Random hypergraphs and property B

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Abstract

In 1964 Erdős proved that (1+o1)eln(2)4k22k edges are sufficient to build a k-graph which is not two colorable. To this day, it is not known whether there exist such k-graphs with smaller number of edges. Erdős’ bound is consequence of the fact that a hypergraph with k22 vertices and M(k)=(1+o1)eln(2)4k22k randomly chosen edges of size k is asymptotically almost surely not two colorable. Our first main result implies that for any ε>0, any k-graph with (1ε)M(k) randomly and uniformly chosen edges is a.a.s. two colorable. The presented proof is an adaptation of the second moment method analogous to the developments of Achlioptas and Moore (2002) who considered the problem with fixed size of edges and number of vertices tending to infinity. In the second part of the paper we consider the problem of algorithmic coloring of random k-graphs. We show that quite simple, and somewhat greedy procedure, a.a.s. finds a proper two coloring for random k-graphs on k22 vertices, with at most Oklnk2k edges. That is of the same asymptotic order as the analogue of the algorithmic barrier defined by Achlioptas and Coja-Oghlan (2008), for the case of fixed k.

Introduction

The smallest number of edges in a k-graph (i.e. k-uniform hypergraph) that is not two colorable is denoted by m(k). Early results by Erdős from 60s [7], [8] determined that the exponential factor of the growth of m(k) is 2k (that is log2(m(k))k). In 1975 Erdős and Lovász in [9] suggested that perhaps k2k is the correct order of magnitude for m(k).

Despite a few improvements on the side of the lower bounds (the most recent one by Radhakrishnan and Srinivasan in [12]) the upper bound of Erdős from 1964 has not been improved since. He proved in [8] that m(k)(1+o1)eln(2)4k22k.

This bound results from the fact that the random hypergraph with that number of edges, over the set of vertices of size about k22, a.a.s. cannot be properly two colored. The chosen number of vertices, turns out to give the smallest number of edges in the random construction. Our motivation for taking a closer look on random k-graphs on k22 vertices stems mainly from this fact. Within these constraints we address two problems. First, we show that the bound of Erdős is tight for random hypergraphs. Then, we focus on random k-graphs with smaller number of edges and discuss the problem of finding a proper coloring efficiently.

Analogous problems have been considered in the context of random constraint satisfaction problems, where the size k of constraints/edges was fixed and the number of variables/vertices n was tending to infinity. In that framework, the straightforward first moment calculation shows that if r2k1ln(2)ln(2)2, then Hk(n,rn) is a.a.s. not two colorable. That observation was complemented by Achlioptas and Moore [3] who proved that, if r<2k1ln(2)(1+ln(2))2+ok(1), then random hypergraph Hk(n,rn) is a.a.s. two colorable. This is the first of the papers that directly inspire our developments (the bound itself has been later improved in [5] and [6]).

Second result from that area which provides a context for our considerations on efficient coloring, was proved by Achlioptas and Coja-Oghlan in [1]. They discussed the evolution of the space of solutions when successive random constraints are added to the instance. They observed that, at some point, the set of solutions, which in certain sense is initially connected, undergoes a sudden change after which it becomes shattered into exponentially many well separated regions. That behavior has been interpreted as a barrier for effective algorithms. Since that time, for some specific problems, algorithms and their analyses were improved up to the threshold of shattering (see e.g. [4]) but none surpassed that barrier.

The case of our interests, when n=Θ(k2), is slightly different, and most proofs for fixed k do not translate literally. The ideas of the proofs however usually can be applied. As shown by our work, the resulting proofs turn out to be technically simpler. Moreover, since the values like 2k1 are no longer constants, we were able to obtain the sharp threshold for two colorability of the random hypergraph in many cases. Note that for fixed k, this question is still open (however the results in [5] give very tight bounds). For a similar problem k-SAT with not too slowly growing k, Frieze and Wormald in [10] proved the existence of satisfiability threshold by analogous methods.

Section snippets

Main results

When discussing two coloring of a hypergraph, we use colors blue and red. We analyze the problem of coloring of random k-graphs with large k. Therefore asymptotic statements shall be understood with k. We say that a property holds asymptotically almost surely if the probability that it holds is 1o1. For positive integer b we denote by ab̲ the falling factorial, i.e. ab̲=j=0,,b1(aj).For positive integers n2k, we define φ=φ(n,k) to be such that (n2)k̲nk̲=φ(n,k)2k.Note that always 2kφ(n,

Proof of Theorem 1

We fix ε>0 and assume that n=n(k) and c=c(n,k) satisfy the assumptions of Theorem 1. In the canonical definition of Hk(n,m), edges of the hypergraph are sampled without replacement. That causes certain purely technical complications in the calculations. In order to avoid them we alter the probabilistic space slightly. Let Hk(n,m) be a random multi-hypergraph in which m edges are sampled with replacement from the set of all k-subsets of a set of size n. Clearly Hk(n,m) conditioned on the event

Algorithmic coloring of random k-graphs

Theorem 2 has been stated in terms of uniform random hypergraph Hk(n,m). However, closely related binomial model allows for much more natural proof. In a random hypergraph Hk(n,p) every k-subset of the set of vertices of size n is independently added to the hypergraph with probability p. We are going to work with the following technical statement.

Proposition 4

For any fixed α<12, and any superlinear polynomially bounded function n=n(k), there exists an efficient algorithm that on random k-graph Hk(n,p) with

References (12)

  • KravstovD. et al.

    Panchromatic 3-colorings of random hypergraphs

    European J. Combin.

    (2019)
  • AchlioptasD. et al.

    Algorithmic barriers from phase transitions

  • AchlioptasD. et al.

    Two-coloring random hypergraphs

    Random Struct. Algorithms

    (2002)
  • AchlioptasD. et al.

    On the 2-colorability of random hypergraphs

  • Coja-OghlanA.

    A better algorithm for random k-SAT

    SIAM J. Comput.

    (2010)
  • Coja-OghlanA. et al.

    Catching the k-NAESAT threshold [extended abstract]

There are more references available in the full text version of this article.

Cited by (0)

1

Research of L. Duraj and J. Kozik was partially supported by Polish National Science Center (2016/21/B/ST6/02165).

2

Research of D. Shabanov was supported by grant of the President of Russian Federation No. MD-1562.2020.1 and by the program ”Leading Scientific Schools ”, Grant No. NSh-2540.2020.1.

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