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The problem of partitioning dense matrices into sets of sub-matrices has received increased attention recently and is crucial when considering dense linear algebra and kernels with similar communication patterns on heterogeneous platforms. The problem of load balancing and minimizing communication is traditionally reducible to an optimization problem that involves partitioning a square into rectangles. This problem has been proven to be NP-Complete for an arbitrary number of partitions. In this paper, we present recent approaches that relax the restriction that all partitions be rectangles. The first approach uses an original mathematical technique to find the exact optimal partitioning. Due to the complexity of the technique, it has been developed for a small number of partitions only. However, even at a small scale, the optimal partitions found by this approach are often non-rectangular and sometimes non-intuitive. The second approach is the study of approximate partitioning methods utilizing recursive partitioning algorithms. In particular we use the work on optimal partitioning to improve pre-existing algorithms. In this paper we discuss the different perspectives this approach opens and present two algorithms, SNRPP which is a $\sqrt{\frac{3}{2}}$ approximation, and NRPP which is a $\frac{2}{\sqrt{3}}$ approximation. While sub-optimal, the NRRP approach works for an arbitrary number of partitions. We use the first exact approach to analyse how close to the known optimal solutions the NRRP algorithm is for small numbers of partitions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of assigning a set of clients with demands to a set of servers with capacities and degree constraints. The goal is to find an allocation such that the number of clients assigned to a server is smaller than the server's degree and their overall demand is smaller than the server's capacity, while maximizing the overall throughput. This problem has several natural applications in the context of independent tasks scheduling or virtual machines allocation. We consider both the offline (when clients are known beforehand) and the online (when clients can join and leave the system at any time) versions of the problem. We first show that the degree constraint on the maximal number of clients that a server can handle is realistic in many contexts. Then, our main contribution is to prove that even if it makes the allocation problem more difficult (NP-Complete), a very small additive resource augmentation on the servers degree is enough to find in polynomial time a solution that achieves at least the optimal throughput. After a set of theoretical results on the complexity of the offline and online versions of the problem, we propose several other greedy heuristics to solve the online problem and we compare the performance (in terms of throughput) and the cost (in terms of disconnections and reconnections) of all proposed algorithms through a set of extensive simulation results. [ABSTRACT FROM AUTHOR]