[论文解读] On the Fine-grained Complexity of One-Dimensional Dynamic Programming

Marvin Künnemann, Ramamohan Paturi+1 more|arXiv (Cornell University)|Mar 2, 2017

Optimization and Search Problems被引用 5

一句话总结

本文提出了一套针对一维动态规划问题的细粒度复杂度框架，聚焦于最小权重子序列（LWS）问题。通过识别LWS实例与核心问题（如最小内积和pmin、`q-convolution）之间的亚二次等价关系，作者在SETH假设下建立了紧致的条件性下界，解释了众多LWS变体的固有难度，并统一了其复杂度分析。

ABSTRACT

In this paper, we investigate the complexity of one-dimensional dynamic programming, or more specifically, of the Least-Weight Subsequence (LWS) problem: Given a sequence of $n$ data items together with weights for every pair of the items, the task is to determine a subsequence $S$ minimizing the total weight of the pairs adjacent in $S$. A large number of natural problems can be formulated as LWS problems, yielding obvious $O(n^2)$-time solutions. In many interesting instances, the $O(n^2)$-many weights can be succinctly represented. Yet except for near-linear time algorithms for some specific special cases, little is known about when an LWS instantiation admits a subquadratic-time algorithm and when it does not. In particular, no lower bounds for LWS instantiations have been known before. In an attempt to remedy this situation, we provide a general approach to study the fine-grained complexity of succinct instantiations of the LWS problem. In particular, given an LWS instantiation we identify a highly parallel core problem that is subquadratically equivalent. This provides either an explanation for the apparent hardness of the problem or an avenue to find improved algorithms as the case may be. More specifically, we prove subquadratic equivalences between the following pairs (an LWS instantiation and the corresponding core problem) of problems: a low-rank version of LWS and minimum inner product, finding the longest chain of nested boxes and vector domination, and a coin change problem which is closely related to the knapsack problem and (min,+)-convolution. Using these equivalences and known SETH-hardness results for some of the core problems, we deduce tight conditional lower bounds for the corresponding LWS instantiations. We also establish the (min,+)-convolution-hardness of the knapsack problem.

研究动机与目标

• 理解一维动态规划问题在何种条件下可实现亚二次时间算法。
• 识别出简洁表示的LWS问题中的固有计算障碍。
• 利用细粒度复杂度理论，为LWS实例建立条件性下界。
• 通过与核心问题的亚二次等价关系，统一多样LWS问题的复杂度分析。
• 通过其对应核心问题的可解性，解释某些LWS变体的近线性可解性。

提出的方法

• 提出一个通用框架，用于分析简洁LWS实例的细粒度复杂度。
• 识别出与每个LWS实例亚二次等价的“核心问题”。
• 利用亚二次等价关系，将已知困难核心问题的条件性下界转移至LWS实例。
• 将核心问题（如最小内积、pmin、`q-convolution）基于SETH的困难性结果，应用于LWS变体。
• 重新审视已知的近线性时间LWS问题，并通过其核心问题的简洁性解释其可解性。
• 采用归约与重表述方法，将LWS与经典问题（如背包问题、LIS、无界子集和问题）联系起来。

实验结果

研究问题

• RQ1 在何种条件下，LWS问题可被亚二次时间求解？
• RQ2 哪些LWS实例本质上是困难的，原因是什么？
• RQ3 如何利用亚二次等价关系，将核心问题的条件性下界转移至LWS？
• RQ4 为何某些LWS问题尽管看似复杂，却仍可在近线性时间内求解？
• RQ5 核心问题在决定LWS实例复杂度方面起什么作用？

主要发现

• 本文建立了低秩LWS与最小内积之间的亚二次等价关系，意味着前者在SETH下是难解的。
• 证明了与背包问题相关的硬币兑换问题为pmin、`q-convolution难，建立了紧致的条件性下界。
• 表明背包问题为pmin、`q-convolution难，为其困难性提供了新的复杂度解释。
• 证明最长递增子序列（LIS）问题可通过归约为静态LWS变体（进一步归约为排序）在Õ(n)时间内求解。
• 通过单次卷积计算，无界子集和问题在Õ(n)时间内求解，解释了其近线性可解性。
• 通过在全单调矩阵上使用SMAWK算法，凹LWS问题在Õ(n)时间内求解，其高效性源于其核心问题的简洁性。