Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Online Bipartite Matching with Decomposable Weights   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 {\em
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.

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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: math
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The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: eess