hypergraphs. In particular, we show that any non-constant monotone property P: {0,1} (3n) → {0,1} of n-vertex 3-uniform hypergraphs has D(P) = Ω (n). Our proof combines the combinatorial approach of Rivest and Vuillemin with the topological approach of Kahn, Saks, and Sturtevant. Interestingly, our proof makes use of Vinogradov's Theorem (weak Gold-bach Conjecture), inspired by its recent use by Babai et. al. [1] in the context of the topological approach. Our work leaves the generalization to k-uniform hypergraphs as an intriguing open question. © Springer-Verlag Berlin Heidelberg 2013