×

Online bipartite matching with decomposable weights. (English) Zbl 1423.68607

Schulz, Andreas S. (ed.) et al., Algorithms – ESA 2014. 22nd annual European symposium, Wrocław, Poland, September 8–10, 2014. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 8737, 260-271 (2014).
Summary: We study a weighted online bipartite matching problem: \(G(V _{1}, V _{2}, E)\) is a weighted bipartite graph where \(V _{1}\) is known beforehand and the vertices of \(V _{2}\) arrive online. The goal is to match vertices of \(V _{2}\) as they arrive to vertices in \(V _{1}\), so as to maximize the sum of weights of edges in the matching. If assignments to \(V _{1}\) cannot be changed, no bounded competitive ratio is achievable. We study the weighted online matching problem with free disposal, where vertices in \(V _{1}\) can be assigned multiple times, but only get credit for the maximum weight edge assigned to them over the course of the algorithm. For this problem, the greedy algorithm is 0.5-competitive and determining whether a better competitive ratio is achievable is a well known open problem.
We identify an interesting special case where the edge weights are decomposable as the product of two factors, one corresponding to each end point of the edge. This is analogous to the well studied related machines model in the scheduling literature, although the objective functions are different. For this case of decomposable edge weights, we design a 0.5664 competitive randomized algorithm in complete bipartite graphs. We show that such instances with decomposable weights are non-trivial by establishing upper bounds of 0.618 for deterministic and 0.8 for randomized algorithms.
A tight competitive ratio of \(1 - 1/e \approx 0.632\) was known previously for both the 0-1 case as well as the case where edge weights depend on the offline vertices only, but for these cases, reassignments cannot change the quality of the solution. Beating 0.5 for weighted matching where reassignments are necessary has been a significant challenge. We thus give the first online algorithm with competitive ratio strictly better than 0.5 for a non-trivial case of weighted matching with free disposal.
For the entire collection see [Zbl 1295.68031].

MSC:

68W27 Online algorithms; streaming algorithms
05C22 Signed and weighted graphs
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C85 Graph algorithms (graph-theoretic aspects)
68R10 Graph theory (including graph drawing) in computer science
68W40 Analysis of algorithms
PDFBibTeX XMLCite
Full Text: DOI arXiv