Seeun William Umboh

A bicriteria approximation for the reordering buffer problem

In the reordering buffer problem (RBP), a server is asked to process a sequence of requests lying in a metric space. To process a request the server must move to the corresponding point in the metric. The requests can be processed slightly out of order; in particular, the server has a buffer of capacity k which can store up to k requests as it reads in the sequence. The goal is to reorder the requests in such a manner that the buffer constraint is satisfied and the total travel cost of the server is minimized. The RBP arises in many applications that require scheduling with a limited buffer capacity, such as scheduling a disk arm in storage systems, switching colors in paint shops of a car manufacturing plant, and rendering 3D images in computer graphics. We study the offline version of RBP and develop bicriteria approximations. When the underlying metric is a tree, we obtain a solution of cost no more than 9 OPT using a buffer of capacity 4k+1 where OPT is the cost of an optimal solution with buffer capacity k. Via randomized tree embeddings, this implies an O(logn) approximation to cost and O(1) approximation to buffer size for general metrics. In contrast, when the buffer constraint is strictly enforced, constant-factor approximations are known only for the uniform metric (Avigdor-Elgrabli et al., 2012); the best known approximation ratio for arbitrary metrics is O(log2k logn) (Englert et al., 2007).

Tight approximation bounds for dominating set on graphs of bounded arboricity

We consider the minimum dominating set problem on graphs with bounded arboricity. For graphs with arboricity a, we give an LP rounding algorithm that achieves an approximation factor of 3a and we complement this with a matching (up to constants) hardness of approximation result.

Online Constrained Forest and Prize-Collecting Network Design

In this paper, we study a very general type of online network design problem, and generalize two different previous algorithms, one for an online network design problem due to Berman and Coulston (Proceedings of the 29th annual ACM symposium on theory of computing, pp 344–353, 1997) and one for (offline) general network design problems due to Goemans and Williamson (SIAM J Comput 24:296–317, 1995); we give an

LAST but not least: online spanners for buy-at-bulk

O(\log k)

Secretary problems with convex costs

O(logk)-competitive algorithm, where k is the number of nodes that must be connected. We also consider a further generalization of the problem that allows us to pay penalties in exchange for violating connectivity constraints; we give an online

Nested convex bodies are chaseable

O(\log k)