A note on a problem in communication complexity

In this note, we prove a version of Tarui's Theorem in communication complexity, namely $PH^{cc} \subseteq BP\cdot PP^{cc}$. Consequently, every measure for $PP^{cc}$ leads to a measure for $PH^{cc}$, subsuming a result of Linial and Shraibman that problems with high mc-rigidity lie outside the polynomial hierarchy. By slightly changing the definition of mc-rigidity (arbitrary instead of uniform distribution), it is then evident that the class $M^{cc}$ of problems with low mc-rigidity equals $BP\cdot PP^{cc}$. As $BP\cdot PP^{cc} \subseteq PSPACE^{cc}$, this rules out the possibility, that had been left open, that even polynomial space is contained in $M^{cc}$

Monotone Projection Lower Bounds from Extended Formulation Lower Bounds

In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our reduction to the seminal extended formulation lower bounds of Fiorini, Massar, Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014; J. ACM, 2017), we obtain the following interesting consequences. 1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size projection of the permanent; this both rules out a natural attempt at a monotone lower bound on the Boolean permanent, and shows that the permanent is not complete for non-negative polynomials in VNP$_{{\mathbb R}}$ under monotone p-projections. 2. The cut polynomials and the perfect matching polynomial (or "unsigned Pfaffian") are not monotone p-projections of the permanent. The latter, over the Boolean and-or semi-ring, rules out monotone reductions in one of the natural approaches to reducing perfect matchings in general graphs to perfect matchings in bipartite graphs. As the permanent is universal for monotone formulas, these results also imply exponential lower bounds on the monotone formula size and monotone circuit size of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18; Received: November 10, 2015, Revised: July 27, 2016, Published: December 22, 201

Fast Computation of Small Cuts via Cycle Space Sampling

We describe a new sampling-based method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the cycle space identifies the cuts of a graph. This leads to simple new linear-time sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph. In the model of distributed computing in a graph G=(V, E) with O(log V)-bit messages, our approach yields faster algorithms for several problems. The diameter of G is denoted by Diam, and the maximum degree by Delta. We obtain simple O(Diam)-time distributed algorithms to find all cut edges, 2-edge-connected components, and cut pairs, matching or improving upon previous time bounds. Under natural conditions these new algorithms are universally optimal --- i.e. a Omega(Diam)-time lower bound holds on every graph. We obtain a O(Diam+Delta/log V)-time distributed algorithm for finding cut vertices; this is faster than the best previous algorithm when Delta, Diam = O(sqrt(V)). A simple extension of our work yields the first distributed algorithm with sub-linear time for 3-edge-connected components. The basic distributed algorithms are Monte Carlo, but they can be made Las Vegas without increasing the asymptotic complexity. In the model of parallel computing on the EREW PRAM our approach yields a simple algorithm with optimal time complexity O(log V) for finding cut pairs and 3-edge-connected components.Comment: Previous version appeared in Proc. 35th ICALP, pages 145--160, 200

Quantum communication complexity of symmetric predicates

We completely (that is, up to a logarithmic factor) characterize the bounded-error quantum communication complexity of every predicate $f(x,y)$ depending only on $|x\cap y|$ ($x,y\subseteq [n]$). Namely, for a predicate $D$ on $\{0,1,...,n\}$ let \ell_0(D)\df \max\{\ell : 1\leq\ell\leq n/2\land D(\ell)\not\equiv D(\ell-1)\} and \ell_1(D)\df \max\{n-\ell : n/2\leq\ell < n\land D(\ell)\not\equiv D(\ell+1)\}. Then the bounded-error quantum communication complexity of $f_D(x,y) = D(|x\cap y|)$ is equal (again, up to a logarithmic factor) to $\sqrt{n\ell_0(D)}+\ell_1(D)$. In particular, the complexity of the set disjointness predicate is $\Omega(\sqrt n)$. This result holds both in the model with prior entanglement and without it.Comment: 20 page

Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover

Set cover, over a universe of size $n$, may be modelled as a data-streaming problem, where the $m$ sets that comprise the instance are to be read one by one. A semi-streaming algorithm is allowed only $O(n\, \mathrm{poly}\{\log n, \log m\})$ space to process this stream. For each $p \ge 1$, we give a very simple deterministic algorithm that makes $p$ passes over the input stream and returns an appropriately certified $(p+1)n^{1/(p+1)}$-approximation to the optimum set cover. More importantly, we proceed to show that this approximation factor is essentially tight, by showing that a factor better than $0.99\,n^{1/(p+1)}/(p+1)^2$ is unachievable for a $p$-pass semi-streaming algorithm, even allowing randomisation. In particular, this implies that achieving a $\Theta(\log n)$-approximation requires $\Omega(\log n/\log\log n)$ passes, which is tight up to the $\log\log n$ factor. These results extend to a relaxation of the set cover problem where we are allowed to leave an $\varepsilon$ fraction of the universe uncovered: the tight bounds on the best approximation factor achievable in $p$ passes turn out to be $\Theta_p(\min\{n^{1/(p+1)}, \varepsilon^{-1/p}\})$. Our lower bounds are based on a construction of a family of high-rank incidence geometries, which may be thought of as vast generalisations of affine planes. This construction, based on algebraic techniques, appears flexible enough to find other applications and is therefore interesting in its own right.Comment: 20 page

Sign rank versus VC dimension

This work studies the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is {three}. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. For $d >2$, similar but slightly less accurate statements hold. {The lower bounds improve over previous ones by Ben-David et al., and the upper bounds are novel.} The lower bounds are obtained by probabilistic constructions, using a theorem of Warren in real algebraic topology. The upper bounds are obtained using a result of Welzl about spanning trees with low stabbing number, and using the moment curve. The upper bound technique is also used to: (i) provide estimates on the number of classes of a given VC dimension, and the number of maximum classes of a given VC dimension -- answering a question of Frankl from '89, and (ii) design an efficient algorithm that provides an $O(N/\log(N))$ multiplicative approximation for the sign rank. We also observe a general connection between sign rank and spectral gaps which is based on Forster's argument. Consider the $N \times N$ adjacency matrix of a $\Delta$ regular graph with a second eigenvalue of absolute value $\lambda$ and $\Delta \leq N/2$. We show that the sign rank of the signed version of this matrix is at least $\Delta/\lambda$. We use this connection to prove the existence of a maximum class $C\subseteq\{\pm 1\}^N$ with VC dimension $2$ and sign rank $\tilde{\Theta}(N^{1/2})$. This answers a question of Ben-David et al.~regarding the sign rank of large VC classes. We also describe limitations of this approach, in the spirit of the Alon-Boppana theorem. We further describe connections to communication complexity, geometry, learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC dimension". Additional results in this version: (i) Estimates on the number of maximum VC classes (answering a question of Frankl from '89). (ii) Estimates on the sign rank of large VC classes (answering a question of Ben-David et al. from '03). (iii) A discussion on the computational complexity of computing the sign-ran

Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems

The control theory of motion-based communication: problems in teaching robots to dance

The paper describes results on two components of a research program focused on motion-based communication mediated by the dynamics of a control system. Specifically we are interested in how mobile agents engaged in a shared activity such as dance can use motion as a medium for transmitting certain types of messages. The first part of the paper adopts the terminology of motion description languages and deconstructs an elementary form of the well-known popular dance, Salsa, in terms of four motion primitives (dance steps). Several notions of dance complexity are introduced. We describe an experiment in which ten performances by an actual pair of dancers are evaluated by judges and then compared in terms of proposed complexity metrics. An energy metric is also defined. Values of this metric are obtained by summing the lengths of motion segments executed by wheeled robots replicating the movements of the human dancers in each of the ten dance performances. Of all the metrics that are considered in this experiment, energy is the most closely correlated with the human judges' assessments of performance quality. The second part of the paper poses a general class of dual objective motion control problems in which a primary objective (artistic execution of a dance step or efficient movement toward a specified terminal state) is combined with a communication objective. Solutions of varying degrees of explicitness can be given in several classes of problems of communicating through the dynamics of finite dimensional linear control systems. In this setting it is shown that the cost of adding a communication component to motions that steer a system between prescribed pairs of states is independent of those states. At the same time, the optimal encoding problem itself is shown to be a problem of packing geometric objects, and it remains open. Current research is aimed at solving such communication-through-action problems in the context of the motion control of mobile robots.Support for this work is gratefully acknowledged to ODDR&E MURI07 Program Grant Number FA9550-07-1-0528, the National Science Founda tion ITR Program Grant Number DMI-0330171, and the Office of Naval Research, and by ODDR&E MURI10 Program Grant Number N00014-10- 1-0952. (FA9550-07-1-0528 - ODDRE MURI07; DMI-0330171 - National Science Foundation ITR Program; Office of Naval Research; N00014-10-1-0952 - ODDRE MURI10

Some Applications of Coding Theory in Computational Complexity

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography. Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs