An ordering constraint satisfaction problem (OCSP) is given by a positive integer $k$ and a constraint predicate $Pi$ mapping permutations on ${1,ldots,k}$ to ${0,1}$. Given an instance of OCSP$(Pi)$ on $n$ variables and $m$ constraints, the goal is
to find an ordering of the $n$ variables that maximizes the number of constraints that are satisfied, where a constraint specifies a sequence of $k$ distinct variables and the constraint is satisfied by an ordering on the $n$ variables if the ordering induced on the $k$ variables in the constraint satisfies $Pi$. OCSPs capture natural problems including Maximum acyclic subgraph (MAS) and Betweenness. In this work we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, where an instance is presented as a stream of constraints. We show that for every $Pi$, OCSP$(Pi)$ is approximation-resistant to $o(n)$-space streaming algorithms. This space bound is tight up to polylogarithmic factors. In the case of MAS our result shows that for every $epsilon>0$, MAS is not $1/2+epsilon$-approximable in $o(n)$ space. The previous best inapproximability result only ruled out a $3/4$-approximation in $o(sqrt n)$ space. Our results build on recent works of Chou, Golovnev, Sudan, Velingker, and Velusamy who show tight, linear-space inapproximability results for a broad class of (non-ordering) constraint satisfaction problems over arbitrary (finite) alphabets. We design a family of appropriate CSPs (one for every $q$) from any given OCSP, and apply their work to this family of CSPs. We show that the hard instances from this earlier work have a particular small-set expansion property. By exploiting this combinatorial property, in combination with the hardness results of the resulting families of CSPs, we give optimal inapproximability results for all OCSPs.
Wooley ({em J. Number Theory}, 1996) gave an elementary proof of a Bezout like theorem allowing one to count the number of isolated integer roots of a system of polynomial equations modulo some prime power. In this article, we adapt the proof to a
slightly different setting. Specifically, we consider polynomials with coefficients from a polynomial ring $mathbb{F}[t]$ for an arbitrary field $mathbb{F}$ and give an upper bound on the number of isolated roots modulo $t^s$ for an arbitrary positive integer $s$. In particular, using $s=1$, we can bound the number of isolated roots of a system of polynomials over an arbitrary field $mathbb{F}$.
