The Query-commit Problem

In the query-commit problem we are given a graph where edges have distinct probabilities of existing. It is possible to query the edges of the graph, and if the queried edge exists then its endpoints are irrevocably matched. The goal is to find a querying strategy which maximizes the expected size of the matching obtained. This stochastic matching setup is motivated by applications in kidney exchanges and online dating. In this paper we address the query-commit problem from both theoretical and experimental perspectives. First, we show that a simple class of edges can be queried without compromising the optimality of the strategy. This property is then used to obtain in polynomial time an optimal querying strategy when the input graph is sparse. Next we turn our attentions to the kidney exchange application, focusing on instances modeled over real data from existing exchange programs. We prove that, as the number of nodes grows, almost every instance admits a strategy which matches almost all nodes. This result supports the intuition that more exchanges are possible on a larger pool of patient/donors and gives theoretical justification for unifying the existing exchange programs. Finally, we evaluate experimentally different querying strategies over kidney exchange instances. We show that even very simple heuristics perform fairly well, being within 1.5% of an optimal clairvoyant strategy, that knows in advance the edges in the graph. In such a time-sensitive application, this result motivates the use of committing strategies

Multicommodity Multicast, Wireless and Fast

We study rumor spreading in graphs, specifically multicommodity multicast problem under the wireless model: given source-destination pairs in the graph, one needs to find the fastest schedule to transfer information from each source to the corresponding destination. Under the wireless model, nodes can transmit to any subset of their neighbors in synchronous time steps, as long as they either transmit or receive from at most one transmitter during the same time step. We improve approximation ratio for this problem from O~(n^(2/3)) to O~(n^((1/2) + epsilon)) on n-node graphs. We also design an algorithm that satisfies p given demand pairs in O(OPT + p) steps, where OPT is the length of an optimal schedule, by reducing it to the well-studied packet routing problem. In the case where underlying graph is an n-node tree, we improve the previously best-known approximation ratio of O((log n)/(log log n)) to 3. One consequence of our proof is a simple constructive rule for optimal broadcasting in a tree under a widely studied telephone model

Spanning trees short or small

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page

Weakening and Shifting of the Saharan Shallow Meridional Circulation During Wet Years of the West African Monsoon

The correlation between increased Sahel rainfall and reduced Saharan surface pressure is well established in observations and global climate models, and has been used to imply that increased Sahel rainfall is caused by a stronger shallow meridional circulation (SMC) over the Sahara. This study uses two atmospheric reanalyses to examine interannual variability of Sahel rainfall and the Saharan SMC, which consists of northward near-surface flow across the Sahel into the Sahara and southward flow near 700 hPa out of the Sahara. During wet Sahel years, the Saharan SMC shifts poleward, producing a drop in low-level geopotential and surface pressure over the Sahara. Statistically removing the effect of the poleward shift from the low-level geopotential eliminates significant correlations between this geopotential and Sahel precipitation. As the Saharan SMC shifts poleward, its mid-tropospheric divergent outflow decreases, indicating a weakening of its overturning mass flux. The poleward shift and weakening of the Saharan SMC during wet Sahel years is reproduced in an idealized model of West Africa; a wide range of imposed sea surface temperature and land surface albedo perturbations in this model produce a much larger range of SMC variations that nevertheless have similar quantitative associations with Sahel rainfall as in the reanalyses. These results disprove the idea that enhanced Sahel rainfall is caused by strengthening of the Saharan SMC. Instead, these results are consistent with the hypothesis that the a stronger SMC inhibits Sahel rainfall, perhaps by advecting mid-tropospheric warm and dry air into the precipitation maximum.Comment: Submitted to Journal of Climat

Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur