Daniel Hathcock

Daniel Hathcock, R. Ravi

Given a bipartite graph that has a perfect matching, a prefect proportional allocation is an assignment of positive weights to the nodes of the right partition so that every left node is fractionally assigned to its neighbors in proportion to their weights, and these assignments define a fractional perfect matching. We prove that a bipartite graph has a perfect proportional allocation if and only if it is matching covered, by using a classical result on matrix scaling. We also present an extension of this result to provide a simple allocation strategy in non-matching covered bipartite graphs.

Daniel Hathcock, Guy Kortsarz, R. Ravi

We consider minimum time multicasting problems in directed and undirected graphs: given a root node and a subset of $t$ terminal nodes, multicasting seeks to find the minimum number of rounds within which all terminals can be informed with a message originating at the root. In each round, the telephone model we study allows the information to move via a matching from the informed nodes to the uninformed nodes. Since minimum time multicasting in digraphs is poorly understood compared to the undirected variant, we study an intermediate problem in undirected graphs that specifies a target $k < t$, and requires that only $k$ of the terminals be informed in the minimum number of rounds. For this problem, we improve the implications of the previous results and obtain a multiplicative approximation factor of $\tilde{O}(t^{1/3})$. For the directed version, we obtain an additive $\tilde{O}(k^{1/2})$ approximation algorithm (with a polylogarithmic multiplicative factor). Our algorithms are based on reductions to the related problems of finding $k$-trees of minimum poise (sum of maximum degree and diameter) and applying a combination of greedy network decomposition techniques and set covering under partition matroid constraints. We also study the problem of bounded degree Directed Steiner Tree, for which we obtain improved polylogarithmic approximations for the special case of bounded treewidth graphs. This extends prior work on the Group Steiner Tree problem.

Daniel Hathcock, R. Ravi, Amitabh Sinha

In e-retail, order fulfillment speed has become one of the most important metrics affecting customer satisfaction. While common wisdom dictates that maintaining a large global fulfillment network maximizes efficiency via economies of scale, recent evidence has shown that breaking up the network into smaller regions can yield significant speed improvements. In this paper, we consider a simple abstract model of order fulfillment by which we explain this phenomenon. We characterize fulfillment assignments satisfying an equilibrium condition based on the greedy fulfillment strategy, and quantify how the resulting fulfillment delay can be decreased by regionalizing the network. Finally, we provide some algorithmic results for computing low delay assignments, and some simulations supporting our equilibrium framework.