52 research outputs found
Catalytic space: Non-determinism and hierarchy
Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation
{Improved Bounds on Fourier Entropy and Min-entropy}
Given a Boolean function , the Fourier distribution assigns probability to . The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai asks if there exist a universal constant C>0 such that , where is the Shannon entropy of the Fourier distribution of and is the total influence of . 1) We consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture. This asks if , where is the min-entropy of the Fourier distribution. We show , where is the minimum parity certificate complexity of . We also show that for every , we have , where is the approximate spectral norm of . As a corollary, we verify the FMEI conjecture for the class of read- s (for constant ). 2) We show that , where is the average unambiguous parity certificate complexity of . This improves upon Chakraborty et al. An important consequence of the FEI conjecture is the long-standing Mansour's conjecture. We show that a weaker version of FEI already implies Mansour's conjecture: is ?, where are the 0- and 1-certificate complexities of , respectively. 3) We study what FEI implies about the structure of polynomials that 1/3-approximate a Boolean function. We pose a conjecture (which is implied by FEI): no "flat" degree- polynomial of sparsity can 1/3-approximate a Boolean function. We prove this conjecture unconditionally for a particular class of polynomials
A separator theorem for hypergraphs and a CSP-SAT algorithm
We show that for every r≥2 there exists ϵr>0 such that any r-uniform hypergraph with m edges and maximum vertex degree o(m−−√) contains a set of at most (12−ϵr)m edges the removal of which breaks the hypergraph into connected components with at most m/2 edges. We use this to give an algorithm running in time d(1−ϵr)m that decides satisfiability of m-variable (d,k)-CSPs in which every variable appears in at most r constraints, where ϵr depends only on r and k∈o(m−−√). Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable (2,k)-CSPs with variable frequency r can be refuted in tree-like resolution in size 2(1−ϵr)m. Furthermore for Tseitin formulas on graphs with degree at most k (which are (2,k)-CSPs) we give a deterministic algorithm finding such a refutation
High Entropy Random Selection Protocols
In this paper, we construct protocols for two parties that do not trust each other,
to generate random variables with high Shannon entropy.
We improve known bounds for the trade off between the number of rounds, length of communication and the entropy of the outcome
Inverting Onto Functions and Polynomial Hierarchy
In this paper we construct an oracle under which
the polynomial hierarchy is infinite but
there are non-invertible polynomial time computable multivalued onto functions
Catalytic space: non-determinism and hierarchy
Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation
Lower bounds for elimination via weak regularity
We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. [1] and later studied by Beimel et al. [4] for its connection to the famous direct sum question. In this problem, let f: {0, 1}2n → {0,1} be any boolean function. Alice and Bob get k inputs x1,⋯, xk and y1,⋯, yk respectively, with xi, yi ∈ {0, 1}n. They want to output a k-bit vector v, such that there exists one index i for which vi = f(xi,yi). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds. To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson [19]. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola [23], to show that Greater-Than is weakly regular
- …