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  1. Free, publicly-accessible full text available August 1, 2022
  2. The tensor rank and border rank of the $3 \times 3$ determinant tensor are known to be $5$ if the characteristic is not two. In characteristic two, the existing proofs of both the upper and lower bounds fail. In this paper, we show that the tensor rank remains $5$ for fields of characteristic two as well.
  3. We investigate the approximability of the following optimization problem. The input is an n× n matrix A=(Aij) with real entries and an origin-symmetric convex body K⊂ ℝn that is given by a membership oracle. The task is to compute (or approximate) the maximum of the quadratic form ∑i=1n∑j=1n Aij xixj=⟨ x,Ax⟩ as x ranges over K. This is a rich and expressive family of optimization problems; for different choices of matrices A and convex bodies K it includes a diverse range of optimization problems like max-cut, Grothendieck/non-commutative Grothendieck inequalities, small set expansion and more. While the literature studied these specialmore »cases using case-specific reasoning, here we develop a general methodology for treatment of the approximability and inapproximability aspects of these questions. The underlying geometry of K plays a critical role; we show under commonly used complexity assumptions that polytime constant-approximability necessitates that K has type-2 constant that grows slowly with n. However, we show that even when the type-2 constant is bounded, this problem sometimes exhibits strong hardness of approximation. Thus, even within the realm of type-2 bodies, the approximability landscape is nuanced and subtle. However, the link that we establish between optimization and geometry of Banach spaces allows us to devise a generic algorithmic approach to the above problem. We associate to each convex body a new (higher dimensional) auxiliary set that is not convex, but is approximately convex when K has a bounded type-2 constant. If our auxiliary set has an approximate separation oracle, then we design an approximation algorithm for the original quadratic optimization problem, using an approximate version of the ellipsoid method. Even though our hardness result implies that such an oracle does not exist in general, this new question can be solved in specific cases of interest by implementing a range of classical tools from functional analysis, most notably the deep factorization theory of linear operators. Beyond encompassing the scenarios in the literature for which constant-factor approximation algorithms were found, our generic framework implies that that for convex sets with bounded type-2 constant, constant factor approximability is preserved under the following basic operations: (a) Subspaces, (b) Quotients, (c) Minkowski Sums, (d) Complex Interpolation. This yields a rich family of new examples where constant factor approximations are possible, which were beyond the reach of previous methods. We also show (under commonly used complexity assumptions) that for symmetric norms and unitarily invariant matrix norms the type-2 constant nearly characterizes the approximability of quadratic maximization.« less
  4. Given a mixture between two populations of coins, “positive” coins that each have unknown and potentially different—bias ≥ 1 + ∆ and “negative” coins with bias ≤ 2 − ∆, we consider the task of estimating the fraction ρ of positive coins to within additive error E. We achieve an upper and lower bound of Θ( ρ log 1 ) samples for a 1 −δ probability of success, where crucially, our lower bound applies to all fully-adaptive algorithms. Thus, our sample complexity bounds have tight dependence for every relevant problem parameter. A crucial component of our lower bound proof ismore »a decomposition lemma (Lemma 5.2) showing how to assemble partially-adaptive bounds into a fully-adaptive bound, which may be of independent interest: though we invoke it for the special case of Bernoulli random variables (coins), it applies to general distributions. We present sim- ulation results to demonstrate the practical efficacy of our approach for realistic problem parameters for crowdsourcing applications, focusing on the “rare events” regime where ρ is small. The fine-grained adaptive flavor of both our algo- rithm and lower bound contrasts with much previous workin distributional testing and learning.« less
  5. A graph G is called {\em self-ordered} (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from G to any graph that is isomorphic to G. We say that G=(VE) is {\em robustly self-ordered}if the size of the symmetric difference between E and the edge-set of the graph obtained by permuting V using any permutation :VV is proportional to the number of non-fixed-points of . In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs. We show that robustly self-ordered bounded-degree graphs exist (in abundance), andmore »that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph). We provide two very different constructions, in tools and structure. The first, a direct construction, is based on proving a sufficient condition for robust self-ordering, which requires that an auxiliary graph, on {\em pairs} of vertices of the original graph, is expanding. In this case the original graph is (not only robustly self-ordered but) also expanding. The second construction proceeds in three steps: It boosts the mere existence of robustly self-ordered graphs, which provides explicit graphs of sublogarithmic size, to an efficient construction of polynomial-size graphs, and then, repeating it again, to exponential-size(robustly self-ordered) graphs that are locally constructible. This construction can yield robustly self-ordered graphs that are either expanders or highly disconnected, having logarithmic size connected components. We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree)exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors with very weak parameters but with some additional natural features. We actually show two reductions, one simpler than the other but yielding a less efficient construction when combined with the known constructions of extractors. We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. Indeed, their robustness offers efficient, local and distance preserving reductions from testing problems on ordered structures (like sequences) to the unordered (effectively unlabeled) graphs. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model. Changes to previous version: We retract the claims made in our initial posting regarding the construction of non-malleable two-source extractors (which are quasi-orthogonal) as well as the claims about the construction of relocation-detecting codes (see Theorems 1.5 and 1.6 in the original version). The source of trouble is a fundamental flaw in the proof of Lemma 9.7 (in the original version), which may as well be wrong. Hence, the original Section 9 was omitted, except that the original Section 9.3 was retained as a new Section 8.3. The original Section 8 appears as Section 8.0 and 8.1, and Section 8.2 is new.« less
  6. One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P 6⊆ NC1). Karchmer, Raz, and Wigderson [KRW95] suggested to approach this problem by proving that depth complexity behaves “as expected” with respect to the composition of functions f ⋄ g. They showed that the validity of this conjecture would imply that P 6⊆ NC^1 . Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function f, but only for few inner functions g. Thus,more »it is an important challenge to prove the KRW conjecture for a wider range of inner functions. In this work, we extend significantly the range of inner functions that can be handled. First, we consider the monotone version of the KRW conjecture. We prove it for every monotone inner function g whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the s-t-connectivity, clique, and generation functions. In order to carry this progress back to the non-monotone setting, we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function f with the monotone complexity of the inner function g. In this setting, we prove the KRW conjecture for a similar selection of inner functions g, but only for a specific choice of the outer function f.« less
  7. We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of SL_n(C)-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions). We develop a general framework, denoted algebraic circuit search problems, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computingmore »succinct encodings for generators for invariant rings.« less
  8. Restarts are a widely-used class of techniques integral to the efficiency of Conflict-Driven Clause Learning (CDCL) Boolean SAT solvers. While the utility of such policies has been well-established empirically, a theoretical understanding of whether restarts are indeed crucial to the power of CDCL solvers is missing. In this paper, we prove a series of theoretical results that characterize the power of restarts for various models of SAT solvers. More precisely, we make the following contributions. First, we prove an exponential separation between a drunk randomized CDCL solver model with restarts and the same model without restarts using a family ofmore »satisfiable instances. Second, we show that the configuration of CDCL solver with VSIDS branching and restarts (with activities erased after restarts) is exponentially more powerful than the same configuration without restarts for a family of unsatisfiable instances. To the best of our knowledge, these are the first separation results involving restarts in the context of SAT solvers. Third, we show that restarts do not add any proof complexity-theoretic power vis-a-vis a number of models of CDCL and DPLL solvers with non-deterministic static variable and value selection.« less