Theory Of Submodular Functions Research Articles

The nature of optimal policies for deterministic finite-horizon inventory models

This paper concerns a general formulation for a longstanding problem. This is the problem of determining a replenishment schedule that minimises the total cost of stocking and holding an inventory in a deterministic finite-horizon inventory model. The formulation permits the treatment of seemingly unrelated models in a single framework. These include classical lot-size models, batching models, repair models, recovery models and others. Admissible control policies are restricted to a partition of some closed interval on the real line. The solution of a mixed integer nonlinear programming problem (MINLP) delivers the optimal partition. It is shown that the MINLP possesses an optimal solution under very mild conditions. The theory of submodular functions on a lattice is the key to handling the integer variable. This theory permits the recovery and generalisations of earlier results on the interleaving property of optimal partitions and a convexity property of the value of the objective function. Past intractable inventory models with demand driven by a general differential equation and in the absence and in the presence of shortages and inflation are solved. This generalises a number of existing results in the literature.

Query-oriented text summarization based on hypergraph transversals

The rise in the amount of textual resources available on the Internet has created the need for tools of automatic document summarization. The main challenges of query-oriented extractive summarization are (1) to identify the topics of the documents and (2) to recover query-relevant sentences of the documents that together cover these topics. Existing graph- or hypergraph-based summarizers use graph-based ranking algorithms to produce individual scores of relevance for the sentences. Hence, these systems fail to measure the topics jointly covered by the sentences forming the summary, which tends to produce redundant summaries. To address the issue of selecting non-redundant sentences jointly covering the main query-relevant topics of a corpus, we propose a new method using the powerful theory of hypergraph transversals. First, we introduce a new topic model based on the semantic clustering of terms in order to discover the topics present in a corpus. Second, these topics are modeled as the hyperedges of a hypergraph in which the nodes are the sentences. A summary is then produced by generating a transversal of nodes in the hypergraph. Algorithms based on the theory of submodular functions are proposed to generate the transversals and to build the summaries. The proposed summarizer outperforms existing graph- or hypergraph-based summarizers by at least 6% of ROUGE-SU4 F-measure on DUC 2007 dataset. It is moreover cheaper than existing hypergraph-based summarizers in terms of computational time complexity.