The Online Graph Exploration Problem on Restricted Graphs

Abstract

SUMMARYThe purpose of the online graph exploration problem is to visit all the nodes of a given graph and come back to the starting node with the minimum total traverse cost. However, unlike the classical Traveling Salesperson Problem, information of the graph is given online. When an online algorithm (called a searcher) visits a node v, then it learns information on nodes and edges adjacent to v. The searcher must decide which node to visit next depending on partial and incomplete information of the graph that it has gained in its searching process. The goodness of the algorithm is evaluated by the competitive analysis. If input graphs to be explored are restricted to trees, the depth-first search always returns an optimal tour. However, if graphs have cycles, the problem is non-trivial. In this paper we consider two simple cases. First, we treat the problem on simple cycles. Recently, Asahiro et al. proved that there is a 1.5-competitive online algorithm, while no online algorithm can be (1.25 − )-competitive for any positive constant . In this paper, we give an optimal online algorithm for this problem; namely, we give a 1+ √ 3 2 ( 1.366)-competitive algorithm, and prove that there is no (2 − )-competitive algorithm for any positive constant . Furthermore, we consider the problem on unweighted graphs. We also give an optimal result; namely we give a 2-competitive algorithm and prove that there is no (2 − )-competitive online algorithm for any positive constant . key words: the graph exploration problem, online algorithm, competitive analysis

Tactics for Evacuating from an Affected Area

“…In this paper, we define a convex polygon as a closed polygonal chain with no interior angle equal and less than 180 degrees [8]. Also an edge of a polygon is defined as a line segment forming a part of the polygonal chain, a vertex of a polygon as a point where two polygon edges meet and the boundary of polygon as a polygonal chain.…”

Directed Graph Exploration

“…While a greedy algorithm achieves a competitive ratio of Θ(log n) [35], it is not known if a constant competitive ratio for general graphs is possible [31]. For cycles there is an algorithm with a sharp competitive ratio of 1+ √ , while for trees depth-first search is optimal [32]. Recently, the best known lower bound for general graphs was improved from 2 − [32] to 5/2 − [16].…”

Online Graph Exploration: New Results on Old and New Algorithms

“…It yields a total tour not larger than twice the size of a minimum spanning tree (MST ), a lower bound on the optimal tour. This is optimal in the unit-weight case [25].…”

“…Additionally, a lower bound of 5/4 for any deterministic online algorithm is proven. Both lower and upper bound are improved in [25]. There, the authors give a more sophisticated algorithm which takes additionally the current total tour length into account.…”

“…In particular, on trees the problem of exploring all vertices is equivalent to exploring all edges. Apart from the aforementioned previous work on our problem in planar graphs [23] and cycles [2,25], it has been studied on un-weighted trees also for multiple synchronously moving searchers [13,14,17].…”