On the Usefulness of Predicates

Motivated by the pervasiveness of strong inapproximability results for Max-CSPs, we introduce a relaxed notion of an approximate solution of a Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly real-valued) constraints, and then only needs to satisfy as many of the new constraints as possible. To be more precise, we introduce the following notion of a predicate $P$ being \emph{useful} for a (real-valued) objective $Q$: given an almost satisfiable Max-$P$ instance, there is an algorithm that beats a random assignment on the corresponding Max-$Q$ instance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a Max-CSP with predicate $P$ is exactly the same as saying that $P$ is useful for $P$ itself. We say that $P$ is useless if it is not useful for any $Q$. This turns out to be equivalent to the following pseudo-randomness property: given an almost satisfiable instance of Max-$P$ it is hard to find an assignment such that the induced distribution on $k$-bit strings defined by the instance is not essentially uniform. Under the Unique Games Conjecture, we give a complete and simple characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is useless if and only if there is a pairwise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain Max-CSPs

On the Power of Many One-Bit Provers

We study the class of languages, denoted by \MIP[k, 1-\epsilon, s], which have $k$-prover games where each prover just sends a \emph{single} bit, with completeness $1-\epsilon$ and soundness error $s$. For the case that $k=1$ (i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson ({\em Computational Complexity'02}) demonstrate that \SZK exactly characterizes languages having 1-bit proof systems with"non-trivial" soundness (i.e., $1/2 < s \leq 1-2\epsilon$). We demonstrate that for the case that $k\geq 2$, 1-bit $k$-prover games exhibit a significantly richer structure: + (Folklore) When $s \leq \frac{1}{2^k} - \epsilon$, \MIP[k, 1-\epsilon, s] = \BPP; + When $\frac{1}{2^k} + \epsilon \leq s < \frac{2}{2^k}-\epsilon$, \MIP[k, 1-\epsilon, s] = \SZK; + When $s \ge \frac{2}{2^k} + \epsilon$, \AM \subseteq \MIP[k, 1-\epsilon, s]; + For $s \le 0.62 k/2^k$ and sufficiently large $k$, \MIP[k, 1-\epsilon, s] \subseteq \EXP; + For $s \ge 2k/2^{k}$, \MIP[k, 1, 1-\epsilon, s] = \NEXP. As such, 1-bit $k$-prover games yield a natural "quantitative" approach to relating complexity classes such as \BPP,\SZK,\AM, \EXP, and \NEXP. We leave open the question of whether a more fine-grained hierarchy (between \AM and \NEXP) can be established for the case when $s \geq \frac{2}{2^k} + \epsilon$

Intermediate problems in modular circuits satisfiability

In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite algebras had been introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties. However the problem for nilpotent (which had not been shown to be NP-hard) but not supernilpotent algebras (which had been shown to be polynomial time) remained open. In this paper we provide a broad class of examples, lying in this grey area, and show that, under the Exponential Time Hypothesis and Strong Exponential Size Hypothesis (saying that Boolean circuits need exponentially many modular counting gates to produce boolean conjunctions of any arity), satisfiability over these algebras have intermediate complexity between $\Omega(2^{c\log^{h-1} n})$ and $O(2^{c\log^h n})$, where $h$ measures how much a nilpotent algebra fails to be supernilpotent. We also sketch how these examples could be used as paradigms to fill the nilpotent versus supernilpotent gap in general. Our examples are striking in view of the natural strong connections between circuits satisfiability and Constraint Satisfaction Problem for which the dichotomy had been shown by Bulatov and Zhuk

Systems of Linear Equations over F2\mathbb{F}_2 and Problems Parameterized Above Average

In the problem Max Lin, we are given a system $Az=b$ of $m$ linear equations with $n$ variables over $\mathbb{F}_2$ in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess. Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC'06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least $k$, where $k$ is the parameter. It is not hard to see that we may assume that no two equations in $Az=b$ have the same left-hand side and $n={\rm rank A}$. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: $m\le 2^{p(n)}$ for an arbitrary fixed function $p(n)=o(n)$. Max $r$-Lin AA is a special case of Max Lin AA, where each equation has at most $r$ variables. In Max Exact $r$-SAT AA we are given a multiset of $m$ clauses on $n$ variables such that each clause has $r$ variables and asked whether there is a truth assignment to the $n$ variables that satisfies at least $(1-2^{-r})m + k2^{-r}$ clauses. Using our maximum excess results, we prove that for each fixed $r\ge 2$, Max $r$-Lin AA and Max Exact $r$-SAT AA can be solved in time $2^{O(k \log k)}+m^{O(1)}.$ This improves $2^{O(k^2)}+m^{O(1)}$-time algorithms for the two problems obtained by Gutin et al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively

A Hypergraph Dictatorship Test with Perfect Completeness

A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan and serves as a key component in their unique games based \PCP construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all their low degree influences are $o(1).$ Their test makes $q\geq3$ queries and has amortized query complexity $1+O(\frac{\log q}{q})$ but has an inherent loss of perfect completeness. In this paper we give an adaptive hypergraph dictatorship test that achieves both perfect completeness and amortized query complexity $1+O(\frac{\log q}{q})$.Comment: Some minor correction

Evolution of ultraviolet vision in the largest avian radiation - the passerines

<p>Abstract</p> <p>Background</p> <p>Interspecific variation in avian colour vision falls into two discrete classes: violet sensitive (VS) and ultraviolet sensitive (UVS). They are characterised by the spectral sensitivity of the most shortwave sensitive of the four single cones, the SWS1, which is seemingly under direct control of as little as one amino acid substitution in the cone opsin protein. Changes in spectral sensitivity of the SWS1 are ecologically important, as they affect the abilities of birds to accurately assess potential mates, find food and minimise visibility of social signals to predators. Still, available data have indicated that shifts between classes are rare, with only four to five independent acquisitions of UV sensitivity in avian evolution.</p> <p>Results</p> <p>We have classified a large sample of passeriform species as VS or UVS from genomic DNA and mapped the evolution of this character on a passerine phylogeny inferred from published molecular sequence data. Sequencing a small gene fragment has allowed us to trace the trait changing from one stable state to another through the radiation of the passeriform birds. Their ancestor is hypothesised to be UVS. In the subsequent radiation, colour vision changed between UVS and VS at least eight times.</p> <p>Conclusions</p> <p>The phylogenetic distribution of SWS1 cone opsin types in Passeriformes reveals a much higher degree of complexity in avian colour vision evolution than what was previously indicated from the limited data available. Clades with variation in the colour vision system are nested among clades with a seemingly stable VS or UVS state, providing a rare opportunity to understand how an ecologically important trait under simple genetic control may co-evolve with, and be stabilised by, associated traits in a character complex.</p

When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the $n$-stage repeated version of the game exist if and only if both players have $\Omega(n)$ random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the $n$-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on "random-like" sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators \emph{do not} exist, then $\Omega(n)$ random bits remain necessary for equilibria to exist