Lower and upper competitive bounds for online directed graph exploration

Abstract We study the problem of exploring all nodes of an unknown directed graph. A searcher has to construct a tour that visits all nodes, but only has information about the parts of the graph it already visited. Analogously to the traveling salesman problem, the goal is to minimize the cost of such a tour. In this article, we present upper and lower bounds for the competitive ratio of both the deterministic and the randomized online version of exploring all nodes of directed graphs. Our bounds are sharp or sharp up to a small constant, depending on the specific model. As it turns out, restricting the diameter, the incoming/outgoing degree, or randomly choosing a starting point does not improve lower bounds beyond a small constant factor. Even supplying the searcher in a planar euclidean graph with the nodes' coordinates does not help. Essentially, exploring a directed graph has a multiplicative overhead linear in the number of nodes. Furthermore, if one wants to search for a specific node in unweighted directed graphs, a greedy algorithm with quadratic multiplicative overhead can only be improved by a small constant factor as well.

[1]  Allan Borodin,et al.  Online computation and competitive analysis , 1998 .

[2]  Ravi Prakash,et al.  Unidirectional links prove costly in wireless ad hoc networks , 1999, DIALM '99.

[3]  Jean-Sébastien Sereni,et al.  Collaborative search on the plane without communication , 2012, PODC '12.

[4]  Krzysztof Diks,et al.  Tree exploration with little memory , 2002, SODA.

[5]  Daniel J. Rosenkrantz,et al.  An analysis of several heuristics for the traveling salesman problem , 2013, Fundamental Problems in Computing.

[6]  Jeffrey Scott Vitter,et al.  Shortest paths in euclidean graphs , 2005, Algorithmica.

[7]  Shay Kutten.  Stepwise construction of an efficient distributed traversing algorithm for general strongly connected directed networks or: Traversing one way streets with no map , 1988, ICCC.

[8]  Pierre Fraigniaud,et al.  Forewords: Special issue on graph searching , 2008, Theor. Comput. Sci..

[9]  Amin Saberi,et al.  An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem , 2010, SODA '10.

[10]  Ning Xu,et al.  Improved analysis of a multirobot graph exploration strategy , 2014, 2014 13th International Conference on Control Automation Robotics & Vision (ICARCV).

[11]  T. D. Parsons,et al.  Pursuit-evasion in a graph , 1978 .

[12]  Andrzej Pelc,et al.  Tree exploration with advice , 2008, Inf. Comput..

[13]  Owen Kaser,et al.  Attribute value reordering for efficient hybrid OLAP , 2003, DOLAP '03.

[14]  Xiaotie Deng,et al.  Exploring an unknown graph , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[15]  Christos H. Papadimitriou,et al.  The complexity of searching a graph , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[16]  Piotr Berman,et al.  On-line Searching and Navigation , 1996, Online Algorithms.

[17]  Donald F. Towsley,et al.  Sampling directed graphs with random walks , 2012, 2012 Proceedings IEEE INFOCOM.

[18]  Wolfram Burgard,et al.  Collaborative multi-robot exploration , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[19]  Euripides Markou,et al.  Online Graph Exploration with Advice , 2012, SIROCCO.

[20]  Rudolf Fleischer,et al.  Competitive Online Approximation of the Optimal Search Ratio , 2008, SIAM J. Comput..

[21]  Nicolas Nisse,et al.  Report on GRASTA 2014 , 2014 .

[22]  Lali Barrière,et al.  Connected graph searching , 2012, Inf. Comput..

[23]  Nicolas Nisse,et al.  Exclusive Graph Searching , 2013, Algorithmica.

[24]  Patrick Bosc,et al.  On a fuzzy bipolar relational algebra , 2013, Inf. Sci..

[25]  Fabian Kuhn,et al.  The Complexity of Data Aggregation in Directed Networks , 2011, DISC.

[26]  Roger Wattenhofer,et al.  Directed Graph Exploration , 2012, OPODIS.

[27]  Susanne Albers,et al.  Exploring unknown environments , 1997, STOC '97.

[28]  Anthony Bonato,et al.  The Game of Cops and Robbers on Graphs , 2011 .

[29]  Lali Barrière,et al.  Capture of an intruder by mobile agents , 2002, SPAA '02.

[30]  Gregor von Bochmann,et al.  A brief history of web crawlers , 2013, CASCON.

[31]  Stephan Kreutzer,et al.  Special Issue on "Theory and Applications of Graph Searching Problems" , 2011, Theor. Comput. Sci..

[32]  Mohit Singh,et al.  Improved Approximation Ratios for Traveling Salesperson Tours and Paths in Directed Graphs , 2007, APPROX-RANDOM.

[33]  Alan M. Frieze,et al.  On the worst-case performance of some algorithms for the asymmetric traveling salesman problem , 1982, Networks.

[34]  Sundar Vishwanathan,et al.  An Approximation Algorithm for the Asymmetric Travelling Salesman Problem with Distances One and Two , 1992, Inf. Process. Lett..

[35]  Amos Korman,et al.  Memory Lower Bounds for Randomized Collaborative Search and Implications for Biology , 2012, DISC.

[36]  Nicola Santoro,et al.  Map construction of unknown graphs by multiple agents , 2007, Theor. Comput. Sci..

[37]  P. Hudson.  Search Games , 1982 .

[38]  Yasuo Okabe,et al.  The Online Graph Exploration Problem on Restricted Graphs , 2009, IEICE Trans. Inf. Syst..

[39]  Dennis Komm,et al.  Treasure Hunt with Advice , 2015, SIROCCO.

[40]  Tobias Langner,et al.  How Many Ants Does It Take to Find the Food? , 2014, SIROCCO.

[41]  Andrzej Pelc,et al.  Graph exploration by a finite automaton , 2005, Theor. Comput. Sci..

[42]  Jasmin Smula,et al.  Information content of online problems: advice versus determinism and randomization , 2015 .

[43]  Pierre Fraigniaud,et al.  Digraphs Exploration with Little Memory , 2004, STACS.

[44]  Dimitrios M. Thilikos,et al.  An annotated bibliography on guaranteed graph searching , 2008, Theor. Comput. Sci..

[45]  Michael A. Bender,et al.  The power of a pebble: exploring and mapping directed graphs , 1998, STOC '98.

[46]  Tobias Langner,et al.  Solving the ANTS Problem with Asynchronous Finite State Machines , 2014, ICALP.

[47]  Christian Schindelhauer,et al.  Why Robots Need Maps , 2007, SIROCCO.

[48]  Nicolas Jozefowiez,et al.  The vehicle routing problem: Latest advances and new challenges , 2007 .

[49]  Giorgio Ausiello,et al.  The on-line asymmetric traveling salesman problem , 2008, J. Discrete Algorithms.

[50]  Bala Kalyanasundaram,et al.  Constructing Competitive Tours from Local Information , 1994, Theor. Comput. Sci..

[51]  Lars Engebretsen.  An Explicit Lower Bound for TSP with Distances One and Two , 1999, STACS.

[52]  Kurt Mehlhorn,et al.  Online graph exploration: New results on old and new algorithms , 2012, Theor. Comput. Sci..

[53]  Rudolf Fleischer,et al.  Exploring an Unknown Graph Efficiently , 2005, ESA.

[54]  Bernard Mans,et al.  Network Exploration by Silent and Oblivious Robots , 2010, WG.

[55]  Moshe Lewenstein,et al.  Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs , 2005, JACM.

[56]  Gerhard J. Woeginger,et al.  On the nearest neighbor rule for the traveling salesman problem , 2004, Oper. Res. Lett..

[57]  Michel Raynal,et al.  Anonymous graph exploration without collision by mobile robots , 2008, Inf. Process. Lett..

[58]  Markus Bläser,et al.  A new approximation algorithm for the asymmetric TSP with triangle inequality , 2003, TALG.

[59]  Rudolf Fleischer,et al.  Experimental Studies of Graph Traversal Algorithms , 2003, WEA.