Message Passing Algorithm for the Generalized Assignment Problem

Abstract

The generalized assignment problem (GAP) is NP-hard. It is even APX-hard to approximate it. The best known approximation algorithm is the LP-rounding algorithm in [1] with a (1− 1 e ) approximation ratio. We investigate the max-product belief propagation algorithm for the GAP, which is suitable for distributed implementation. The basic algorithm passes an exponential number of real-valued messages in each iteration. We show that the algorithm can be simplified so that only a linear number of real-valued messages are passed in each iteration. In particular, the computation of the messages from machines to jobs decomposes into two knapsack problems, which are also present in each iteration of the LP-rounding algorithm. The messages can be computed in parallel at each iteration. We observe that for small instances of GAP where the optimal solution can be computed, the message passing algorithm converges to the optimal solution when it is unique. We then show how to add small deterministic perturbations to ensure the uniqueness of the optimum. Finally, we prove GAP remains strongly NP-hard even if the optimum is unique.

Survey on Multi-Objective Task Allocation Algorithms for IoT Networks

“…While solving the TAP is already an NP-hard challenge by itself [ 2 ], the TAP for the IoT presents an even more complex problem due to the following dynamic elements it presents.…”

“…Because the performance of the network regarding those metrics is highly dependent on where in the network tasks are allocated, solving the Task Allocation Problem (TAP) is a critical challenge in the IoT. Due to the inherently heterogeneous and dynamic nature of IoT networks, finding a task allocation that is optimal in terms of a single optimization goal is already challenging because of the basic NP-hard Task Allocation Problem [ 2 ]. However, with the increasing range of application and user requirements, optimizing for multiple objectives and providing multiple solutions to adjust the system in real time becomes increasingly important.…”

A two-phase method for operating room scheduling

“…Tabu search metaheuristic is used by Swangnop and Chaovalitwongse (2014) and Mckendall et al (2015). Quick heuristic algorithms the assignment problem are described by Srivastava and Bullo (2014), Yuan et al (2014) and Topcuoglu et al (2014). Lagrangian and dynamic programming are associated in the method proposed by Posta et al (2012).…”

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“…Posta et al (2012) combinam relaxação lagrangeana e programação dinâmica. Srivastava e Bullo (2014), Yuan et al (2014), Topcuoglu et al (2014) e Sethanan e Pitakaso (2016) descrevem algoritmos aproximados e rápidos.…”