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The Power of Deferral: Maintaining a ConstantCompetitive Steiner Tree Online
"... In the online Steiner tree problem, a sequence of points is revealed onebyone: when a point arrives, we only have time to add a single edge connecting this point to the previous ones, and we want to minimize the total length of edges added. Here, a tight bound has been known for two decades: the g ..."
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In the online Steiner tree problem, a sequence of points is revealed onebyone: when a point arrives, we only have time to add a single edge connecting this point to the previous ones, and we want to minimize the total length of edges added. Here, a tight bound has been known for two decades: the greedy algorithm maintains a tree whose cost is O(log n) times the Steiner tree cost, and this is best possible. But suppose, in addition to the new edge we add, we have time to change a single edge from the previous set of edges: can we do much better? Can we, e.g., maintain a tree that is constantcompetitive? We answer this question in the affirmative. We give a primaldual algorithm that makes only a single swap per step (in addition to adding the edge connecting the new point to the previous ones), and such that the tree’s cost is only a constant times the optimal cost. Our dualbased analysis is quite different from previous primalonly analyses. In particular, we give a correspondence between radii of dual balls and lengths of tree edges; since dual balls are associated with points and hence do not move around (in contrast to edges), we can closely monitor the edge lengths based on the dual radii. Showing that these dual radii cannot change too rapidly is the technical heart of the paper, and allows us to give a hard bound on the number of swaps per arrival, while maintaining a constantcompetitive tree at all times. Previous results for this problem gave an algorithm that performed an amortized constant number of swaps: for each n, the number of swaps in the first n steps was O(n). We also give a simpler tight analysis for this amortized case.
A robust afptas for online bin packing with polynomial migration
 In Proc. 40th International Colloquium on Automata, Languages, and Programming (ICALP
, 2013
"... In this paper we develop general LP and ILP techniques to find an approximate solution with improved objective value close to an existing solution. The task of improving an approximate solution is closely related to a classical theorem of Cook et al. [7] in the sensitivity analysis for LPs and ILPs ..."
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In this paper we develop general LP and ILP techniques to find an approximate solution with improved objective value close to an existing solution. The task of improving an approximate solution is closely related to a classical theorem of Cook et al. [7] in the sensitivity analysis for LPs and ILPs. This result is often applied in designing robust algorithms for online problems. We apply our new techniques to the online bin packing problem, where it is allowed to reassign a certain number of items, measured by the migration factor. The migration factor is defined by the total size of reassigned items divided by the size of the arriving item. We obtain a robust asymptotic fully polynomial time approximation scheme (AFPTAS) for the online bin packing problem with migration factor bounded by a polynomial in 1 . This answers an open question stated by Epstein and Levin [10] in the affirmative. As a byproduct we prove an approximate variant of the sensitivity theorem by Cook at el. [7] for linear programs. 1
Changing Bases: Multistage Optimization for Matroids and Matchings
"... Abstract. This paper is motivated by the fact that many systems need to be maintained continually while the underlying costs change over time. The challenge then is to continually maintain nearoptimal solutions to the underlying optimization problems, without creating too much churn in the solutio ..."
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Abstract. This paper is motivated by the fact that many systems need to be maintained continually while the underlying costs change over time. The challenge then is to continually maintain nearoptimal solutions to the underlying optimization problems, without creating too much churn in the solution itself. We model this as a multistage combinatorial optimization problem where the input is a sequence of cost functions (one for each time step); while we can change the solution from step to step, we incur an additional cost for every such change. We first study the multistage matroid maintenance problem, where we need to maintain a base of a matroid in each time step under the changing cost functions and acquisition costs for adding new elements. The online version of this problem generalizes onine paging, and is a wellstructured case of the metrical task systems. E.g., given a graph, we need to maintain a spanning tree T t at each step: we pay c t (T t ) for the cost of the tree at time t, and also Tt \ Tt−1 for the number of edges changed at this step. Our main result is a polynomial time O(log m log r)approximation to the online multistage matroid maintenance problem, where m is the number of elements/edges and r is the rank of the matroid. This improves on results of Buchbinder et al.