ACM Transactions on Computation Theory 

About: ACM Transactions on Computation Theory is an academic journal published by Association for Computing Machinery. The journal publishes majorly in the area(s): Upper and lower bounds & Parameterized complexity. It has an ISSN identifier of 1942-3454. Over the lifetime, 209 publications have been published receiving 2826 citations. The journal is also known as: TOCT & ToCT.

Algebrization: A New Barrier in Complexity Theory

Abstract: Any proof of P ≠ NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (e.g., that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory.In this article, we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a low-degree extension of A over a finite field or ring.We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetization---both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP ⊄ P/poly---do indeed algebrize. Second, we show that almost all of the major open problems---including P versus NP, P versus RP, and NEXP versus P/poly---will require non-algebrizing techniques. In some cases, algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP.Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this article and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MA-protocol for the Inner Product function with O (Snlogn) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL ≠ NP.

Complexity Hierarchies beyond Elementary

Abstract: We introduce a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many nonelementary problems. This hierarchy allows the classification of many decision problems with a nonelementary complexity, which occur naturally in areas such as logic, combinatorics, formal languages, and verification, with complexities ranging from simple towers of exponentials to Ackermannian and beyond.

Directed Planar Reachability Is in Unambiguous Log-Space

Abstract: We make progress in understanding the complexity of the graph reachability problem in the context of unambiguous logarithmic space computation; a restricted form of nondeterminism. As our main result, we show a new upper bound on the directed planar reachability problem by showing that it can be decided in the class unambiguous logarithmic space (UL). We explore the possibility of showing the same upper bound for the general graph reachability problem. We give a simple reduction showing that the reachability problem for directed graphs with thickness two is complete for the class nondeterministic logarithmic space (NL). Hence an extension of our results to directed graphs with thickness two will unconditionally collapse NL to UL.

On multiway cut parameterized above lower bounds

Abstract: We introduce a concept of parameterizing a problem above the optimum solution of its natural linear programming relaxation and prove that the node multiway cut problem is fixed-parameter tractable (FPT) in this setting As a consequence we prove that node multiway cut is FPT, when parameterized above the maximum separating cut, resolving an open problem of RazgonOur results imply Oa(4k) algorithms for vertex cover above maximum matching and almost 2-SAT as well as an Oa(2k) algorithm for node multiway cut with a standard parameterization by the solution size, improving previous bounds for these problems

Optimal Lower Bounds for Locality-Sensitive Hashing (Except When q is Tiny)

Abstract: We study lower bounds for Locality-Sensitive Hashing (LSH) in the strongest setting: point sets in {0,1}d under the Hamming distance. Recall that H is said to be an (r, cr, p, q)-sensitive hash family if all pairs x, y ∈ {0,1}d with dist(x, y) ≤ r have probability at least p of collision under a randomly chosen h ∈ H, whereas all pairs x, y ∈ {0, 1}d with dist(x, y) ≥ cr have probability at most q of collision. Typically, one considers d → ∞, with c > 1 fixed and q bounded away from 0.For its applications to approximate nearest-neighbor search in high dimensions, the quality of an LSH family H is governed by how small its ρparameterρ = ln(1/p)/ln(1/q) is as a function of the parameter c. The seminal paper of Indyk and Motwani [1998] showed that for each c ≥ 1, the extremely simple family H = {x ↦ xi : i ∈ [d]} achieves ρ ≤ 1/c. The only known lower bound, due to Motwani et al. [2007], is that ρ must be at least ( e1/c - 1)/(e1/c + 1) ≥ .46/c (minus od(1)). The contribution of this article is twofold. (1) We show the “optimal” lower bound for ρ: it must be at least 1/c (minus od(1)). Our proof is very simple, following almost immediately from the observation that the noise stability of a boolean function at time t is a log-convex function of t. (2) We raise and discuss the following issue: neither the application of LSH to nearest-neighbor search nor the known LSH lower bounds hold as stated if the q parameter is tiny. Here, “tiny” means q = 2-Θ(d), a parameter range we believe is natural.