We investigate computational problems involving large weights through the lens of kernelization, which is a framework of polynomial-time preprocessing aimed at compressing the instance size. Our main focus is the weighted Clique problem, where we are given an edge-weighted graph and the goal is to detect a clique of total weight equal to a prescribed value. We show that the weighted variant, parameterized by the number of vertices $n$, is significantly harder than the unweighted problem by presenting an $O(n^{3 - \varepsilon})$ lower bound on the size of the kernel, under the assumption that NP $\not \subseteq$ coNP/poly. This lower bound is essentially tight: we show that we can reduce the problem to the case with weights bounded by $2^{O(n)}$, which yields a randomized kernel of $O(n^3)$ bits. We generalize these results to the weighted $d$-Uniform Hyperclique problem, Subset Sum, and weighted variants of Boolean Constraint Satisfaction Problems (CSPs). We also study weighted minimization problems and show that weight compression is easier when we only want to preserve the collection of optimal solutions. Namely, we show that for node-weighted Vertex Cover on bipartite graphs it is possible to maintain the set of optimal solutions using integer weights from the range $[1, n]$, but if we want to maintain the ordering of the weights of all inclusion-minimal solutions, then weights as large as $2^{\Omega(n)}$ are necessary.
翻译:我们通过内核化的透镜来调查涉及大重量的计算问题。 内核化是一个多元时预处理框架, 旨在压缩实例大小。 我们的主要焦点是加权 clique 问题, 给我们一个边加权图, 目标是检测一个总重量与规定值相等的球状。 我们显示, 加权变量, 以螺旋数为参数, 比未加权问题要困难得多得多 。 我们将这些结果概括为美元( n% 3 -\varepsilon} ) 内核重量的比值低, 假设其重量的比值是 NP $\ not\ subseqeq$ comple/polear。 这个更下限基本上很紧 : 我们显示, 以2 ⁇ (n) $ min) 来减少总重量的总重量的球质。 我们将这些结果概括为美元- Uncial- clocial 的比值解决方案的比值, 我们想要将所有 美元 的比值 的比值 继续 。 将 MI 的比值 问题, suse Sum sult sult 和 等 重量的比值变值的比值的比值 。 当 我们的比值 最重的比值 的比值的比值