[论文解读] Generalized Metric Repair on Graphs

Chenglin Fan, Anna C. Gilbert+3 more|arXiv (Cornell University)|Aug 21, 2019

Complexity and Algorithms in Graphs参考文献 19被引用 4

一句话总结

本文提出了图度量修复（GMR）问题，将度量修复推广至仅部分距离有定义的加权图。针对仅允许增加权重和一般修复两种情形，提出了近似算法，证明了其NP难性，并基于环路亏量与贪心命中集技术，给出了$O(\kappa \log n)$-近似解；同时证明了在$\varsigma$-弦图上具有固定参数可追踪性，并通过归约至MULTICUT证明了其困难性。

ABSTRACT

Many modern data analysis algorithms either assume or are considerably more efficient if the distances between the data points satisfy a metric. These algorithms include metric learning, clustering, and dimension reduction. As real data sets are noisy, distances often fail to satisfy a metric. For this reason, Gilbert and Jain and Fan et al. introduced the closely related sparse metric repair and metric violation distance problems. The goal of these problems is to repair as few distances as possible to ensure they satisfy a metric. Three variants were considered, one admitting a polynomial time algorithm. The other variants were shown to be APX-hard, and an $O(OPT^{1/3})$-approximation was given, where $OPT$ is the optimal solution size. In this paper, we generalize these problems to no longer consider all distances between the data points. That is, we consider a weighted graph $G$ with corrupted weights $w$, and our goal is to find the smallest number of weight modifications so that the resulting weighted graph distances satisfy a metric. This is a natural generalization and is more flexible as it takes into account different relationships among the data points. As in previous work, we distinguish among the types of repairs permitted and focus on the increase only and general versions. We demonstrate the inherent combinatorial structure of the problem, and give an approximation-preserving reduction from MULTICUT. Conversely, we show that for any fixed constant $ς$, for the large class of $ς$-chordal graphs, the problems are fixed parameter tractable. Call a cycle broken if it contains an edge whose weight is larger than the sum of all its other edges, and call the amount of this difference its deficit. We present approximation algorithms, one which depends on the maximum number of edges in a broken cycle, and one which depends on the number of distinct deficit values.

研究动机与目标

• 将度量修复从完整的距离矩阵推广至稀疏加权图，以捕捉具有部分关系的现实世界数据。
• 研究图上度量修复的计算复杂性，区分仅允许增加权重与一般权重修改的情形。
• 为GMR问题设计高效的近似算法，尤其基于环路亏量与命中集归约。
• 通过证明$\varsigma$-弦图上的固定参数可追踪性，解决一个开放问题。
• 通过从MULTICUT和LB-CUT的归约，建立强的不可近似性结果。

提出的方法

• 将图度量修复问题归约为具有正亏量的环路的命中集问题，其中亏量用于度量度量违反程度。
• 在边-环路关联矩阵上使用贪心算法选择修复方案，每个亏量组的近似保证为$O(\log n)$。
• 采用分层方法：按亏量值递增处理图，为每组亏量维护一个命中集实例。
• 通过Floyd-Warshall算法计算所有点对最短路径（APSP），以计算距离并检测度量违反。
• 利用动态数据结构以期提升运行时间，尽管在$N_l(e,\delta)$更新方面仍存在挑战。
• 通过保持近似比不变的变换，将仅允许增加权重的版本归约为一般版本，但代价是图的规模增大。

实验结果

研究问题

• RQ1 当仅允许增加边权重时，图度量修复问题是否为NP难？
• RQ2 该问题能否在如$\varsigma$-弦图等结构化图类上高效求解？
• RQ3 GMR的一般版本与仅允许增加权重版本的最佳可能近似比是多少？
• RQ4 是否可以消除或减少近似因子中对不同亏量值数量$\kappa$的依赖？
• RQ5 GMR的不可近似阈值是多少？其与MULTICUT等已知难问题有何关联？

主要发现

• 仅减少权重的GMR版本可在$O(n^3)$时间内求解，而仅增加权重的版本为NP完全。
• 为一般GMR问题提出了$O(\kappa \log n)$-近似算法，其中$\kappa$为不同亏量值的数量。
• 对于$\varsigma$-弦图，问题具有固定参数可追踪性，解决了先前工作中的一个开放问题。
• 在唯一游戏假设下，通过从MULTICUT的归约，该问题难以在任意常数因子内近似。
• 通过从$L$-LB-CUT的归约，可推导出$\Omega(\sqrt{L})$-近似下界。
• 主算法的运行时间为$O(n^3 + m^2) \cdot OPT \cdot \kappa \log n$，使用动态APSP数据结构可进一步优化。