Computer Science

The prototypical construction of error correcting codes is based on linear codes over finite fields. In this work, we make first steps in the study of codes defined over integers. We focus on Maximum Distance Separable (MDS) codes, and show that MDS codes with linear rate and distance can be realized over the integers with a constant alphabet size. This is in contrast to the situation over finite fields, where a linear size finite field is needed. The core of this paper is a new result on the singularity probability of random matrices. We show that for a random n×n matrix with entries chosen independently from the range {−m,…,m} , the probability that it is singular is at most m −cn for some absolute constant c>0 .

We consider probabilistic circuits working over the real numbers, and using arbitrary semialgebraic functions of bounded description complexity as gates. In particular, such circuits can use all arithmetic operations +, -, x, /, optimization operations min and max, conditional branching (if-then-else), and many more. We show that probabilistic circuits using any of these operations as gates can be simulated by deterministic circuits with only about a quadratical blowup in size. A not much larger blow up in circuit size is also shown when derandomizing approximating circuits. The algorithmic consequence, motivating the title, is that randomness cannot substantially speed up dynamic programming algorithms.

In this note we compare two measures of the complexity of a class F of Boolean functions studied in (unconditional) pseudorandomness: F 's ability to distinguish between biased and uniform coins (the coin problem), and the norms of the different levels of the Fourier expansion of functions in F (the Fourier growth). We show that for coins with low bias ε=o(1/n) , a function's distinguishing advantage in the coin problem is essentially equivalent to ε times the sum of its level 1 Fourier coefficients, which in particular shows that known level 1 and total influence bounds for some classes of interest (such as constant-width read-once branching programs) in fact follow as a black-box from the corresponding coin theorems, thereby simplifying the proofs of some known results in the literature. For higher levels, it is well-known that Fourier growth bounds on all levels of the Fourier spectrum imply coin theorems, even for large ε , and we discuss here the possibility of a converse.

We study the communication complexity of computing functions F:{0,1 } n ×{0,1 } n →{0,1} in the memoryless communication model. Here, Alice is given x∈{0,1 } n , Bob is given y∈{0,1 } n and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x; the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study space-bounded communication models. We establish the following: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose complexity; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c>1 for which the memoryless communication complexity is at least clogn ? Note that c≥2+ε would imply a Ω( n 2+ε ) lower bound for general formula size, improving upon the best lower bound by Nečiporuk in 1966.

Automata networks are mappings of the form f : Q Z → Q Z , where Q is a finite alphabet and Z is a set of entities; they generalise Cellular Automata and Boolean networks. An update schedule dictates when each entity updates its state according to its local function fi : Q Z → Q. One major question is to study the behaviour of a given automata networks under different update schedules. In this paper, we study automata networks that are invariant under many different update schedules. This gives rise to two definitions, locally commutative and globally commu-tative networks. We investigate the relation between commutativity and different forms of locality of update functions; one main conclusion is that globally commutative networks have strong dynamical properties, while locally commutative networks are much less constrained. We also give a complete classification of all globally commutative Boolean networks.

Computational pseudorandomness studies the extent to which a random variable Z looks like the uniform distribution according to a class of tests F . Computational entropy generalizes computational pseudorandomness by studying the extent which a random variable looks like a \emph{high entropy} distribution. There are different formal definitions of computational entropy with different advantages for different applications. Because of this, it is of interest to understand when these definitions are equivalent. We consider three notions of computational entropy which are known to be equivalent when the test class F is closed under taking majorities. This equivalence constitutes (essentially) the so-called \emph{dense model theorem} of Green and Tao (and later made explicit by Tao-Zeigler, Reingold et al., and Gowers). The dense model theorem plays a key role in Green and Tao's proof that the primes contain arbitrarily long arithmetic progressions and has since been connected to a surprisingly wide range of topics in mathematics and computer science, including cryptography, computational complexity, combinatorics and machine learning. We show that, in different situations where F is \emph{not} closed under majority, this equivalence fails. This in turn provides examples where the dense model theorem is \emph{false}.

Hypertree decompositions (HDs), as well as the more powerful generalized hypertree decompositions (GHDs), and the yet more general fractional hypertree decompositions (FHDs) are hypergraph decomposition methods successfully used for answering conjunctive queries and for solving constraint satisfaction problems. Every hypergraph H has a width relative to each of these methods: its hypertree width hw(H) , its generalized hypertree width ghw(H) , and its fractional hypertree width fhw(H) , respectively. It is known that hw(H)≤k can be checked in polynomial time for fixed k , while checking ghw(H)≤k is NP-complete for k≥3 . The complexity of checking fhw(H)≤k for a fixed k has been open for over a decade. We settle this open problem by showing that checking fhw(H)≤k is NP-complete, even for k=2 . The same construction allows us to prove also the NP-completeness of checking ghw(H)≤k for k=2 . After that, we identify meaningful restrictions which make checking for bounded ghw or fhw tractable or allow for an efficient approximation of the fhw .

We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a few others. We show that these complexity measures are polynomially related for the symmetric group and for many other domains. To show that all measures but sensitivity are polynomially related, we generalize classical arguments of Nisan and others. To add sensitivity to the mix, we reduce to Huang's sensitivity theorem using "pseudo-characters", which witness the degree of a function. Using similar ideas, we extend the characterization of Boolean degree 1 functions on the symmetric group due to Ellis, Friedgut and Pilpel to the perfect matching scheme. As another application of our ideas, we simplify the characterization of maximum-size t -intersecting families in the symmetric group and the perfect matching scheme.

In this paper, we prove intractability results about sampling from the set of partitions of a planar graph into connected components. Our proofs are motivated by a technique introduced by Jerrum, Valiant, and Vazirani. Moreover, we use gadgets inspired by their technique to provide families of graphs where the "flip walk" Markov chain used in practice for this sampling task exhibits exponentially slow mixing. Supporting our theoretical results we present some empirical evidence demonstrating the slow mixing of the flip walk on grid graphs and on real data. Inspired by connections to the statistical physics of self-avoiding walks, we investigate the sensitivity of certain popular sampling algorithms to the graph topology. Finally, we discuss a few cases where the sampling problem is tractable. Applications to political redistricting have recently brought increased attention to this problem, and we articulate open questions about this application that are highlighted by our results.

This paper studies complexity theoretic aspects of quantum refereed games, which are abstract games between two competing players that send quantum states to a referee, who performs an efficiently implementable joint measurement on the two states to determine which of the player wins. The complexity class QRG(1) contains those decision problems for which one of the players can always win with high probability on yes-instances and the other player can always win with high probability on no-instances, regardless of the opposing player's strategy. This class trivially contains QMA∪co-QMA and is known to be contained in PSPACE . We prove stronger containments on two restricted variants of this class. Specifically, if one of the players is limited to sending a classical (probabilistic) state rather than a quantum state, the resulting complexity class CQRG(1) is contained in ∃⋅PP (the nondeterministic polynomial-time operator applied to PP ); while if both players send quantum states but the referee is forced to measure one of the states first, and incorporates the classical outcome of this measurement into a measurement of the second state, the resulting class MQRG(1) is contained in P⋅PP (the unbounded-error probabilistic polynomial-time operator applied to PP ).