Denis Pankratov

From information to exact communication

We develop a new local characterization of the zero-error information complexity function for two-party communication problems, and use it to compute the exact internal and external information complexity of the 2-bit AND function: IC(AND,0) = C_∧≅ 1.4923 bits, and IC^ext(AND,0) = log₂ 3 ≅ 1.5839 bits. This leads to a tight (upper and lower bound) characterization of the communication complexity of the set intersection problem on subsets of {1,...,n} (the player are required to compute the intersection of their sets), whose randomized communication complexity tends to C_∧⋅ n pm o(n) as the error tends to zero. The information-optimal protocol we present has an infinite number of rounds. We show this is necessary by proving that the rate of convergence of the r-round information cost of AND to IC(AND,0)=C_∧ behaves like Θ(1/r²), i.e. that the r-round information complexity of AND is C_∧+Θ(1/r²). We leverage the tight analysis obtained for the information complexity of AND to calculate and prove the exact communication complexity of the <i>set disjointness</i> function Disj_n(X,Y) = - v_i=1ⁿ AND(x_i,y_i) with error tending to 0, which turns out to be = C_DISJ⋅ n pm o(n), where C_DISJ≅ 0.4827. Our rate of convergence results imply that an asymptotically optimal protocol for set disjointness will have to use ω(1) rounds of communication, since every r-round protocol will be sub-optimal by at least Ω(n/r²) bits of communication. We also obtain the tight bound of 2/ln2 k pm o(k) on the communication complexity of disjointness of sets of size ≤ k. An asymptotic bound of Θ(k) was previously shown by Hastad and Wigderson.

Calibree: Calibration-Free Localization Using Relative Distance Estimations

Existing localization algorithms, such as centroid or fingerprinting, compute the location of a mobile device based on measurements of signal strengths from radio base stations. Unfortunately, these algorithms require tedious and expensive off-line calibration in the target deployment area before they can be used for localization. In this paper, we present Calibree, a novel localization algorithm that does not require off-line calibration. The algorithm starts by computing relative distances between pairs of mobile phones based on signatures of their radio environment. It then combines these distances with the known locations of a small number of GPS-equipped phones to estimate absolute locations of all phones, effectively spreading location measurements from phones with GPS to those without. Our evaluation results show that Calibree performs better than the conventional centroid algorithm and only slightly worse than fingerprinting, without requiring off-line calibration. Moreover, when no phones report their absolute locations, Calibree can be used to estimate relative distances between phones.

Variations on the Sensitivity Conjecture

We present a selection of known as well as new variants of the Sensitivity Conjecture and point out some weaker versions that are also open.

Information Lower Bounds via Self-reducibility

We use self-reduction methods to prove strong information lower bounds on two of the most studied functions in the communication complexity literature: Gap Hamming Distance (GHD) and Inner Product (IP). In our first result we affirm the conjecture that the information cost of GHD : is linear even under the uniform distribution, which strengthens the Ω(n) bound recently shown by [15], and answering an open problem from [10]. In our second result we prove that the information cost of IP n is arbitrarily close to the trivial upper bound n as the permitted error tends to zero, again strengthening the Ω(n) lower bound recently proved by [9].

On the relative merits of simple local search methods for the MAX-SAT problem

Algorithms based on local search are popular for solving many optimization problems including the maximum satisfiability problem (MAX-SAT). With regard to MAX-SAT, the state of the art in performance for universal (i.e. non specialized solvers) seems to be variants of Simulated Annealing (SA) and MaxWalkSat (MWS), stochastic local search methods. Local search methods are conceptually simple, and they often provide near optimal solutions. In contrast, it is relatively rare that local search algorithms are analyzed with respect to the worst-case approximation ratios. In the first part of the paper, we build on Mastrolilli and Gambardella’s work [14] and present a worst-case analysis of tabu search for the MAX-k-SAT problem. In the second part of the paper, we examine the experimental performance of determinstic local search algorithms (oblivious and non-oblivious local search, tabu search) in comparison to stochastic methods (SA and MWS) on random 3-CNF and random k-CNF formulas and on benchmarks from MAX-SAT competitions. For random MAX-3-SAT, tabu search consistently outperforms both oblivious and non-oblivious local search, but does not match the performance of SA and MWS. Initializing with non-oblivious local search improves both the performance and the running time of tabu search. The better performance of the various methods that escape local optima in comparison to the more basic oblivious and non-oblivious local search algorithms (that stop at the first local optimum encountered) comes at a cost, namely a significant increase in complexity (which we measure in terms of variable flips). The performance results observed for the unweighted MAX-3-SAT problem carry over to the weighted version of the problem, but now the better performance of MWS is more pronounced. In contrast, as we consider MAX-k-SAT as k is increased, MWS loses its advantage. Finally, on benchmark instances, it appears that simulated annealing and tabu search initialized with non-oblivious local search outperform the other methods on most instances.

Random Θ(log n)-CNFs Are Hard for Cutting Planes

The random k-SAT model is the most important and well-studied distribution over k-SAT instances. It is closely connected to statistical physics and is a benchmark for satisfiability algorithms. We show that when k = &#x398;(log n), any Cutting Planes refutation for random k-SAT requires exponential size in the interesting regime where the number of clauses guarantees that the formula is unsatisfiable with high probability.

Information Lower Bounds via Self-Reducibility

We use self-reduction methods to prove strong information lower bounds on two of the most studied functions in the communication complexity literature: Gap Hamming Distance (GHD) and Inner Product (IP). In our first result we affirm the conjecture that the information cost of GHD is linear even under the uniform distribution, which strengthens the Ω(n) bound recently shown by Kerenidis et al. (2012), and answers an open problem from Chakrabarti et al. (2012). In our second result we prove that the information cost of IPn is arbitrarily close to the trivial upper bound n as the permitted error tends to zero, again strengthening the Ω(n) lower bound recently proved by Braverman and Weinstein (Electronic Colloquium on Computational Complexity (ECCC) 18, 164 2011). Our proofs demonstrate that self-reducibility makes the connection between information complexity and communication complexity lower bounds a two-way connection. Whereas numerous results in the past (Chakrabarti et al. 2001; Bar-Yossef et al. J. Comput. Syst. Sci. 68(4), 702–732 2004; Barak et al. 2010) used information complexity techniques to derive new communication complexity lower bounds, we explore a generic way in which communication complexity lower bounds imply information complexity lower bounds in a black-box manner.

A Simple PTAS for the Dual Bin Packing Problem and Advice Complexity of Its Online Version

Recently, Renault (2016) studied the dual bin packing problem in the per-request advice model of online algorithms. He showed that given

On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

O(1/\epsilon)

Direct Sum Questions in Classical Communication Complexity

advice bits for each input item allows approximating the dual bin packing problem online to within a factor of