Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On the Approximability of Presidential Type Predicates (1907.04451v2)

Published 9 Jul 2019 in cs.CC

Abstract: Given a predicate $P: {-1, 1}k \to {-1, 1}$, let $CSP(P)$ be the set of constraint satisfaction problems whose constraints are of the form $P$. We say that $P$ is approximable if given a nearly satisfiable instance of $CSP(P)$, there exists a probabilistic polynomial time algorithm that does better than a random assignment. Otherwise, we say that $P$ is approximation resistant. In this paper, we analyze presidential type predicates, which are balanced linear threshold functions where all of the variables except the first variable (the president) have the same weight. We show that almost all presidential-type predicates $P$ are approximable. More precisely, we prove the following result: for any $\delta_0 > 0$, there exists a $k_0$ such that if $k \geq k_0$, $\delta \in (\delta_0,1 - 2/k]$, and ${\delta}k + k - 1$ is an odd integer then the presidential type predicate $P(x) = sign({\delta}k{x_1} + \sum_{i=2}{k}{x_i})$ is approximable. To prove this, we construct a rounding scheme that makes use of biases and pairwise biases. We also give evidence that using pairwise biases is necessary for such rounding schemes.

Citations (3)

Summary

We haven't generated a summary for this paper yet.