Approximation Algorithms for Graph Burning

Abstract

Numerous approaches study the vulnerability of networks against social contagion. Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchronous, discrete rounds. In each round, a fire breaks out at a vertex, and the fire spreads to all vertices that are adjacent to a burning vertex. The selection of vertices where fires start defines a schedule that indicates the number of rounds required to burn all vertices. Given a graph, the objective of an algorithm is to find a schedule that minimizes the number of rounds to burn graph. Finding the optimal schedule is known to be NP-hard, and the problem remains NP-hard when the graph is a tree or a set of disjoint paths. The only known algorithm is an approximation algorithm for disjoint paths, which has an approximation ratio of 1.5.We present approximation algorithms for graph burning. For general graphs, we introduce an algorithm with an approximation ratio of 3. When the graph is a tree, we present another algorithm with approximation ratio 2. Moreover, we consider a setting where the graph is a forest of disjoint paths. In this setting, when the number of paths is constant, we provide an optimal algorithm which runs in polynomial time. When the number of paths is more than a constant, we provide two approximation schemes: first, under a regularity condition where paths have asymptotically equal lengths, we show the problem admits an approximation scheme which is fully polynomial. Second, for a general setting where the regularity condition does not necessarily hold, we provide another approximation scheme which runs in time polynomial in the size of the graph.

Heuristics for Spreading Alarm throughout a Network

“…Until now, the only known approximation algorithm generating a burning sequence for the general graphs was an algorithm by Bonato and Kamali [3]. This algorithm has an approximation ratio of 3, which means, that the optimal, i.e., the shortest possible burning sequence is guaranteed not to be shorter than one third of the burning sequence length provided by the approximation algorithm.…”

“…The Bonato algorithm [3] starts with a guess, that the burning number equals 3g − 3. The "shortened" burning sequence is initially empty.…”

“…In the same paper [3], the authors also present an algorithm, which is optimized for trees with an approximation ratio of 2 and for disjoint paths with an approximation ratio of 1.5. In the results in Section 4 of this paper, we compare our results only to the results of the approximation algorithm for the general graphs [3], even though a few of the tested networks are trees. However, this is consistent with our further described algorithms, which are also intended for general graphs and are not optimized for trees.…”

“…It was proved that computing a "burning number" is NP-complete [2]. The problem was further studied in papers [3,4], where the bounds for the burning number (i.e., the minimum number of time steps, equal to the minimum size of the set of nodes, which are informed or alarmed "from the outside", not from their neighbor) are analyzed. The papers provided both the lower and the upper bounds, where the upper bound was accompanied by the actual sequence of the set of nodes to be informed from the outside.…”

From Two to One Index Isomorphism in Optimization Program for Quarterly Disaggregation of Annual Times Series

“…The efficient solvability of the problem (P) needs a variable indexation which must be done judiciously in order to optimize the cost of the numerical resolution algorithm with exponential complexity in case of simplex method, as the running time may be an exponential function of the dimension of the problem [5], roughly speaking. Indeed, it is always possible to improve the running time of an algorithm without increasing the cost as it is set up by Guy [6], Ji-Bo and Jian-Jun [7] in order to have polynomial running time as raised by Anthony and Shahin [8], Jon through theorem 4.3.1 [9] or theorem 3.1 [10]. Hence, for an algorithm of running time O(n) of the problem (P), one can find another algorithm with reduced execution time O(logn) which solves the same problem [6].…”