We study the wireless scheduling problem in the SINR model. More specifically, given a set of @math links, each a sender-receiver pair, we wish to partition (or ) the links into the minimum number of slots, each satisfying interference constraints allowing simultaneous transmission. In the basic problem, all senders transmit with the same uniform power. We give a distributed @math -approximation algorithm for the scheduling problem, matching the best ratio known for centralized algorithms. It holds in arbitrary metric space and for every length-monotone and sublinear power assignment. It is based on an algorithm of Kesselheim and V "ocking, whose analysis we improve by a logarithmic factor. We show that every distributed algorithm uses @math slots to schedule certain instances that require only two slots, which implies that the best possible absolute performance guarantee is logarithmic.
For distributed scheduling, the only work we are aware of remains the interesting paper by Kesselheim and V "ocking @cite_11 , who give a @math distributed approximation algorithm for the scheduling problem with any fixed length-monotone and sub-linear power assignment. They consider the model with no free acknowledgements, however their results do not improve if free acknowledgements are assumed. Thus in all cases considered, our results constitute a @math factor improvement.
@cite_11
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We study the wireless scheduling problem in the SINR model. More specifically, given a set of @math links, each a sender-receiver pair, we wish to partition (or ) the links into the minimum number of slots, each satisfying interference constraints allowing simultaneous transmission. In the basic problem, all senders transmit with the same uniform power. We give a distributed @math -approximation algorithm for the scheduling problem, matching the best ratio known for centralized algorithms. It holds in arbitrary metric space and for every length-monotone and sublinear power assignment. It is based on an algorithm of Kesselheim and V "ocking, whose analysis we improve by a logarithmic factor. We show that every distributed algorithm uses @math slots to schedule certain instances that require only two slots, which implies that the best possible absolute performance guarantee is logarithmic. In 1966, Graham showed that a simple procedure called list scheduling yields a 2-approximation algorithm for the central problem of scheduling precedence constrained jobs on identical machines to minimize makespan. To date this has remained the best algorithm, and whether it can or cannot be improved has become a major open problem in scheduling theory. We address this problem by establishing a quite surprising relation between the approximability of the considered problem and that of scheduling precedence constrained jobs on a single machine to minimize weighted completion time. More specifically, we prove that if the single machine problem is hard to approximate within a factor of @math , then the considered parallel machine problem, even in the case of unit processing times, is hard to approximate within a factor of @math , where @math tends to 0 as @math tends to 0. Combining this with Bansal and Khot's recent hardness result for the single machine problem gives that it is NP-hard to improve upon the approximation ratio obtained by Graham, assuming a new variant of the unique games conjecture. One of the classic results in scheduling theory is the @math -approximation algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to minimize makespan on unrelated machines; i.e., job @math requires time @math if processed on machine @math . More than two decades after its introduction it is still the algorithm of choice even in the restricted model where processing times are of the form @math . This problem, also known as the restricted assignment problem, is NP-hard to approximate within a factor less than @math , which is also the best known lower bound for the general version. Our main result is a polynomial time algorithm that estimates the optimal makespan of the restricted assignment problem within a factor @math , where @math is an arbitrarily small constant. The result is obtained by upper bounding the integrality gap of a certain strong linear program, known as the configuration LP, that was previously successfu... In this paper the problem of wireless transmissions scheduling in the path-loss SINR model is considered for different fixed power schemes. The lower bounds for the optimum schedule length proven by Kesselheim and Voecking are considered. It is shown that the lower bound for the linear power scheme is tight, by presenting a constant-factor approximation algorithm. On the other hand, it is shown that the lower bounds proven for many other interesting power schemes can be essentially far from the optima.