168 research outputs found
Evasive Properties of Sparse Graphs and Some Linear Equations in Primes
We give an unconditional version of a conditional, on the Extended Riemann
Hypothesis, result of L. Babai, A. Banerjee, R. Kulkarni and V. Naik (2010) on
the evasiveness of sparse graphs.Comment: This version corrects a mistake made in the previous version, which
was pointed out to the author by Laszlo Baba
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
Collapsibility of CAT(0) spaces
Collapsibility is a combinatorial strengthening of contractibility. We relate
this property to metric geometry by proving the collapsibility of any complex
that is CAT(0) with a metric for which all vertex stars are convex. This
strengthens and generalizes a result by Crowley. Further consequences of our
work are:
(1) All CAT(0) cube complexes are collapsible.
(2) Any triangulated manifold admits a CAT(0) metric if and only if it admits
collapsible triangulations.
(3) All contractible d-manifolds () admit collapsible CAT(0)
triangulations. This discretizes a classical result by Ancel--Guilbault.Comment: 27 pages, 3 figures. The part on collapsibility of convex complexes
has been removed and forms a new paper, called "Barycentric subdivisions of
convexes complex are collapsible" (arXiv:1709.07930). The part on enumeration
of manifolds has also been removed and forms now a third paper, called "A
Cheeger-type exponential bound for the number of triangulated manifolds"
(arXiv:1710.00130
Evasiveness of Graph Properties and Topological Fixed-Point Theorems
Many graph properties (e.g., connectedness, containing a complete subgraph)
are known to be difficult to check. In a decision-tree model, the cost of an
algorithm is measured by the number of edges in the graph that it queries. R.
Karp conjectured in the early 1970s that all monotone graph properties are
evasive -- that is, any algorithm which computes a monotone graph property must
check all edges in the worst case. This conjecture is unproven, but a lot of
progress has been made. Starting with the work of Kahn, Saks, and Sturtevant in
1984, topological methods have been applied to prove partial results on the
Karp conjecture. This text is a tutorial on these topological methods. I give a
fully self-contained account of the central proofs from the paper of Kahn,
Saks, and Sturtevant, with no prior knowledge of topology assumed. I also
briefly survey some of the more recent results on evasiveness.Comment: Book version, 92 page
Counting induced subgraphs: a topological approach to #W[1]-hardness
We investigate the problem of counting all induced subgraphs of size in a graph that satisfy a given property . This continues the work of Jerrum and Meeks who proved the problem to be -hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties , the problem is hard for if the reduced Euler characteristic of the associated simplicial (graph) complex of is non-zero. This observation links to Karp's famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the "topological approach to evasiveness" which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS 83], we prove that is -hard for every monotone property that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not -edge-connected for . Moreover, we show that for those properties can not be solved in time for any computable function unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that is -hard if is any non-trivial modularity constraint on the number of edges with respect to some prime or if enforces the presence of a fixed isolated subgraph
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
Decision trees, monotone functions, and semimatroids
We define decision trees for monotone functions on a simplicial complex. We
define homology decidability of monotone functions, and show that various
monotone functions related to semimatroids are homology decidable. Homology
decidability is a generalization of semi-nonevasiveness, a notion due to
Jonsson. The motivating example is the complex of bipartite graphs, whose Betti
numbers are unknown in general.
We show that these monotone functions have optimum decision trees, from which
we can compute relative Betti numbers of related pairs of simplicial complexes.
Moreover, these relative Betti numbers are coefficients of evaluations of the
Tutte polynomial, and every semimatroid collapses onto its broken circuit
complex.Comment: 16 page
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