68 research outputs found
Online Minimum Cost Matching with Recourse on the Line
In online minimum cost matching on the line, n requests appear one by one and have to be matched immediately and irrevocably to a given set of servers, all on the real line. The goal is to minimize the sum of distances from the requests to their respective servers. Despite all research efforts, it remains an intriguing open question whether there exists an O(1)-competitive algorithm. The best known online algorithm by Raghvendra [S. Raghvendra, 2018] achieves a competitive factor of ?(log n). This result matches a lower bound of ?(log n) [A. Antoniadis et al., 2018] that holds for a quite large class of online algorithms, including all deterministic algorithms in the literature.
In this work, we approach the problem in a recourse model where we allow to revoke online decisions to some extent, i.e., we allow to reassign previously matched edges. We show an O(1)-competitive algorithm for online matching on the line with amortized recourse of O(log n). This is the first non-trivial result for min-cost bipartite matching with recourse. For so-called alternating instances, with no more than one request between two servers, we obtain a near-optimal result. We give a (1+?)-competitive algorithm that reassigns any request at most O(?^{-1.001}) times. This special case is interesting as the aforementioned quite general lower bound ?(log n) holds for such instances
Dynamic Maxflow via Dynamic Interior Point Methods
In this paper we provide an algorithm for maintaining a
-approximate maximum flow in a dynamic, capacitated graph
undergoing edge additions. Over a sequence of -additions to an -node
graph where every edge has capacity our algorithm runs in
time . To obtain this result we
design dynamic data structures for the more general problem of detecting when
the value of the minimum cost circulation in a dynamic graph undergoing edge
additions obtains value at most (exactly) for a given threshold . Over a
sequence -additions to an -node graph where every edge has capacity
and cost we solve this thresholded
minimum cost flow problem in . Both of our algorithms
succeed with high probability against an adaptive adversary. We obtain these
results by dynamizing the recent interior point method used to obtain an almost
linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst
Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for
maintaining minimum ratio cycles in an undirected graph that succeeds with high
probability against adaptive adversaries.Comment: 30 page
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