This paper proposes a new framework for providing approximation guarantees of local search algorithms. Local search is a basic algorithm design technique and is widely used for various combinatorial optimization problems. To analyze local search algorithms for set function maximization, we propose a new notion called localizability of set functions, which measures how effective local improvement is. Moreover, we provide approximation guarantees of standard local search algorithms under various combinatorial constraints in terms of localizability. The main application of our framework is sparse optimization, for which we show that restricted strong concavity and restricted smoothness of the objective function imply localizability, and further develop accelerated versions of local search algorithms. We conduct experiments in sparse regression and structure learning of graphical models to confirm the practical efficiency of the proposed local search algorithms.
Arizona State University（亚利桑那州立大学）；Yuan Ze University（元智大学）；Lockheed Martin（洛克希德·马丁公司）
Requirements driven search-based testing (also known as falsification) has proven to be a practical and effective method for discovering erroneous behaviors in Cyber-Physical Systems. Despite the constant improvements on the performance and applicability of falsification methods, they all share a common characteristic. Namely, they are best-effort methods which do not provide any guarantees on the absence of erroneous behaviors (falsifiers) when the testing budget is exhausted. The absence of finite time guarantees is a major limitation which prevents falsification methods from being utilized …
This paper studies the important problem of finding all $k$-nearest neighbors to points of a query set $Q$ in another reference set $R$ within any metric space. Our previous work defined compressed cover trees and corrected the key arguments in several past papers for challenging datasets. In 2009 Ram, Lee, March, and Gray attempted to improve the time complexity by using pairs of cover trees on the query and reference sets. In 2015 Curtin with the above co-authors used extra parameters to finally prove a time complexity for $k=1$. The current work fills all previous gaps and improves the near…
We study the weighted $k$-Set Packing problem: Given a collection $S$ of sets, each of cardinality at most $k$, together with a positive weight function $w:\mathcal{S}\rightarrow\mathbb{Q}_{>0}$, the task is to compute a disjoint sub-collection $A\subseteq \mathcal{S}$ of maximum total weight. For $k\leq 2$, the weighted $k$-Set Packing problem can be solved in polynomial time, but for $k\geq 3$, it becomes $NP$-hard. Recently, Neuwohner has shown how to obtain approximation guarantees of $\frac{k+ε_k}{2}$ with $\lim_{k\rightarrow\infty}ε_k=0$. She further showed her result to be asymptoticall…
Embedding into hyperbolic space is emerging as an effective representation technique for datasets that exhibit hierarchical structure. This development motivates the need for algorithms that are able to effectively extract knowledge and insights from datapoints embedded in negatively curved spaces. We focus on the problem of nearest neighbor search, a fundamental problem in data analysis. We present efficient algorithmic solutions that build upon established methods for nearest neighbor search in Euclidean space, allowing for easy adoption and integration with existing systems. We prove theore…
Searching the Nodes of a Graph: Theory and Algorithms
一个或多个搜索器必须捕获隐藏在图的节点中的不可见的逃避者。我们研究这个图搜索问题，我们强调，我们研究了捕获的节点定位逃避者，这比边搜索得到较少的关注。我们表明，在一般的图的节点搜索的问题是容易比边搜索（然而，节点搜索是NP完全的，就像边搜索）。我们专注于内部单调连接（IMC）的节点搜索的树，并表明它本质上是等价的IMC边搜索。对于一般图上的IMC节点搜索，我们提出了一种新的算法：GSST（Guaranteed Search by Spanning Tree），它通过沿着图G的一棵生成树T执行图G的所有清除移动来清除图G。我们证明了GSST的概率完全变量的存在性。我们的实验还表明，GSST可以有效地节点清除大图，只有一个小的运行时间。还提供了GSST的实现，并公开提供。
Graph-based approaches are empirically shown to be very successful for the nearest neighbor search (NNS). However, there has been very little research on their theoretical guarantees. We fill this gap and rigorously analyze the performance of graph-based NNS algorithms, specifically focusing on the low-dimensional (d << \log n) regime. In addition to the basic greedy algorithm on nearest neighbor graphs, we also analyze the most successful heuristics commonly used in practice: speeding up via adding shortcut edges and improving accuracy via maintaining a dynamic list of candidates. We believe …
