By Akiyoshi Shioura (auth.), Camil Demetrescu, Magnús M. Halldórsson (eds.)

This ebook constitutes the refereed court cases of the nineteenth Annual ecu Symposium on Algorithms, ESA 2011, held in Saarbrücken, Germany, in September 2011 within the context of the mixed convention ALGO 2011.

The sixty seven revised complete papers offered have been conscientiously reviewed and chosen from 255 preliminary submissions: fifty five out of 209 in music layout and research and 12 out of forty six in tune engineering and purposes. The papers are prepared in topical sections on approximation algorithms, computational geometry, online game conception, graph algorithms, solid matchings and auctions, optimization, on-line algorithms, exponential-time algorithms, parameterized algorithms, scheduling, facts constructions, graphs and video games, allotted computing and networking, strings and sorting, in addition to neighborhood seek and set systems.

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**Additional info for Algorithms – ESA 2011: 19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011. Proceedings**

**Example text**

2 with full optimization (ﬂag -O4). 04 Server Edition. The main focus of the experiments in this paper is the quality of the 2VCSS returned by each algorithm. Therefore we did not put much eﬀort in 20 L. Georgiadis optimizing our source code. ) For completeness we do mention the running times for a subset of the experiments, but leave a thorough experimental study of the running times for the full version of the paper. 2 Instances We considered three types of instances: bicliques, internet peer-to-peer networks taken from the Stanford Large Network Dataset Collection [23], and random digraphs.

Note that given B, we are unable to identify clusters contained exclusively in L (or R), but this will not aﬀect the cost. We therefore take the harmless decision that single-side clusters are always singletons. We will say that a pair e = ( , r) is violated if e ∈ (E \ EB ) ∪ (EB \ E). , xG,B (e) = 1 if e is violated and 0 otherwise. We will also simply use x(e) when it is obvious to which graph G and clustering B it refers. The cost of a clustering solution is deﬁned to be costG (B) = e∈L×R xG,B (e).

Ailon et al. The Combinatorial 4-Approximation Algorithm Notation Before describing the framework we give some general facts and notations. Let the input graph again be G = (L, R, E) where L and R are the sets of left and right nodes and E be a subset of L × R. Each element ( , r) ∈ L × R will be referred to as a pair. A solution to our combinatorial problem is a clustering C1 , C2 , . . , Cm of the set L ∪ R. We identify such a clustering with a bipartite graph B = (L, R, EB ) for which ( , r) ∈ EB if and only if ∈ L and r ∈ R are in the same cluster Ci for some i.