1,082 research outputs found
Cross-intersecting families of vectors
Given a sequence of positive integers , let
denote the family of all sequences of positive integers
such that for all . Two families of sequences (or vectors),
, are said to be -cross-intersecting if no matter how we
select and , there are at least distinct indices
such that . We determine the maximum value of over all
pairs of - cross-intersecting families and characterize the extremal pairs
for , provided that . The case 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
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