Polynomial auction algorithms for shortest paths

Includes bibliographical references (p. 24-25).Supported by NSF. DDM-8903385 Supported by the ARO. C DAAL03-86-K-0171 Supported by a CNR-GNIM grant, and by a Fullbright grant.by Dimitri P. Bertsekas, Stefano Pallottino, and Maria Grazia Scutella

Optimal Approximation Algorithms for Multi-agent Combinatorial Problems with Discounted Price Functions

Submodular functions are an important class of functions in combinatorial optimization which satisfy the natural properties of decreasing marginal costs. The study of these functions has led to strong structural properties with applications in many areas. Recently, there has been significant interest in extending the theory of algorithms for optimizing combinatorial problems (such as network design problem of spanning tree) over submodular functions. Unfortunately, the lower bounds under the general class of submodular functions are known to be very high for many of the classical problems. In this paper, we introduce and study an important subclass of submodular functions, which we call discounted price functions. These functions are succinctly representable and generalize linear cost functions. In this paper we study the following fundamental combinatorial optimization problems: Edge Cover, Spanning Tree, Perfect Matching and Shortest Path, and obtain tight upper and lower bounds for these problems. The main technical contribution of this paper is designing novel adaptive greedy algorithms for the above problems. These algorithms greedily build the solution whist rectifying mistakes made in the previous steps

On distributed virtual network embedding with guarantees

To provide wide-area network services, resources from different infrastructure providers are needed. Leveraging the consensus-based resource allocation literature, we propose a general distributed auction mechanism for the (NP-hard) virtual network (VNET) embedding problem. Under reasonable assumptions on the bidding scheme, the proposed mechanism is proven to converge, and it is shown that the solutions guarantee a worst-case efficiency of (1-(1/e)) relative to the optimal node embedding, or VNET embedding if virtual links are mapped to exactly one physical link. This bound is optimal, that is, no better polynomial-time approximation algorithm exists, unless P=NP. Using extensive simulations, we confirm superior convergence properties and resource utilization when compared to existing distributed VNET embedding solutions, and we show how by appropriate policy design, our mechanism can be instantiated to accommodate the embedding goals of different service and infrastructure providers, resulting in an attractive and flexible resource allocation solution.CNS-0963974 - National Science Foundationhttp://www.cs.bu.edu/fac/matta/Papers/ToN-CAD.pdfAccepted manuscrip

Matrix bids in combinatorial auctions: expressiveness and micro-economic properties

A combinatorial auction is an auction where multiple items are for sale simultaneously to a set of buyers. Furthermore, buyers are allowed to place bids on subsets of the available items. This paper focuses on a combinatorial auction where a bidder can express his preferences by means of a so-called ordered matrix bid. Ordered matrix bids are a bidding language that allows a compact representation of a bidder''s preferences, and was developed by Day (2004). We give an overview of how a combinatorial auction with matrix bids works. We elaborate on the relevance of the matrix bid auction and we develop methods to verify whether a given matrix bid satisfies properties related to micro-economic theory as free disposal, subadditivity, submodularity and the gross substitutes property. Finally, we investigate how a collection of arbitrary bids can be represented as a matrix bid.microeconomics ;

Learning to Prune: Speeding up Repeated Computations

It is common to encounter situations where one must solve a sequence of similar computational problems. Running a standard algorithm with worst-case runtime guarantees on each instance will fail to take advantage of valuable structure shared across the problem instances. For example, when a commuter drives from work to home, there are typically only a handful of routes that will ever be the shortest path. A naive algorithm that does not exploit this common structure may spend most of its time checking roads that will never be in the shortest path. More generally, we can often ignore large swaths of the search space that will likely never contain an optimal solution. We present an algorithm that learns to maximally prune the search space on repeated computations, thereby reducing runtime while provably outputting the correct solution each period with high probability. Our algorithm employs a simple explore-exploit technique resembling those used in online algorithms, though our setting is quite different. We prove that, with respect to our model of pruning search spaces, our approach is optimal up to constant factors. Finally, we illustrate the applicability of our model and algorithm to three classic problems: shortest-path routing, string search, and linear programming. We present experiments confirming that our simple algorithm is effective at significantly reducing the runtime of solving repeated computations