We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms (PIT) for read-once oblivious algebraic branching programs (ABPs). This class has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), but prior to this work had no known such black-box algorithm. Here we obtain the first quasi-polynomial sized hitting sets for this class, when the order of the variables is known. This work can be seen as an algebraic analogue of the results of Nisan [Nis92] and Impagliazzo-Nisan-Wigderson [INW94] for space-bounded pseudorandom generators.We also show that several other circuit classes can be black-box reduced to read-once oblivious ABPs, including set-multilinear ABPs (a generalization of depth 3 set-multilinear formulas), non-commutative ABPs (generalizing non-commutative formulas), and (semi-)diagonal depth-4 circuits (as introduced by Saxena [Sax08], and recently shown by Mulmuley [Mul12] to have implications for derandomizing Noether's Normalization Lemma). For set-multilinear ABPs and non-commutative ABPs, we give quasi-polynomial-time black-box PIT algorithms, where the latter case involves evaluations over the algebra of (D + 1) × (D + 1) matrices, where D is the depth of the ABP. For (semi-)diagonal depth-4 circuits, we obtain a black-box PIT algorithm (over any characteristic) whose run-time is quasi-polynomial in the runtime of Saxena's whitebox algorithm, matching the concurrent work of Agrawal, Saha, and Saxena [ASS12]. Finally, by combining our results with the reconstruction algorithm of Klivans and Shpilka [KS06], we obtain deterministic reconstruction algorithms for the above circuit classes.

We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in n O(lg 2 n) time.Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the model of read-once oblivious boolean branching programs with unknown order.We obtain our results by recasting, and improving upon, the ideas of Agrawal, Saha and Saxena [ASS13]. We phrase the ideas in terms of rank condensers and Wronskians, and show that our results improve upon the classical multivariate Wronskian, which may be of independent interest.In addition, we give the first n O(lg lg n) black-box polynomial identity testing algorithm for the so called model of diagonal circuits. This result improves upon the n Θ(lg n) -time algorithms given by Agrawal, Saha and Saxena [ASS13], and Forbes and Shpilka [FS13b] for this class. More generally, our result holds for any model computing polynomials whose partial derivatives (of all orders) span a low dimensional linear space.

Abstract. The results of Strassen [Str73] and Raz [Raz10] show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds.We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent explicit 0/1 tensors T : [n] d → F with rank at least 2n ⌊d/2⌋ + n − Θ(d lg n). This matches (over F2) or improves (all other fields) known lower bounds for d = 3 and improves (over any field) for odd d > 3.We also explore a generalization of permutation matrices, which we denote permutation tensors. We show, by counting, that there exists an order-3 permutation tensor with super-linear rank. We also explore a natural class of permutation tensors, which we call group tensors. For any group G, we define the group tensor. We give two upper bounds for the rank of these tensors. The first uses representation theory and works over large fields F, showing (among other things) that rank. We also show that if this upper bound is tight, then super-linear tensor rank lower bounds would follow. The second upper bound uses interpolation and only works for abelian G, showing that over any field F that rankIn either case, this shows that many permutation tensors have far from maximal rank, which is very different from the matrix case and thus eliminates many natural candidates for high tensor rank.We also explore monotone tensor rank. We give explicit 0/1 tensors T : [n] d → F that have tensor rank at most dn but have monotone tensor rank exactly n d−1 . This is a nearly optimal separation.

Covering arrays are structures for well-representing extremely large input spaces and are used to efficiently implement blackbox testing for software and hardware. This paper proposes refinements over the In-Parameter-Order strategy (for arbitrary t). When constructing homogeneous-alphabet covering arrays, these refinements reduce runtime in nearly all cases by a factor of more than 5 and in some cases by factors as large as 280. This trend is increasing with the number of columns in the covering array. Moreover, the resulting covering arrays are about 5 % smaller. Consequently, this new algorithm has constructed many covering arrays that are the smallest in the literature. A heuristic variant of the algorithm sometimes produces comparably sized covering arrays while running significantly faster.

Consider a possibly non-linear (n, K, d) q code. Coordinate i has locality r if its value is determined by some r other coordinates. A recent line of work obtained an optimal trade-off between information locality of codes and their redundancy. Further, for linear codes meeting this trade-off, structure theorems were derived. In this work we give a new proof of the locality / redundancy trade-off and generalize structure theorems to non-linear codes.

Mulmuley [Mul12a] recently gave an explicit version of Noether's Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley's reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors.As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation.We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model.Finally

No abstract

No abstract