On the Configuration LP for Maximum Budgeted Allocation

We study the Maximum Budgeted Allocation problem, i.e., the problem of selling a set of $m$ indivisible goods to $n$ players, each with a separate budget, such that we maximize the collected revenue. Since the natural assignment LP is known to have an integrality gap of $\frac{3}{4}$, which matches the best known approximation algorithms, our main focus is to improve our understanding of the stronger configuration LP relaxation. In this direction, we prove that the integrality gap of the configuration LP is strictly better than $\frac{3}{4}$, and provide corresponding polynomial time roundings, in the following restrictions of the problem: (i) the Restricted Budgeted Allocation problem, in which all the players have the same budget and every item has the same value for any player it can be sold to, and (ii) the graph MBA problem, in which an item can be assigned to at most 2 players. Finally, we improve the best known upper bound on the integrality gap for the general case from $\frac{5}{6}$ to $2\sqrt{2}-2\approx 0.828$ and also prove hardness of approximation results for both cases.Comment: 29 pages, 4 figures. To appear in the 17th Conference on Integer Programming and Combinatorial Optimization (IPCO), 201

Population extremal optimisation for discrete multi-objective optimisation problems

The power to solve intractable optimisation problems is often found through population based evolutionary methods. These include, but are not limited to, genetic algorithms, particle swarm optimisation, differential evolution and ant colony optimisation. While showing much promise as an effective optimiser, extremal optimisation uses only a single solution in its canonical form – and there are no standard population mechanics. In this paper, two population models for extremal optimisation are proposed and applied to a multi-objective version of the generalised assignment problem. These models use novel intervention/interaction strategies as well as collective memory in order to allow individual population members to work together. Additionally, a general non-dominated local search algorithm is developed and tested. Overall, the results show that improved attainment surfaces can be produced using population based interactions over not using them. The new EO approach is also shown to be highly competitive with an implementation of NSGA-II.No Full Tex

Online Mixed Packing and Covering

In many problems, the inputs arrive over time, and must be dealt with irrevocably when they arrive. Such problems are online problems. A common method of solving online problems is to first solve the corresponding linear program, and then round the fractional solution online to obtain an integral solution. We give algorithms for solving linear programs with mixed packing and covering constraints online. We first consider mixed packing and covering linear programs, where packing constraints are given offline and covering constraints are received online. The objective is to minimize the maximum multiplicative factor by which any packing constraint is violated, while satisfying the covering constraints. No prior sublinear competitive algorithms are known for this problem. We give the first such --- a polylogarithmic-competitive algorithm for solving mixed packing and covering linear programs online. We also show a nearly tight lower bound. Our techniques for the upper bound use an exponential penalty function in conjunction with multiplicative updates. While exponential penalty functions are used previously to solve linear programs offline approximately, offline algorithms know the constraints beforehand and can optimize greedily. In contrast, when constraints arrive online, updates need to be more complex. We apply our techniques to solve two online fixed-charge problems with congestion. These problems are motivated by applications in machine scheduling and facility location. The linear program for these problems is more complicated than mixed packing and covering, and presents unique challenges. We show that our techniques combined with a randomized rounding procedure give polylogarithmic-competitive integral solutions. These problems generalize online set-cover, for which there is a polylogarithmic lower bound. Hence, our results are close to tight

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

A bi-level model of dynamic traffic signal control with continuum approximation

This paper proposes a bi-level model for traffic network signal control, which is formulated as a dynamic Stackelberg game and solved as a mathematical program with equilibrium constraints (MPEC). The lower-level problem is a dynamic user equilibrium (DUE) with embedded dynamic network loading (DNL) sub-problem based on the LWR model (Lighthill and Whitham, 1955; Richards, 1956). The upper-level decision variables are (time-varying) signal green splits with the objective of minimizing network-wide travel cost. Unlike most existing literature which mainly use an on-and-off (binary) representation of the signal controls, we employ a continuum signal model recently proposed and analyzed in Han et al. (2014), which aims at describing and predicting the aggregate behavior that exists at signalized intersections without relying on distinct signal phases. Advantages of this continuum signal model include fewer integer variables, less restrictive constraints on the time steps, and higher decision resolution. It simplifies the modeling representation of large-scale urban traffic networks with the benefit of improved computational efficiency in simulation or optimization. We present, for the LWR-based DNL model that explicitly captures vehicle spillback, an in-depth study on the implementation of the continuum signal model, as its approximation accuracy depends on a number of factors and may deteriorate greatly under certain conditions. The proposed MPEC is solved on two test networks with three metaheuristic methods. Parallel computing is employed to significantly accelerate the solution procedure