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Dirk Oliver Theis
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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An $ntimes n$ matrix $M$ is called a textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,ell} M_{ell,k} = 0$ for every $k e ell$. Dietzfelbinger, Hromkovi{v{c}}, and Schnitger (1996) showed that $n le (mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $mbox{rk} M$ can be improved. We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = binom{mbox{rk} M+1}{2}$. In nonzero characteristic, we construct an infinite family of matrices with $n= (1+o(1))(mbox{rk} M)^2$.
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An ntimes n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k e ell we have M_{k,ell} M_{ell,k} = 0. Dietzfelbinger, Hromkoviv{c}, and Schnitger (1996) showed that n le (rk M)^2, regardless of over which field the rank is computed, and asked whether the exponent on rk M can be improved. We settle this question for nonzero characteristic by constructing a family of matrices for which the bound is asymptotically tight. The construction uses linear recurring sequences.
Say that A is a Hadamard factorization of the identity I_n of size n if the entrywise product of A and the transpose of A is I_n. It can be easily seen that the rank of any Hadamard factorization of the identity must be at least sqrt{n}. Dietzfelbinger et al. raised the question if this bound can be achieved, and showed a boolean Hadamard factorization of the identity of rank n^{0.792}. More recently, Klauck and Wolf gave a construction of Hadamard factorizations of the identity of rank n^{0.613}. Over finite fields, Friesen and Theis resolved the question, showing for a prime p and r=p^t+1 a Hadamard factorization of the identity A of size r(r-1)+1 and rank r over F_p. Here we resolve the question for fields of zero characteristic, up to a constant factor, giving a construction of Hadamard factorizations of the identity of rank r and size (r+1)r/2. The matrices in our construction are blockwise Toeplitz, and have entries whose magnitudes are binomial coefficients.
We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in $mathbb{R}^d$ that is covered by constant-sized sets of parallel hyperplanes, there exists an affine subspace that accounts for a large (i.e., $2^{-{operatorname{polylog}(d)}}$) fraction of the incidences. Alternatively, our conjecture may be interpreted linear-algebraically as follows: Any rank-$d$ matrix containing at most $O(1)$ distinct entries in each column contains a submatrix of fractional size $2^{-{operatorname{polylog}(d)}}$, in which each column contains one distinct entry. We prove that our conjecture is equivalent to the log-rank conjecture. Motivated by the connections above, we revisit well-studied questions in point-hyperplane incidence geometry without structural assumptions (i.e., the existence of partitions). We give an elementary argument for the existence of complete bipartite subgraphs of density $Omega(epsilon^{2d}/d)$ in any $d$-dimensional configuration with incidence density $epsilon$. We also improve an upper-bound construction of Apfelbaum and Sharir (SIAM J. Discrete Math. 07), yielding a configuration whose complete bipartite subgraphs are exponentially small and whose incidence density is $Omega(1/sqrt d)$. Finally, we discuss various constructions (due to others) which yield configurations with incidence density $Omega(1)$ and bipartite subgraph density $2^{-Omega(sqrt d)}$. Our framework and results may help shed light on the difficulty of improving Lovetts $tilde{O}(sqrt{operatorname{rank}(f)})$ bound (J. ACM 16) for the log-rank conjecture; in particular, any improvement on this bound would imply the first bipartite subgraph size bounds for parallel $3$-partitioned configurations which beat our generic bounds for unstructured configurations.
In the study of extensions of polytopes of combinatorial optimization problems, a notorious open question is that for the size of the smallest extended formulation of the Minimum Spanning Tree problem on a complete graph with $n$ nodes. The best known lower bound is $Omega(n^2)$, the best known upper bound is $O(n^3)$. In this note we show that the venerable fooling set method cannot be used to improve the lower bound: every fooling set for the Spanning Tree polytope has size $O(n^2)$.
A set $Ssubseteq 2^E$ of subsets of a finite set $E$ is emph{powerful} if, for all $Xsubseteq E$, the number of subsets of $X$ in $S$ is a power of 2. Each powerful set is associated with a non-negative integer valued function, which we call the rank function. Powerful sets were introduced by Farr and Wang as a generalisation of binary matroids, as the cocircuit space of a binary matroid gives a powerful set with the corresponding matroid rank function. In this paper we investigate how structural properties of a powerful set can be characterised in terms of its rank function. Powerful sets have four types of degenerate elements, including loops and coloops. We show that certain evaluations of the rank function of a powerful set determine the degenerate elements. We introduce powerful multisets and prove some fundamental results on them. We show that a powerful set corresponds to a binary matroid if and only if its rank function is subcardinal. This paper answers the two conjectures made by Farr and Wang in the affirmative.
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