Computer Science

Since their introduction in the 80s, sandpile models have raised interest for their simple definition and their surprising dynamical properties. In this survey we focus on the computational complexity of the prediction problem, namely, the complexity of knowing, given a finite configuration c and a cell x in c , if cell x will eventually become unstable. This is an attempt to formalize the intuitive notion of "behavioral complexity" that one easily observes in simulations. However, despite many efforts and nice results, the original question remains open: how hard is it to predict the two-dimensional sandpile model of Bak, Tang and Wiesenfeld?

The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, S n , one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of \emph{global functions} on S n , which are functions whose 2 -norm remains small when restricting O(1) coordinates of the input, and assert that low-degree, global functions have small q -norms, for q>2 . As applications, we show: 1. An analog of the level- d inequality on the hypercube, asserting that the mass of a global function on low-degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group A n . 2. Isoperimetric inequalities on the transposition Cayley graph of S n for global functions, that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube. 3. Hypercontractive inequalities on the multi-slice, and stability versions of the Kruskal--Katona Theorem in some regimes of parameters.

Hypergraphs were introduced in 1973 by Bergé. This review aims at giving some hints on the main results that we can find in the literature, both on the mathematical side and on their practical usage. Particularly, different definitions of hypergraphs are compared, some unpublished work on the visualisation of large hypergraphs done by the author. This review does not pretend to be exhaustive.

As data structures and mathematical objects used for complex systems modeling, hypergraphs sit nicely poised between on the one hand the world of network models, and on the other that of higher-order mathematical abstractions from algebra, lattice theory, and topology. They are able to represent complex systems interactions more faithfully than graphs and networks, while also being some of the simplest classes of systems representing topological structures as collections of multidimensional objects connected in a particular pattern. In this paper we discuss the role of (undirected) hypergraphs in the science of complex networks, and provide a mathematical overview of the core concepts needed for hypernetwork modeling, including duality and the relationship to bicolored graphs, quantitative adjacency and incidence, the nature of walks in hypergraphs, and available topological relationships and properties. We close with a brief discussion of two example applications: biomedical databases for disease analysis, and domain-name system (DNS) analysis of cyber data.

Strong spatial mixing (SSM) is a form of correlation decay that has played an essential role in the design of approximate counting algorithms for spin systems. A notable example is the algorithm of Weitz (2006) for the hard-core model on weighted independent sets. We study SSM for the q -colorings problem on the infinite (d+1) -regular tree. Weak spatial mixing (WSM) captures whether the influence of the leaves on the root vanishes as the height of the tree grows. Jonasson (2002) established WSM when q>d+1 . In contrast, in SSM, we first fix a coloring on a subset of internal vertices, and we again ask if the influence of the leaves on the root is vanishing. It was known that SSM holds on the (d+1) -regular tree when q>αd where α≈1.763... is a constant that has arisen in a variety of results concerning random colorings. Here we improve on this bound by showing SSM for q>1.59d . Our proof establishes an L 2 contraction for the BP operator. For the contraction we bound the norm of the BP Jacobian by exploiting combinatorial properties of the coloring of the tree.

A d -dimensional polycube is a facet-connected set of cells (cubes) on the d -dimensional cubical lattice Z d . Let A d (n) denote the number of d -dimensional polycubes (distinct up to translations) with n cubes, and λ d denote the limit of the ratio A d (n+1)/ A d (n) as n→∞ . The exact value of λ d is still unknown rigorously for any dimension d≥2 ; the asymptotics of λ d , as d→∞ , also remained elusive as of today. In this paper, we revisit and extend the approach presented by Klarner and Rivest in 1973 to bound A 2 (n) from above. Our contributions are: Using available computing power, we prove that λ 2 ≤4.5252 . This is the first improvement of the upper bound on λ 2 in almost half a century; We prove that λ d ≤(2d−2)e+o(1) for any value of d≥2 , using a novel construction of a rational generating function which dominates that of the sequence ( A d (n)) ; For d=3 , this provides a subtantial improvement of the upper bound on λ 3 from 12.2071 to 9.8073; However, we implement an iterative process in three dimensions, which improves further the upper bound on λ 3 to 9.3835 .

We show that, for every k≥2 , every k -uniform hypergaph of degree Δ and girth at least 5 is efficiently (1+o(1))(k−1)(Δ/lnΔ ) 1/(k−1) -list colorable. As an application we obtain the currently best deterministic algorithm for list-coloring random hypergraphs of bounded average degree.

Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant C such that for every positive integers a,b and a graph G , if the treedepth of G is at least Cab , then the treewidth of G is at least a or G contains a subcubic (i.e., of maximum degree at most 3 ) tree of treedepth at least b as a subgraph. As a direct corollary, we obtain that every graph of treedepth Ω( k 3 ) is either of treewidth at least k , contains a subdivision of full binary tree of depth k , or contains a path of length 2 k . This improves the bound of Ω( k 5 log 2 k) of Kawarabayashi and Rossman [SODA 2018]. We also show an application of our techniques for approximation algorithms of treedepth: given a graph G of treedepth k and treewidth t , one can in polynomial time compute a treedepth decomposition of G of width O(kt log 3/2 t) . This improves upon a bound of O(k t 2 logt) stemming from a tradeoff between known results. The main technical ingredient in our result is a proof that every tree of treedepth d contains a subcubic subtree of treedepth at least d⋅ log 3 ((1+ 5 – √ )/2) .

In 1964 Erd?s proved, by randomized construction, that the minimum number of edges in a k -graph that is not two colorable is O( k 2 2 k ) . To this day, it is not known whether there exist such k -graphs with smaller number of edges. Known deterministic constructions use much larger number of edges. The most recent one by Gebauer requires 2 k+?( k 2/3 ) edges. Applying derandomization technique we reduce that number to 2 k+ ? ? ( k 1/2 ) .

The T -tour problem is a natural generalization of TSP and Path TSP. Given a graph G=(V,E) , edge cost c:E→ R ≥0 , and an even cardinality set T⊆V , we want to compute a minimum-cost T -join connecting all vertices of G (and possibly containing parallel edges). In this paper we give an 11 7 -approximation for the T -tour problem and show that the integrality ratio of the standard LP relaxation is at most 11 7 . Despite much progress for the special case Path TSP, for general T -tours this is the first improvement on Sebő's analysis of the Best-of-Many-Christofides algorithm (Sebő [2013]).