Quasi-polynomial Hitting-set for Set-depth-Delta Formulas

We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition (X_1,...,X_d) of the variable indices [n] that the top product layer respects, i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a sparse polynomial in F[x_{X_j}]. Extending this definition to any depth - we call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in C respects a (unknown) partition on the variables; if Delta is even then the product gates of the bottom-most Pi-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-Delta formulas (over any field) with running time polynomial in exp(({Delta}^2 log s)^{Delta - 1}), where s is the size bound on the input set-depth-Delta formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of Delta=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995). Our work settles this question, not only for depth-3 but, up to depth epsilon.log s / loglog s, for a fixed constant epsilon < 1. The technique is to investigate depth-Delta formulas via depth-(Delta-1) formulas over a Hadamard algebra, after applying a `shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-Delta formulas.Comment: 22 page

On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT

Hitting Set is a classic problem in combinatorial optimization. Its input consists of a set system F over a finite universe U and an integer t; the question is whether there is a set of t elements that intersects every set in F. The Hitting Set problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of Hitting Set under various structural parameterizations of the input. Our starting point is the folklore result that Hitting Set is polynomial-time solvable if there is a tree T on vertex set U such that the sets in F induce connected subtrees of T. We consider the case that there is a treelike graph with vertex set U such that the sets in F induce connected subgraphs; the parameter of the problem is a measure of how treelike the graph is. Our main positive result is an algorithm that, given a graph G with cyclomatic number k, a collection P of simple paths in G, and an integer t, determines in time 2^{5k} (|G| +|P|)^O(1) whether there is a vertex set of size t that hits all paths in P. It is based on a connection to the 2-SAT problem in multiple valued logic. For other parameterizations we derive W[1]-hardness and para-NP-completeness results.Comment: Presented at the 41st International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2015. (The statement of Lemma 4 was corrected in this update.

Identity Testing for Constant-Width, and Commutative, Read-Once Oblivious ABPs

We give improved hitting-sets for two special cases of Read-once Oblivious Arithmetic Branching Programs (ROABP). First is the case of an ROABP with known variable order. The best hitting-set known for this case had cost (nw)^{O(log(n))}, where n is the number of variables and w is the width of the ROABP. Even for a constant-width ROABP, nothing better than a quasi-polynomial bound was known. We improve the hitting-set complexity for the known-order case to n^{O(log(w))}. In particular, this gives the first polynomial time hitting-set for constant-width ROABP (known-order). However, our hitting-set works only over those fields whose characteristic is zero or large enough. To construct the hitting-set, we use the concept of the rank of partial derivative matrix. Unlike previous approaches whose starting point is a monomial map, we use a polynomial map directly. The second case we consider is that of commutative ROABP. The best known hitting-set for this case had cost d^{O(log(w))}(nw)^{O(log(log(w)))}, where d is the individual degree. We improve this hitting-set complexity to (ndw)^{O(log(log(w)))}. We get this by achieving rank concentration more efficiently

A shortcut to (sun)flowers: Kernels in logarithmic space or linear time

We investigate whether kernelization results can be obtained if we restrict kernelization algorithms to run in logarithmic space. This restriction for kernelization is motivated by the question of what results are attainable for preprocessing via simple and/or local reduction rules. We find kernelizations for d-Hitting Set(k), d-Set Packing(k), Edge Dominating Set(k) and a number of hitting and packing problems in graphs, each running in logspace. Additionally, we return to the question of linear-time kernelization. For d-Hitting Set(k) a linear-time kernelization was given by van Bevern [Algorithmica (2014)]. We give a simpler procedure and save a large constant factor in the size bound. Furthermore, we show that we can obtain a linear-time kernel for d-Set Packing(k) as well.Comment: 18 page