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 deg M (u) = 1 for every u ∈ U. A semi-matching M is called optimal if cost(M): = ∑ v ∈ V (deg M (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 ∈ V d(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).
The research supported by grant N N206 565740.
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Czygrinow, A., Hanćkowiak, M., Krzywdziński, K., Szymańska, E., Wawrzyniak, W.: Distributed approximation algorithm for the semi-matching problem (manuscript)
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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. https://doi.org/10.1007/978-3-642-24100-0_18
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DOI: https://doi.org/10.1007/978-3-642-24100-0_18
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