Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0912.4884

An Invariance Principle for Polytopes

Prahladh Harsha·Adam Klivans·Raghu Meka
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let X be randomly chosen from {-1,1}^n, and let Y be randomly chosen from the standard spherical Gaussian on R^n. For any (possibly unbounded) polytope P formed by the intersection of k halfspaces, we prove that |Pr [X belongs to P] - Pr [Y belongs to P]| < log^{8/5}k * Delta, where Delta is a parameter that is small for polytopes formed by the intersection of "regular" halfspaces (i.e., halfspaces with low influence). The novelty of our invariance principle is the polylogarithmic dependence on k. Previously, only bounds that were at least linear in k were known. We give two important applications of our main result: (1) A polylogarithmic in k bound on the Boolean noise sensitivity of intersections of k "regular" halfspaces (previous work gave bounds linear in k). (2) A pseudorandom generator (PRG) with seed length O((log n)*poly(log k,1/delta)) that delta-fools all polytopes with k faces with respect to the Gaussian distribution. We also obtain PRGs with similar parameters that fool polytopes formed by intersection of regular halfspaces over the hypercube. Using our PRG constructions, we obtain the first deterministic quasi-polynomial time algorithms for approximately counting the number of solutions to a broad class of integer programs, including dense covering problems and contingency tables.

Explore related subjects

cs.CCcs.CGcs.DMcs.LGmath.PR

Keep this discovery

BibTeXRIS

Prahladh Harsha, Adam Klivans, Raghu Meka. 2009-12-24. An Invariance Principle for Polytopes. https://doi.org/10.1145/2395116.2395118

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