Cross-intersecting families of vectors

Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \subseteq S_p$, are said to be $r$-cross-intersecting if no matter how we select $x \in A$ and $y \in B$, there are at least $r$ distinct indices $i$ such that $x_i = y_i$. We determine the maximum value of $|A|\cdot|B|$ over all pairs of $r$- cross-intersecting families and characterize the extremal pairs for $r \ge 1$, provided that $\min p_i >r+1$. The case $\min p_i \le r+1$ is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and answers a question of Zhang

On the communication complexity of sparse set disjointness and exists-equal problems

In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching lower and upper bounds (up to constant factors) for any number of rounds for both of these problems. In the sparse set disjointness problem, each player receives a k-subset of [m] and the goal is to determine whether the sets intersect. For this problem, we give a protocol that communicates a total of O(k\log^{(r)}k) bits over r rounds and errs with very small probability. Here we can take r=\log^{*}k to obtain a O(k) total communication \log^{*}k-round protocol with exponentially small error probability, improving on the O(k)-bits O(\log k)-round constant error probability protocol of Hastad and Wigderson from 1997. In the exist-equal problem, the players receive vectors x,y\in [t]^n and the goal is to determine whether there exists a coordinate i such that x_i=y_i. Namely, the exists-equal problem is the OR of n equality problems. Observe that exists-equal is an instance of sparse set disjointness with k=n, hence the protocol above applies here as well, giving an O(n\log^{(r)}n) upper bound. Our main technical contribution in this paper is a matching lower bound: we show that when t=\Omega(n), any r-round randomized protocol for the exists-equal problem with error probability at most 1/3 should have a message of size \Omega(n\log^{(r)}n). Our lower bound holds even for super-constant r <= \log^*n, showing that any O(n) bits exists-equal protocol should have \log^*n - O(1) rounds

Composable and Efficient Mechanisms

We initiate the study of efficient mechanism design with guaranteed good properties even when players participate in multiple different mechanisms simultaneously or sequentially. We define the class of smooth mechanisms, related to smooth games defined by Roughgarden, that can be thought of as mechanisms that generate approximately market clearing prices. We show that smooth mechanisms result in high quality outcome in equilibrium both in the full information setting and in the Bayesian setting with uncertainty about participants, as well as in learning outcomes. Our main result is to show that such mechanisms compose well: smoothness locally at each mechanism implies efficiency globally. For mechanisms where good performance requires that bidders do not bid above their value, we identify the notion of a weakly smooth mechanism. Weakly smooth mechanisms, such as the Vickrey auction, are approximately efficient under the no-overbidding assumption. Similar to smooth mechanisms, weakly smooth mechanisms behave well in composition, and have high quality outcome in equilibrium (assuming no overbidding) both in the full information setting and in the Bayesian setting, as well as in learning outcomes. In most of the paper we assume participants have quasi-linear valuations. We also extend some of our results to settings where participants have budget constraints

A constructive proof of the general Lovasz Local Lemma

The Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Simplifications of his procedure and relaxations of its restrictions were subsequently exhibited in several publications [Alo91, MR98, CS00, Mos06, Sri08, Mos08]. In [Mos09], a constructive proof was presented that works under negligible restrictions, formulated in terms of the Bounded Occurrence Satisfiability problem. In the present paper, we reformulate and improve upon these findings so as to directly apply to almost all known applications of the general Local Lemma.Comment: 8 page