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Series: Combinatorics Seminar

Joint work with Micha Sharir (Tel-Aviv University).

Following a recent improvement of Cardinal etal. on the complexity of a linear decision tree for k-SUM, resulting in O(n^3 \log^3{n}) linear queries, we present a further improvement to O(n^2 \log^2{n}) such queries. Our approach exploits a point-location mechanism in arrangements of hyperplanes in high dimensions, and, in fact, brings a new view to such mechanisms. In this talk I will first present a background on the k-SUM problem, and then discuss bottom-vertex triangulation and vertical decomposition of arrangements of hyperplanes and how they serve our analysis.

Series: Combinatorics Seminar

A graph is ``strongly regular'' (SRG) if it is $k$-regular, and every pair of adjacent (resp. nonadjacent) vertices has exactly $\lambda$ (resp. $\mu$) common neighbors. Paradoxically, the high degree of regularity in SRGs inhibits their symmetry. Although the line-graphs of the complete graph and complete bipartite graph give examples of SRGs with $\exp(\Omega(\sqrt{n}))$ automorphisms, where $n$ is the number of vertices, all other SRGs have much fewer---the best bound is currently $\exp(\tilde{O}(n^{9/37}))$ (Chen--Sun--Teng, 2013), and Babai conjectures that in fact all primitive SRGs besides the two exceptional line-graph families have only quasipolynomially-many automorphisms. In joint work with Babai, Chen, Sun, and Teng, we make progress toward this conjecture by giving a quasipolynomial bound on the number of automorphisms for valencies $k > n^{5/6}$. Our proof relies on bounds on the vertex expansion of SRGs to show that a polylogarithmic number of randomly chosen vertices form a base for the automorphism group with high probability.

Series: Combinatorics Seminar

The computational
complexity of many geometric problems depends on the dimension of the
input space. We study algorithmic problems on spaces of low fractal
dimension. There are several well-studied notions of fractal dimension
for sets and measures in Euclidean
space. We consider a definition of fractal dimension for finite metric
spaces, which agrees with standard notions used to empirically estimate
the fractal dimension of various sets.
When the fractal dimension of the input is lower than the ambient
dimension, we obtain faster algorithms for a plethora of classical
problems, including TSP, Independent Set, R-Cover, and R-Packing.
Interestingly, the dependence of the performance of these
algorithms on the fractal dimension closely resembles the currently
best-known dependence on the standard Euclidean dimension. For example,
our algorithm for TSP has running time 2^O(n^(1-1/delta) * log(n)), on
sets of fractal dimension delta; in comparison,
the best-known algorithm for sets in d-dimensional Euclidean space has
running time 2^O(n^(1-1/d)).

Series: Combinatorics Seminar

Joint work with Will Perkins and Prasad Tetali.

We consider the extremal counting problem which asks what d-regular, r-uniform hypergraph on n vertices has the largest number of (strong) independent sets. Our goal is to generalize known results for number of matchings and independent sets in regular graphs to give a general bound in the hypergraph case. In particular, we propose an adaptation to the hypergraph setting of the occupancy fraction method pioneered by Davies et al. (2016) for use in the case of graph matchings. Analysis of the resulting LP leads to a new bound for the case r=3 and suggests a method for tackling the general case.

Series: Combinatorics Seminar

One of the most interesting features of Erdös-Rényi random graphs is the `percolation phase transition', where the global structure intuitively changes from only small components to a single giant component plus small ones.
In this talk we discuss the percolation phase transition in the random d-process, which corresponds to a natural algorithmic model for generating random regular graphs (starting with an empty graph on n vertices, it evolves by sequentially adding new random edges so that the maximum degree remains at most d).
Our results on the phase transition solve a problem of Wormald from 1997, and verify a conjecture of Balinska and Quintas from 1990.
Based on joint work with Nick Wormald (Monash University).

Series: Combinatorics Seminar

A classical theorem of Spencer shows that any set system with n sets and n elements admits a coloring of discrepancy O(n^1/2). Recent exciting work of Bansal, Lovett and Meka shows that such colorings can be found in polynomial time. In fact, the Lovett-Meka algorithm finds a half integral point in any "large enough" polytope. However, their algorithm crucially relies on the facet structure and does not apply to general convex sets. We show that for any symmetric convex set K with measure at least exp(-n/500), the following algorithm finds a point y in K \cap [-1,1]^n with Omega(n) coordinates in {-1,+1}: (1) take a random Gaussian vector x; (2) compute the point y in K \cap [-1,1]^n that is closest to x. (3) return y. This provides another truly constructive proof of Spencer's theorem and the first constructive proof of a Theorem of Gluskin and Giannopoulos.

Series: Combinatorics Seminar

The Frankl union-closed sets conjecture states that there exists an element
present in at least half of the sets forming a union-closed family. We
reformulate the conjecture as an optimization problem and present an
integer program to model it. The computations done with this program lead
to a new conjecture: we claim that the maximum number of sets in a
non-empty union-closed family in which each element is present at most a
times is independent of the number n of elements spanned by the sets if n
is greater or equal to log_2(a)+1. We prove that this is true when n is
greater or equal to a. We also discuss the impact that this new conjecture
would have on the Frankl conjecture if it turns out to be true.
This is joint work with Jonad Pulaj and Dirk Theis.

Series: Combinatorics Seminar

Joint work with Yinon Spinka.

Consider a random coloring of a bounded domain in the bipartite graph Z^d with the probability of each color configuration proportional to exp(-beta*N(F)), where beta>0, and N(F) is the number of nearest neighboring pairs colored by the same color. This model of random colorings biased towards being proper, is the antiferromagnetic 3-state Potts model from statistical physics, used to describe magnetic interactions in a spin system. The Kotecky conjecture is that in such a model with d >= 3, Fixing the boundary of a large even domain to take the color $0$ and high enough beta, a sampled coloring would typically exhibits long-range order. In particular a single color occupies most of either the even or odd vertices of the domain. This is in contrast with the situation for small beta, when each bipartition class is equally occupied by the three colors. We give the first rigorous proof of the conjecture for large d. Our result extends previous works of Peled and of Galvin, Kahn, Randall and Sorkin, who treated the zero beta=infinity case, where the coloring is chosen uniformly for all proper three-colorings. In the talk we shell give a glimpse into the combinatorial methods used to tackle the problem. These rely on structural properties of odd-boundary subsets of Z^d. No background in statistical physics will be assumed and all terms will be thoroughly explained.

Series: Combinatorics Seminar

Both for random words or random permutations, I will present a panoramic view of results on the (asymptotic) behavior of the length of the longest common subsequences . Starting with, now, classical results on expectations dating back to the nineteen-seventies I will move to recent results obtained by Ümit Islak and myself giving the asymptotic laws of this length and as such answering a decades-old well know question.

Series: Combinatorics Seminar

We present an algebraic framework which simultaneously generalizes
the notion of linear subspaces, matroids, valuated matroids, and oriented
matroids. We call the resulting objects matroids over hyperfields. We give
"cryptomorphic" axiom systems for such matroids in terms of circuits,
Grassmann-Plucker functions, and dual pairs, and establish some basic
duality theorems.