Paper Info
Title
On Fractional Block Sensitivity.
Abstract
In this paper we study the fractional block sensitivity of Boolean functions. Recently, Tal [Tal13] and Gilmer, Saks, and Srinivasan [GSS13] independently introduced this complexity measure, denoted by fbs(f), and showed that it is equal (up to a constant factor) to the randomized certificate complexity, denoted by RC(f), which was introduced by Aaronson [Aar03]. In this paper, we relate the fractional block sensitivity to other complexity measures such as sensitivity s(f) and approximate degree deg(f). As a consequence we obtain the following results: 1. We show that deg(f) = Ω( √ RC(f)), solving an open question posed by Aaronson [Aar03]. This also implies that deg(f) = Ω(QC(f)), where QC(f) is the quantum certificate complexity of f. As both deg(f) and QC(f) serve as lower bounds for the bounded error quantum query complexity, this shows that deg(f) is always a tighter lower bound compared to QC(f). 2. (a) We show that every transitive function on n variables must have RC(f) = Ω(n), QC(f) = Ω(n) and deg(f) = Ω(n), and note that all these bounds are tight. This is a strengthening of the previous lower bounds given by [SYZ04] and [Sun07]. (b) We show that Chakraborty’s [Cha11] example of a transitive function with s(f) = O(n) is optimal unless there is better than quadratic separation between the block sensitivity and the sensitivity. 3. Using fractional block sensitivity, we show that the zero error randomized decision tree complexity, R0(f), is upper bounded by O(R2(f) 2 · logR2(f)) where R2(f) is the two-sided bounded error randomized decision tree complexity of f . This improves the previous best relation between these two complexity measures given by Midrijanis [Mid05] of R0(f) = O(R2(f) 2 · log n) (where n is the number of variables). 4. We show that the (non-negative weight) adversary methods to lower bound the bounded error quantum query complexity of f can not give better bounds than √ RC(f)RC(f). This refines the earlier bound of √ C0(f)C1(f) by Spalek and Szegedy [SS06] and strengthens the so called certificate complexity barrier to its randomized analogue. ∗Centre for Quantum Technologies, Singapore Email: kulraghav@gmail.com †The Weizmann Institute of Science, Rehovot, Israel. Email: avishay.tal@weizmann.ac.il. Research supported by an Adams Fellowship of the Israel Academy of Sciences and Humanities, by an Israel Science Foundation grant and by the Israeli Centers of Research Excellence (I-CORE) program. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 168 (2013)
Year
Venue
DocType
2016
Chicago J. Theor. Comput. Sci.
Journal
Volume
Citations 
PageRank 
2016
1
0.37
References 
Authors
0
2
Name
Order
Citations
PageRank
Raghav Kulkarni117219.48
Avishay Tal25811.83