更快的图形燃烧启发式算法
Faster Heuristics for Graph Burning
摘要：
图燃烧是由代理以离散步骤通过网络传播信息的过程。问题是要找到一个最佳的节点序列，必须给予信息，使网络覆盖在最少的步骤。图燃烧问题是一个NP难问题，已有文献提出了两种近似算法和一些算法。在这项工作中，我们提出了三种启发式算法，即基于骨干的贪婪启发式算法（BBGH），改进的切角启发式算法（ICCH）和基于组件的递归启发式算法（CBRH）。这些方法主要基于特征向量中心性测度。BBGH找到网络的主干，并从主干的顶点中选择要烧绿的顶点。ICCH是一种基于最短路径的启发式算法，它从最佳中心节点中选择要燃烧的顶点。不连通图上的烧数问题比连通图上的烧数问题更难。例如，烧数问题在路径上很容易，而在不相交的路径上是NP难的。在实践中，大型网络通常是断开的，而且即使输入图是连接的，在燃烧过程中，未燃烧的顶点之间的图也可能断开。对于不连通图，组件的排序是至关重要的。我们的CBRH在断开连接的图上工作得很好，因为它优先考虑组件。所有的算法都已经在几个基准网络上实现和测试，包括规模超过$50$ K个节点的大型网络。实验还包括与近似算法的比较。我们的算法的优点是，它们是更简单的实现，也比文献中提出的算法快几个数量级。
英文摘要：
Graph burning is a process of information spreading through the network by an agent in discrete steps. The problem is to find an optimal sequence of nodes which have to be given information so that the network is covered in least number of steps. Graph burning problem is NP-Hard for which two approximation algorithms and a few heuristics have been proposed in the literature. In this work, we propose three heuristics, namely, Backbone Based Greedy Heuristic (BBGH), Improved Cutting Corners Heuristic (ICCH) and Component Based Recursive Heuristic (CBRH). These are mainly based on Eigenvector centrality measure. BBGH finds a backbone of the network and picks vertex to be burned greedily from the vertices of the backbone. ICCH is a shortest path based heuristic and picks vertex to burn greedily from best central nodes. The burning number problem on disconnected graphs is harder than on the connected graphs. For example, burning number problem is easy on a path where as it is NP-Hard on disjoint paths. In practice, large networks are generally disconnected and moreover even if the input graph is connected, during the burning process the graph among the unburned vertices may be disconnected. For disconnected graphs, ordering of the components is crucial. Our CBRH works well on disconnected graphs as it prioritizes the components. All the heuristics have been implemented and tested on several bench-mark networks including large networks of size more than $50$K nodes. The experimentation also includes comparison to the approximation algorithms. The advantages of our algorithms are that they are much simpler to implement and also several orders faster than the heuristics proposed in the literature.
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