13 research outputs found
Online Perfect Matching and Mobile Computing
The original publication is available at www.springerlink.comWe present a natural online perfect matching problem moti-
vated by problems in mobile computing. A total of n customers connect
and disconnect sequentially, and each customer has an associated set of
stations to which it may connect. Each station has a capacity limit. We
allow the network to preemptively switch a customer between allowed
stations to make room for a new arrival. We wish to minimize the total
number of switches required to provide service to every customer. Equiv-
alently, we wish to maintain a perfect matching between customers and
stations and minimize the lengths of the augmenting paths. We measure
performance by the worst case ratio of the number of switches made to
the minimum number required.
When each customer can be connected to at most two stations:
{ Some intuitive algorithms have lower bounds of
(n) and
(n= log n).
{ When the station capacities are 1, there is an upper bound of O(pn).
{ When customers do not disconnect and the station capacity is 1, we
achieve a competitive ratio of O(log n).
{ There is a lower bound of
(pn) when the station capacities are 2.
{ We present optimal algorithms when the station capacity is arbitrary
in special cases
A Tight Bound for Shortest Augmenting Paths on Trees
The shortest augmenting path technique is one of the fundamental ideas used
in maximum matching and maximum flow algorithms. Since being introduced by
Edmonds and Karp in 1972, it has been widely applied in many different
settings. Surprisingly, despite this extensive usage, it is still not well
understood even in the simplest case: online bipartite matching problem on
trees. In this problem a bipartite tree is being revealed
online, i.e., in each round one vertex from with its incident edges
arrives. It was conjectured by Chaudhuri et. al. [K. Chaudhuri, C. Daskalakis,
R. D. Kleinberg, and H. Lin. Online bipartite perfect matching with
augmentations. In INFOCOM 2009] that the total length of all shortest
augmenting paths found is . In this paper, we prove a tight upper bound for the total length of shortest augmenting paths for
trees improving over bound [B. Bosek, D. Leniowski, P.
Sankowski, and A. Zych. Shortest augmenting paths for online matchings on
trees. In WAOA 2015].Comment: 22 pages, 10 figure
Shortest augmenting paths for online matchings on trees
The shortest augmenting path (Sap) algorithm is one of the most classical approaches to the maximum matching and maximum flow problems, e.g., using it Edmonds and Karp (J. ACM 19(2), 248–264 1972) have shown the first strongly polynomial time algorithm for the maximum flow problem. Quite astonishingly, although it has been studied for many years already, this approach is far from being fully understood. This is exemplified by the online bipartite matching problem. In this problem a bipartite graph G = (W ⊎ B, E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. After arrival of this vertex we augment the current matching by using shortest augmenting path. It was conjectured by Chaudhuri et al. (INFOCOM’09) that the total length of all augmenting paths found by Sap is O(nlogn). However, no better bound than O(n2) is known even for trees. In this paper we prove an O(nlog2n) upper bound for the total length of augmenting paths for trees