17 research outputs found
Tighter Relations Between Sensitivity and Other Complexity Measures
Sensitivity conjecture is a longstanding and fundamental open problem in the
area of complexity measures of Boolean functions and decision tree complexity.
The conjecture postulates that the maximum sensitivity of a Boolean function is
polynomially related to other major complexity measures. Despite much attention
to the problem and major advances in analysis of Boolean functions in the past
decade, the problem remains wide open with no positive result toward the
conjecture since the work of Kenyon and Kutin from 2004.
In this work, we present new upper bounds for various complexity measures in
terms of sensitivity improving the bounds provided by Kenyon and Kutin.
Specifically, we show that deg(f)^{1-o(1)}=O(2^{s(f)}) and C(f) < 2^{s(f)-1}
s(f); these in turn imply various corollaries regarding the relation between
sensitivity and other complexity measures, such as block sensitivity, via known
results. The gap between sensitivity and other complexity measures remains
exponential but these results are the first improvement for this difficult
problem that has been achieved in a decade.Comment: This is the merged form of arXiv submission 1306.4466 with another
work. Appeared in ICALP 2014, 14 page
Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma
We study the structure of sets with small sensitivity.
The well-known Simon's lemma says that any of
sensitivity must be of size at least . This result has been useful
for proving lower bounds on sensitivity of Boolean functions, with applications
to the theory of parallel computing and the "sensitivity vs. block sensitivity"
conjecture.
In this paper, we take a deeper look at the size of such sets and their
structure. We show an unexpected "gap theorem": if has
sensitivity , then we either have or . This is shown via classifying such sets into sets that can be
constructed from low-sensitivity subsets of for and
irreducible sets which cannot be constructed in such a way and then proving a
lower bound on the size of irreducible sets.
This provides new insights into the structure of low sensitivity subsets of
the Boolean hypercube
Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers
The Sensitivity Conjecture and the Log-rank Conjecture are among the most
important and challenging problems in concrete complexity. Incidentally, the
Sensitivity Conjecture is known to hold for monotone functions, and so is the
Log-rank Conjecture for and with monotone
functions , where and are bit-wise AND and XOR,
respectively. In this paper, we extend these results to functions which
alternate values for a relatively small number of times on any monotone path
from to . These deepen our understandings of the two conjectures,
and contribute to the recent line of research on functions with small
alternating numbers
Pseudorandom Generators for Low Sensitivity Functions
A Boolean function is said to have maximal sensitivity s if s is the largest number of Hamming neighbors of a point which differ from it in function value. We initiate the study of pseudorandom generators fooling low-sensitivity functions as an intermediate step towards settling the sensitivity conjecture. We construct a pseudorandom generator with seed-length 2^{O(s^{1/2})} log(n) that fools Boolean functions on n variables with maximal sensitivity at most s. Prior to our work, the (implicitly) best pseudorandom generators for this class of functions required seed-length 2^{O(s)} log(n)
On the Sensitivity Complexity of k-Uniform Hypergraph Properties
In this paper we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n^{ceil(k/3)}) for any k >= 3, where n is the number of vertices. Moreover, we can do better when k = 1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n^{ceil(k/3)-1/2}). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Omega(n^{k/2}). We also investigate the sensitivity complexity of other weakly symmetric functions and show that for many classes of transitive-invariant Boolean functions the minimum achievable sensitivity complexity can be O(N^{1/3}), where N is the number of variables. Finally, we give a lower bound for sensitivity of k-uniform hypergraph properties, which implies the sensitivity conjecture of k-uniform hypergraph properties for any constant k
On the Sensitivity Conjecture for Read-k Formulas
Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision tree depth, and degree over R, are all polynomially related to one another. The sensitivity conjecture states that there is also a polynomial relationship between sensitivity and block sensitivity, thus supplying the "missing link".
Since its introduction in 1991, the sensitivity conjecture has remained a challenging open question in the study of Boolean functions. One natural approach is to prove it for special classes of functions. For instance, the conjecture is known to be true for monotone functions, symmetric functions, and
functions describing graph properties.
In this paper, we consider the conjecture for Boolean functions computable by read-k formulas. A read-k formula is a tree in which each variable appears at most k times among the leaves and has Boolean gates at its internal nodes. We show that the sensitivity conjecture holds for read-once formulas with gates computing symmetric functions. We next consider regular formulas with OR and AND gates. A formula is regular if it is a leveled tree with all gates at a given level having the same fan-in and computing the same function. We prove the sensitivity conjecture for constant depth regular read-k formulas for constant k