DISC 2011: Distributed Computing pp 200-201

# Brief Announcement: Distributed Approximations for the Semi-matching Problem

• Andrzej Czygrinow
• Michal Hanćkowiak
• Krzysztof Krzywdziński
• Edyta Szymańska
• Wojciech Wawrzyniak
Conference paper

DOI: 10.1007/978-3-642-24100-0_18

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6950)
Cite this paper as:
Czygrinow A., Hanćkowiak M., Krzywdziński K., Szymańska E., Wawrzyniak W. (2011) Brief Announcement: Distributed Approximations for the Semi-matching Problem. In: Peleg D. (eds) Distributed Computing. DISC 2011. Lecture Notes in Computer Science, vol 6950. Springer, Berlin, Heidelberg

## 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).

© 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