Optimal Polynomial-time Compression for Boolean Max CSP

IF 0.8 Q3 COMPUTER SCIENCE, THEORY & METHODS
Bart M.P. Jansen, Michał Włodarczyk
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引用次数: 1

Abstract

In the Boolean maximum constraint satisfaction problem – Max CSP ( Γ ) – one is given a collection of weighted applications of constraints from a finite constraint language Γ , over a common set of variables, and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized. There exists a concise dichotomy theorem providing a criterion on Γ for the problem to be polynomial-time solvable and stating that otherwise it becomes NP-hard. We study the NP-hard cases through the lens of kernelization and provide a complete characterization of Max CSP ( Γ ) with respect to the optimal compression size. Namely, we prove that Max CSP ( Γ ) parameterized by the number of variables n is either polynomial-time solvable, or there exists an integer d ≥ 2 depending on Γ , such that: (1) An instance of Max CSP ( Γ ) can be compressed into an equivalent instance with \(\mathcal {O}(n^d\log n) \) bits in polynomial time, (2) Max CSP( Γ ) does not admit such a compression to \(\mathcal {O}(n^{d-\varepsilon }) \) bits unless NP⊆co-NP/poly. Our reductions are based on interpreting constraints as multilinear polynomials combined with the framework of ‘constraint implementations’, formerly used in the context of APX-hardness. As another application of our reductions, we reveal tight connections between optimal running times for solving Max CSP( Γ ) . More precisely, we show that obtaining a running time of the form \(\mathcal {O}(2^{(1-\varepsilon)n}) \) for particular classes of Max CSP s is as hard as breaching this barrier for Max d - SAT for some d .
布尔Max CSP的最优多项式时间压缩
在布尔最大约束满足问题Max CSP (Γ)中,给定了来自有限约束语言Γ的约束加权应用的集合,在一组公共变量上,目标是为变量分配布尔值,以便使满足约束的总权重最大化。存在一个简洁的二分定理,在Γ上给出了问题是多项式时间可解的判据,否则问题就变成np困难。我们通过核化透镜研究了NP-hard情况,并提供了关于最佳压缩大小的Max CSP (Γ)的完整表征。即证明由变量数n参数化的Max CSP(Γ)是多项式时间可解的,或者存在依赖于Γ的整数d≥2,使得:(1)Max CSP(Γ)的一个实例可以在多项式时间内压缩成一个具有\(\mathcal {O}(n^d\log n) \) bits的等价实例,(2)Max CSP(Γ)不允许压缩到\(\mathcal {O}(n^{d-\varepsilon }) \) bits,除非NP co-NP/poly。我们的缩减是基于将约束解释为与“约束实现”框架相结合的多线性多项式,以前在apx硬度上下文中使用。作为我们缩减的另一个应用,我们揭示了解决Max CSP的最佳运行时间之间的紧密联系(Γ)。更准确地说,我们表明,对于特定类别的Max CSP,获得\(\mathcal {O}(2^{(1-\varepsilon)n}) \)形式的运行时间与突破Max d - SAT在某些d中的这个障碍一样困难。
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来源期刊
ACM Transactions on Computation Theory
ACM Transactions on Computation Theory COMPUTER SCIENCE, THEORY & METHODS-
CiteScore
2.30
自引率
0.00%
发文量
10
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