图的扩张搜索比
The expanding search ratio of a graph
摘要：
我们研究的问题，搜索隐藏的目标在一个环境中，由边加权图建模。从给定的根顶点开始选择一系列边，使得每条边与先前选择的边相邻。这种搜索范式，被称为扩展搜索，最近由Wynn和Lidbetter [2013]引入，用于建模问题，例如寻找煤炭或扫雷，其中重新勘探的成本可以忽略不计。例如，它也可以用来模拟一组搜索者在搜索隐藏的对手或爆炸装置时相继分开。我们将扩展搜索的搜索比定义为到达顶点所需时间与从根到顶点的最短路径成本之比的最大值。这可以被解释为在搜索中产生的乘法遗憾的措施，和类似的目标以前已经研究了传统的（路径）搜索的背景下。在本文中，我们解决的算法和计算问题，最大限度地减少搜索比在所有扩展搜索，各种搜索环境，包括一般的图形，树和星形图。我们的主要结果集中在寻找具有最小期望搜索比的随机扩展搜索问题上，这等价于解决搜索者和隐藏者之间的零和对策。我们解决了这些问题的某些类的图，并获得常数因子的近似。
英文摘要：
We study the problem of searching for a hidden target in an environment that is modeled by an edge-weighted graph. A sequence of edges is chosen starting from a given root vertex such that each edge is adjacent to a previously chosen edge. This search paradigm, known as expanding search was recently introduced by Apern and Lidbetter [2013] for modeling problems such as searching for coal or minesweeping in which the cost of re-exploration is negligible. It can also be used to model a team of searchers successively splitting up in the search for a hidden adversary or explosive device, for example. We define the search ratio of an expanding search as the maximum over all vertices of the ratio of the time taken to reach the vertex and the shortest-path cost to it from the root. This can be interpreted as a measure of the multiplicative regret incurred in searching, and similar objectives have previously been studied in the context of conventional (pathwise) search. In this paper we address algorithmic and computational issues of minimizing the search ratio over all expanding searches, for a variety of search environments, including general graphs, trees and star-like graphs. Our main results focus on the problem of finding the randomized expanding search with minimum expected search ratio, which is equivalent to solving a zero-sum game between a Searcher and a Hider. We solve these problems for certain classes of graphs, and obtain constant-factor approximations for others.
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