On the de-randomization of space-bounded approximate counting problems

Noga Alon, Rina Panigrahy‎, Sergey Yekhanin

The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v in F_2^n and a linear space L in F_2^n of dimension k NCP asks to find a point l in L that minimizes the (Hamming) distance from v. It is well-known that the nearest codeword problem is NP-hard. Therefore approximation algorithms are of interest. The best effcient approximation algorithms for the NCP to date are due to Berman and Karpinski. They are a deterministic algorithm that achieves an approximation ratio of O(k/c) for an arbitrary constant c; and a randomized algorithm that achieves an approximation ratio of O(k/ log n). In this paper we present new deterministic algorithms for approximating the NCP that improve substantially upon the earlier work, (almost) de-randomizing the randomized algorithm of Berman and Karpinski. We also initiate a study of the following Remote Point Problem (RPP). Given a linear space L in F_2^n of dimension k RPP asks to find a point v in F_2^n that is far from L. We say that an algorithm achieves a remoteness of r for the RPP if it always outputs a point v that is at least r-far from L. In this paper we present a deterministic polynomial time algorithm that achieves a remoteness of Omega(n log k / k) for all k < n/2. We motivate the remote point problem by relating it to both the nearest codeword problem and the matrix rigidity approach to circuit lower bounds in computational complexity theory.

Martin Dyer, Leslie Ann Goldberg, Catherine Greenhill +1 more

Elad Verbin, Wei Yu

In this paper we introduce a new technique for proving streaming lower bounds (and one-way communication lower bounds), by reductions from a problem called the Boolean Hidden Hypermatching problem (BHH). BHH is a problem that we introduce and prove the first lower bound for, but it is a generalization of a well-known problem called the Boolean Hidden Matching, that was used by Gavinsky et al. to prove separations between quantum communication complexity and one-way randomized communication complexity. The hardness of the BHH problem is inherently one-way: it is easy to solve using logarithmic two-way communication, but requires √ n communication if Alice is only allowed to send messages to Bob, and not vice-versa. This one-wayness allows us to prove lower bounds, via reductions, for streaming problems and related communication problems whose hardness is also inherently one-way. By designing reductions from BHH, we prove lower bounds for the streaming complexity of approximating the sorting by reversal distance, for approximately counting the number of cycles in a 2-regular graph, and for other problems. For example, here is one lower bound that we prove, for a cycle-counting problem: Alice gets a perfect matching EA on a set of n nodes, and Bob gets a perfect matching EB on the same set of nodes. The union EA ∪ EB is a collection of cycles, and the goal is to approximate the number of cycles in this collection. We prove that if Alice is allowed to send o ( √ n) bits to Bob (and Bob is not allowed to send anything to Alice), then the number of cycles cannot be approximated to within a factor of 1.999, even using a randomized protocol. We prove that it is not even possible to distinguish the case where all cycles are of length 4, from the case where all cycles are of length 8. This lower bound is “natively ” one-way: With 4 rounds of communication, it is easy to distinguish these two cases. 1

Larry J. Stockmeyer

Elad Verbin, Wei Yu

In this paper we introduce a new technique for proving streaming lower bounds (and one-way communication lower bounds), by reductions from a problem called the Boolean Hidden Hypermatching problem (BHH). BHH is a generalization of the well-known Boolean Hidden Matching problem, which was used by Gavinsky et al. to prove an exponential separation between quantum communication complexity and one-way randomized communication complexity. We are the first to introduce BHH, and to prove a lower bound for it. The hardness of the BHH problem is inherently one-way: it is easy to solve BHH using logarithmic two-way communication, but it requires √n communication if Alice is only allowed to send messages to Bob, and not vice-versa. This one-wayness allows us to prove lower bounds, via reductions, for streaming problems and related communication problems whose hardness is also inherently one-way. By designing reductions from BHH, we prove lower bounds for the streaming complexity of approximating the sorting by reversal distance, of approximately counting the number of cycles in a 2-regular graph, and of other problems. For example, here is one lower bound that we prove, for a cycle-counting problem: Alice gets a perfect matching EA on a set of n nodes, and Bob gets a perfect matching EB on the same set of nodes. The union EA ∪ EB is a collection of cycles, and the goal is to approximate the number of cycles in this collection. We prove that if Alice is allowed to send o(√n) bits to Bob (and Bob is not allowed to send anything to Alice), then the number of cycles cannot be approximated to within a factor of 1.999, even using a randomized protocol. We prove that it is not even possible to distinguish the case where all cycles are of length 4, from the case where all cycles are of length 8. This lower bound is “natively” one-way: With 4 rounds of communication, it is easy to distinguish these two cases.

Shinichi Yamada