Brief Announcement: Distributed Approximations for the Semi-matching Problem

  • Andrzej Czygrinow
  • Michal Hanćkowiak
  • Krzysztof Krzywdziński
  • Edyta Szymańska
  • Wojciech Wawrzyniak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6950)

Abstract

We consider the semi-matching problem in bipartite graphs. The network is represented by a bipartite graph G = (U ∪ V, E), where U corresponds to clients, V to servers, and E is the set of available connections between them. The goal is to find a set of edges M ⊆ E such that every vertex in U is incident to exactly one edge in M. The load of a server v ∈ V is defined as the square of its degree in M and the problem is to find an optimal semi-matching, i.e. a semi-matching that minimizes the sum of the loads of the servers. Formally, given a bipartite graph G = (U ∪ V,E), a semi-matching in G is a subgraph M such that degM(u) = 1 for every u ∈ U. A semi-matching M is called optimal if cost(M): = ∑ v ∈ V (degM(v))2 is minimal. It is not difficult to see that for any semi-matching M, \(\tfrac{|U|^2}{|V|} \leq{\rm cost}(M) \leq \Delta |U|\) where Δ is such that max v ∈ Vd(v) ≤ Δ. Consequently, if M* is optimal and M is arbitrary, then \({\rm cost}(M) \leq \tfrac{\Delta |V| {\rm cost}(M*)}{|U|}.\) Our main result shows that in some networks the \(\tfrac{\Delta |V|}{|U|}\) factor can be reduced to a constant (Theorem 1).

References

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Andrzej Czygrinow
    • 1
  • Michal Hanćkowiak
    • 2
  • Krzysztof Krzywdziński
    • 2
  • Edyta Szymańska
    • 2
  • Wojciech Wawrzyniak
    • 2
  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland

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