For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$ that have equal multisets of isomorphism types of proper subgroups and proper quotient groups, isomorphic character tables, and power maps. This obstructs recent speculation concerning a path towards efficient isomorphism tests for general finite groups. These groups have a special purpose polylogarithmic-time isomorphism test.

$ \newcommand{\ensuremath}[1]{#1} \newcommand{\eps}{\ensuremath{\varepsilon}} $ A Boolean $k$-monotone function defined over a finite poset domain $\mathcal{D}$ alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in $\mathcal{D}$. Therefore, $k$-monotone functions are natural generalizations of the classical monotone functions, which are the $1$-monotone functions. Motivated by the recent interest in $k$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of $k$-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are $k$-monotone (or are close to being $k$-monotone) from functions that are far from being $k$-monotone. Our results include the following. We demonstrate a separation between testing $k$-monotonicity and testing monotonicity, on the hypercube domain $\{0,1\}^d$, for $k\geq 3$; We demonstrate a separation between testing and learning on $\{0,1\}^d$, for $k=\omega(\log d)$: testing $k$-monotonicity can be performed with $\exp(O(\sqrt d \cdot \log d\cdot \log(1/\eps)))$ queries, while learning $k$-monotone functions requires $\exp(\Omega(k\cdot \sqrt d\cdot{1/\eps}))$ queries (Blais et al. (RANDOM 2015)); We present a tolerant test for $k$-monotonicity of functions $f\colon[n]^d\to \{0,1\}$ with complexity independent of $n$. The test implies a tolerant test for monotonicity of functions $f\colon[n]^d\to [0,1]$ in $\ell_1$ distance, which makes progress on a problem left open by Berman et al. (STOC 2014). Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid $[n]^d$, and draw connections to distribution testing techniques. ---------- An extended abstract of this paper appeared in the Proceedings of the 8th Innovations in Theoretical Computer Science conference, 2017.

We present a classical interactive protocol that verifies the validity of a quantum witness state for the local Hamiltonian problem. It follows from this protocol that approximating the non-local value of a multi-player one-round game to inverse polynomial precision is QMA-hard. Our work makes an interesting connection between the theory of QMA-completeness and Hamiltonian complexity on one hand and the study of non-local games and Bell inequalities on the other.

This paper gives the first separation between the power of {\em formulas} and {\em circuits} of equal depth in the $\mathrm{AC}^0[\oplus]$ basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show, for all $d(n) \le O(\frac{\log n}{\log\log n})$, that there exist {\em polynomial-size depth-$d$ circuits} that are not equivalent to {\em depth-$d$ formulas of size $n^{o(d)}$} (moreover, this is optimal in that $n^{o(d)}$ cannot be improved to $n^{O(d)}$). This result is obtained by a combination of new lower and upper bounds for {\em Approximate Majorities}, the class of Boolean functions $\{0,1\}^n \to \{0,1\}$ that agree with the Majority function on $3/4$ fraction of inputs. $\mathrm{AC}^0[\oplus]$ formula lower bound: We show that every depth-$d$ $\mathrm{AC}^0[\oplus]$ formula of size $s$ has a {\em $1/8$-error polynomial approximation} over $\mathbb{F}_2$ of degree $O(\frac{1}{d}\log s)^{d-1}$. This strengthens a classic $O(\log s)^{d-1}$ degree approximation for \underline{circuits} due to Razborov. Since the Majority function has approximate degree $\Theta(\sqrt n)$, this result implies an $\exp(\Omega(dn^{1/2(d-1)}))$ lower bound on the depth-$d$ $\mathrm{AC}^0[\oplus]$ formula size of all Approximate Majority functions for all $d(n) \le O(\log n)$. Monotone $\mathrm{AC}^0$ circuit upper bound: For all $d(n) \le O(\frac{\log n}{\log\log n})$, we give a randomized construction of depth-$d$ monotone $\mathrm{AC}^0$ circuits (without NOT or MOD$_2$ gates) of size $\exp(O(n^{1/2(d-1)}))$ that compute an Approximate Majority function. This strengthens a construction of \underline{formulas} of size $\exp(O(dn^{1/2(d-1)}))$ due to Amano.

Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ gates (for some constant $m > 1$), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen as a midpoint between $ACC$ (where we know nontrivial lower bounds) and depth-two linear threshold circuits (where nontrivial lower bounds remain open). We give an algorithm for evaluating an arbitrary symmetric function of $2^{n^{o(1)}}$ $ACC \circ THR$ circuits of size $2^{n^{o(1)}}$, on all possible inputs, in $2^n \cdot poly(n)$ time. Several consequences are derived: $\bullet$ The number of satisfying assignments to an $ACC \circ THR$ circuit of subexponential size can be computed in $2^{n-n^{\varepsilon}}$ time (where $\varepsilon > 0$ depends on the depth and modulus of the circuit). $\bullet$ $NEXP$ does not have quasi-polynomial size $ACC \circ THR$ circuits, nor does $NEXP$ have quasi-polynomial size $ACC \circ SYM$ circuits. Nontrivial size lower bounds were not known even for $AND \circ OR \circ THR$ circuits. $\bullet$ Every 0-1 integer linear program with $n$ Boolean variables and $s$ linear constraints is solvable in $2^{n-\Omega(n/((\log M)(\log s)^{5}))}\cdot poly(s,n,M)$ time with high probability, where $M$ upper bounds the bit complexity of the coefficients. (For example, 0-1 integer programs with weights in $[-2^{poly(n)},2^{poly(n)}]$ and $poly(n)$ constraints can be solved in $2^{n-\Omega(n/\log^6 n)}$ time.) We also present an algorithm for evaluating depth-two linear threshold circuits (a.k.a., $THR \circ THR$) with exponential weights and $2^{n/24}$ size on all $2^n$ input assignments, running in $2^n \cdot poly(n)$ time. This is evidence that non-uniform lower bounds for $THR \circ THR$ are within reach.

