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Advice Complexity of Maximum Independent set in Sparse and Bipartite Graphs

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Abstract

We study the advice complexity of the online version of the Maximum Independent Set problem, restricted to the sparse, and bipartite graphs, respectively. We show that for sparse graphs, constant-sized advice is sufficient to obtain a constant competitive ratio, whereas for bipartite graphs, only competitive ratio Ω(log(n/a)/loglog(n/a)) can be obtained with an advice of size a > loglogn. However, competitive ratio O(logn) can be achieved with advice O(loglogn).

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Notes

  1. Actually, the paper proves inapproximability of Maximum-Clique problem, where the aim is to find the subset of vertices with largest cardinality that form a clique. However, the two problems are obviously equivalent w.r.t. approximation.

  2. Some works use a slightly relaxed definition by allowing an additive constant, i.e., the algorithm is c-competitive if there exists a constant α such that the cost of the worst-case output (for randomized algorithms the worst-case expected output) is at least 1/c⋅o p t − α.

  3. A σ-bounded disc graph is an intersection graph of a set of discs in a plane, where the ratio of the radii of any two discs is at most σ.

  4. although without any restrictions on the structure or size of the supergraph, one can always construct a “universal” supergraph that fools any deterministic algorithm; with randomized algorithms, however, the situation is more subtle (see [2])

  5. As a remark we mention the lower bound 1.13747 logn colors needed for online coloring of bipartite graphs due to [3].

  6. Since n is not known, a self-delimited encoding will be used, at a cost of small increase in the number of bits used.

  7. note, however, that A d e t may behave differently afterwards

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Acknowledgments

The authors acknowledge the support of the VEGA agency under the grants 1/0671/11, and 2/0136/12. The research originated at the “Mountains and Algorithms” workshop organized by Juraj Hromkovič in August, 2011. Preliminary version of the paper has been presented at WAOA 2012. We thank the anonymous reviewer for fruitful comments.

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Authors and Affiliations

  1. Institute of Mathematics, Slovak Academy of Sciences, Bratislava, Slovakia

    Stefan Dobrev

  2. Department of Computer Science, Comenius University, Bratislava, Slovakia

    Rastislav Královič

  3. Department of Computer Science, ETH Zurich, Zurich, Switzerland

    Richard Královič

  4. Google Inc., Zurich, Switzerland

    Richard Královič

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  1. Stefan Dobrev
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  2. Rastislav Královič
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Correspondence to Rastislav Královič.

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Dobrev, S., Královič, R. & Královič, R. Advice Complexity of Maximum Independent set in Sparse and Bipartite Graphs. Theory Comput Syst 56, 197–219 (2015). https://doi.org/10.1007/s00224-014-9592-2

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