Aspects of Submodularity Workshop Titles and Abstracts Title: Learning Submodular Mixtures; and Active/semi-supervised Learning Joint Work with Title: Multicommodity Flows and Cuts in Polymatroidal Networks

We discuss several recent applications of submodularity to machine learning. First, we present a class of submodular functions useful for document summarization. We show the best ever results on for both generic and query-focused document summarization on widely used and standardized evaluations. We then further improve on these results using a new method to learn submodular mixtures in a large-margin setting. The method uses a submodular loss function, and has risk-bound guarantees in terms of the loss function optimization's approximation ratio. Second, we present recent results for joint active semi-supervised learning with submodular functions. Here, a submodular function parameterizes an objective that consists of an activelearning and a semi-supervised learning part, and that when optimized minimizes a deterministic upper bound on the error. We present an algorithm that approximately minimizes this upper bound. Joint work with Andrew Guillory and Hui Lin. 2. Chandra Chekuri, UIUC Title: Multicommodity Flows and Cuts in Polymatroidal Networks Abstract: The well-known maxflow-mincut theorem for single-commodity flows in directed networks is of fundamental importance in combinatorial optimization. Flow-cut equivalence does not hold in the multicommodity case. A substantial amount of research in the algorithms community has focused on understanding the gap between multicommodity flows and associated cuts. Poly-logarithmic gap results have been established in various settings via tools from network decomposition and metric embeddings. In this talk we describe work that obtains flow-cut gap results in *polymatroidal* networks. In these networks there are submodular capacity constraints on the edges incident to a node. Such networks were introduced by Lawler & Martel and Hassin in the single-commodity setting and are closely related to the submodular flow model of Edmonds and Giles. The well-known maxflow-mincut theorem for a singlecommodity holds in polymatroidal networks, however, the multicommodity case has not been previously explored. Our work is primarily motivated by applications to information flow in wireless networks although the technical results we obtain are of independent interest as they generalize known results in standard networks. The results highlight the utility of line embeddings suggested by Matousek and Rabinovich to round the dual of the flow-relaxation rewritten with a convex objective function (obtained from the Lovaszextension of a submodular function). Joint work with Sreeram Kannan, Adnan Raja and Pramod Viswanath from UIUC ECE Department.