Independent Set with Advice: The Impact of Graph Knowledge - (Extended Abstract)

We are interested in online graph problems where the knowledge of the underlying graph G (all arriving vertices are from G) has a profound impact on the size of the advice needed to solve the problem efficiently. On one hand, we show that, for sparse graphs, constant-size advice is sufficient to solve the maximum independent set problem with constant competitive ratio, even with no knowledge of the underlying graph. On the other hand, we show a lower bound of Ω(log(n/a)/loglog(n/a)) on the competitive ratio of finding a maximum independent set in bipartite graphs if no knowledge of the underlying graph is available and if the advice is of size a. We complement the lower bounds by providing corresponding upper bounds.

[1]  Richard J. Lipton,et al.  Online interval scheduling , 1994, SODA '94.

[2]  Magnús M. Halldórsson.  Online coloring known graphs , 1999, SODA '99.

[3]  Pierre Fraigniaud,et al.  Online computation with advice , 2011, Theor. Comput. Sci..

[4]  Ioannis Caragiannis,et al.  Randomized on-line algorithms and lower bounds for computing large independent sets in disk graphs , 2007, Discret. Appl. Math..

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

[6]  Andrzej Pelc,et al.  Communication algorithms with advice , 2010, J. Comput. Syst. Sci..

[7]  Magnús M. Halldórsson,et al.  Online independent sets , 2002, Theor. Comput. Sci..

[8]  Juraj Hromkovic,et al.  Information Complexity of Online Problems , 2010, MFCS.

[9]  Andrzej Pelc,et al.  Distributed computing with advice: information sensitivity of graph coloring , 2007, Distributed Computing.

[10]  George Christodoulou,et al.  On-line maximum independent set in chordal graphs , 2005 .

[11]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[12]  Iwona Cieślik.  On-line graph coloring , 2006 .

[13]  Dennis Komm,et al.  On the Advice Complexity of the k-Server Problem , 2011, ICALP.

[14]  Andrzej Pelc,et al.  Fast radio broadcasting with advice , 2008, Theor. Comput. Sci..

[15]  Andrzej Pelc,et al.  Drawing maps with advice , 2010, J. Parallel Distributed Comput..

[16]  Ingo Schiermeyer,et al.  Vertex Colouring and Forbidden Subgraphs – A Survey , 2004, Graphs Comb..

[17]  Dennis Komm,et al.  On the Advice Complexity of Online Problems , 2009, ISAAC.

[18]  William I. Grosky,et al.  SOFSEM 2002: Theory and Practice of Informatics , 2002, Lecture Notes in Computer Science.

[19]  Vangelis Th. Paschos,et al.  On-line models and algorithms for max independent set , 2006, RAIRO Oper. Res..

[20]  Amos Fiat,et al.  Lower Bounds for On-line Graph Problems with Application to On-line Circuit and Optical Routing , 2006, SIAM J. Comput..

[21]  Vangelis Th. Paschos,et al.  On-Line Maximum-Order Induces Hereditary Subgraph Problems , 2000, SOFSEM.

[22]  Petr Hliněný,et al.  Mathematical Foundations of Computer Science 2010, 35th International Symposium, MFCS 2010, Brno, Czech Republic, August 23-27, 2010. Proceedings , 2010, MFCS.

[23]  Stefan Dobrev,et al.  Measuring the problem-relevant information in input , 2009, RAIRO Theor. Informatics Appl..

[24]  Andrzej Pelc,et al.  Trade-offs Between the Size of Advice and Broadcasting Time in Trees , 2008, SPAA '08.

[25]  Pierre Fraigniaud,et al.  Local MST Computation with Short Advice , 2007, SPAA '07.

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

[27]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.