$ \newcommand{\MCSP}{\mathrm{MCSP}} \newcommand{\NMCSP}{\mathrm{NMCSP}} \def \AC {{\sf AC}} \def \poly {\mathrm{poly}} \def \P {{\sf P}} \def \ZPP {{\sf ZPP}} \def \EXP {{\sf EXP}} \def \NP {{\sf NP}} \def \eps {{\varepsilon}} \def \E {{\sf E}} \def \SIZE {{\sf SIZE}} \def \BPP {{\sf BPP}} \def \Ppoly {{\P/\poly}} \def \io {\mathrm{i.o.}{\text{-}}} $ The Minimum Circuit Size Problem ($\MCSP$) is: given the truth table of a Boolean function $f$ and a size parameter $k$, is the circuit complexity of $f$ at most $k$? This is the definitive problem of circuit synthesis, and it has been studied since the 1950s. Unlike many problems of its kind, $\MCSP$ is not known to be $\NP$-hard, yet an efficient algorithm for this problem also seems very unlikely: for example, $\MCSP\in \P$ would imply there are no pseudorandom functions. Although most $\NP$-complete problems are complete under strong “local” reduction notions such as polylogarithmic-time projections, we show that $\MCSP$ is provably not $\NP$-hard under $O(n^{1/2-\eps})$-time projections, for every $\eps > 0$, and is not $\NP$-hard under randomized $O(n^{1/5-\eps})$-time projections, for every $\eps > 0$. We prove that the $\NP$-hardness of $\MCSP$ under (logtime-uniform) $\AC0$ reductions would imply extremely strong lower bounds: $\NP \not\subset \Ppoly$ and $\E \not\subset \io\SIZE(2^{\delta n})$ for some $\delta > 0$ (hence $\P = \BPP$ also follows). We show that even the $\NP$-hardness of $\MCSP$ under general polynomial-time reductions would separate complexity classes: $\EXP \neq \NP \cap \Ppoly$, which implies $\EXP \neq \ZPP$. These results help explain why it has been so difficult to prove that $\MCSP$ is $\NP$-hard. We also consider the nondeterministic generalization of $\MCSP$: the Nondeterministic Minimum Circuit Size Problem ($\NMCSP$), where one wishes to compute the nondeterministic circuit complexity of a given function. We prove that the $\Sigma_2 \P$-hardness of $\NMCSP$, even under arbitrary polynomial-time reductions, would imply $\EXP \not\subset \Ppoly$. A preliminary version of this paper appeared in the Proceedings of the 30th IEEE Conference on Computational Complexity, 2015.

Motivated by the fact that in many game-theoretic settings, the game analyzed is only an approximation to the game being played, in this work we analyze equilibrium computation for the broad and natural class of bimatrix games that are stable under perturbations. We specifically focus on games with the property that small changes in the payoff matrices do not cause the Nash equilibria of the game to fluctuate wildly. For such games we show how one can compute approximate Nash equilibria more efficiently than the general result of Lipton et al. (EC'03), by an amount that depends on the degree of stability of the game and that reduces to their bound in the worst case. Additionally, we show that for stable games, the approximate equilibria found will be close in variation distance to true equilibria, and moreover this holds even if we are given as input only a perturbation of the actual underlying stable game.

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 order of the variables. The best previously known hitting set for this case had size $(nw)^{O(\log n)}$ 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 size for the known-order case to $n^{O(\log w)}$. In particular, this gives the first polynomial-size hitting set for constant-width ROABP (known-order). However, our hitting set only works when the characteristic of the field is zero or large enough. To construct the hitting set, we use the concept of the rank of the partial derivative matrix. Unlike previous approaches which build up from mapping variables to monomials, we map variables to polynomials directly. The second case we consider is that of polynomials computable by width-$w$ ROABPs in any order of the variables. The best previously known hitting set for this case had size $d^{O(\log w)}(nw)^{O(\log \log w)}$, where $d$ is the individual degree. We improve the hitting-set size to $(ndw)^{O(\log \log w)}$.

We give a deterministic algorithm for solving the $(1+\eps)$-approximate Closest Vector Problem (CVP) on any $n$-dimensional lattice and in any near-symmetric norm in $2^{O(n)}(1+1/\eps)^n$ time and $2^n\poly(n)$ space. Our algorithm builds on the lattice point enumeration techniques of Micciancio and Voulgaris (STOC 2010, SICOMP 2013) and Dadush, Peikert and Vempala (FOCS 2011), and gives an elegant, deterministic alternative to the AKS Sieve-based algorithms for $(1+\eps)$-CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002). Furthermore, assuming the existence of a $\poly(n)$-space and $2^{O(n)}$-time algorithm for exact CVP in the $\ell_2$ norm, the space complexity of our algorithm can be reduced to polynomial. Our main technical contribution is a method for sparsifying any input lattice while approximately maintaining its metric structure. To this end, we employ the idea of random sublattice restrictions, which was first employed by Khot (FOCS 2003, J. Comp. Syst. Sci. 2006) for the purpose of proving hardness for the Shortest Vector Problem (SVP) under $\ell_p$ norms. A preliminary version of this paper appeared in the Proc. 24th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA'13) (http://dx.doi.org/10.1137/1.9781611973105.78).