Showing papers by "Alexander A. Razborov published in 2017"
TL;DR: In this paper, the problem of minimizing the number of triangles in a graph of given order and size was considered and the asymptotic structure of extremal graphs was described by characterizing the set of flag algebra homomorphisms that minimize the triangle density.
Abstract: We consider the problem of minimizing the number of triangles in a graph of given order and size, and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.
32 citations
TL;DR: In this article, it was shown that for every fixed k, the number of occurrences of the transitive tournament Trk of order k in a tournament Tn on n vertices is asymptotically minimized when Tn is random.
Abstract: We prove that for every fixed k, the number of occurrences of the transitive tournament Trk of order k in a tournament Tn on n vertices is asymptotically minimized when Tn is random. In the opposite direction, we show that any sequence of tournaments {Tn} achieving this minimum for any fixed k⩾4 is necessarily quasirandom. We present several other characterizations of quasirandom tournaments nicely complementing previously known results and relatively easily following from our proof techniques.
28 citations
TL;DR: Ccol(P) coincides with the tree-width of the pattern P within a logarithmic factor, and it is shown that the previously known upper bound by Alon, Yuster, Zwick is almost tight, which strongly suggests that the colorful version of the subgraph isomorphism problem is much better structured and well-behaved than the standard (worstcase, uncolored) one.
Abstract: Let $P$ be a fixed graph (hereafter called a “pattern''), and let ${\sc Subgraph}(P)$ denote the problem of deciding whether a given graph $G$ contains a subgraph isomorphic to $P$. We are interested in $AC^0$-complexity of this problem, determined by the smallest possible exponent $C(P)$ for which ${\sc Subgraph}(P)$ possesses bounded-depth circuits of size $n^{C(P)+o(1)}$. Motivated by the previous research in the area, we also consider its “colorful” version ${\sc Subgraph}_\mathsf{col}(P)$ in which the target graph $G$ is $V(P)$-colored, and the average-case version ${\sc Subgraph}_\mathsf{ave}(P)$ under the distribution $G(n,n^{-\theta(P)})$, where $\theta(P)$ is the threshold exponent of $P$. Defining $C_\mathsf{col}(P)$ and $C_\mathsf{ave}(P)$ analogously to $C(P)$, our main contributions can be summarized as follows: (1) $C_\mathsf{col}(P)$ coincides with the treewidth of the pattern $P$ up to a logarithmic factor. This shows that the previously known upper bound by Alon, Yuster, and Zwick [J. ACM...
25 citations
TL;DR: An “ultimate” trade-off between width and rank is established, that is give an example in which small width proofs are possible but require exponentially many rounds to perform them.
Abstract: In this paper we study width of semialgebraic proof systems and various cut-based procedures in integer programming. We focus on two important systems: Gomory-Chvatal cutting planes and Lovasz-Schrijver lift-and-project procedures. We develop general methods for proving width lower bounds and apply them to random k-CNFs and several popular combinatorial principles, like the perfect matching principle and Tseitin tautologies. We also show how to apply our methods to various combinatorial optimization problems. We establish a “supercritical” trade-off between width and rank, that is we give an example in which small width proofs are possible but require exponentially many rounds to perform them.
13 citations