Given n jobs, where job j has size pj, and m machines, we want to assign...
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Given n jobs, where job j has size pj, and m machines, we want to assign the jobs to the machines and minimize the maximum load of any machine (the sum of the p, for jobs j assigned to it). Furthermore, the jobs are "picky": there is a bipartite graph (J, M, E) with J the jobs, M the machines, and a job j can be assigned to machine m iff (j, m) € E. Denote by OPT the cost of the optimal assignment (the maximum load of a machine). (a) Show that the greedy algorithm, which considers the jobs in an arbitrary order and assignes the job to the least loaded machine (so far) that can take it, can output a solution of cost log n OPT (for some input graph, job order and and job sizes). (b) Consider the following linear program (with variables Z€R and TRE): min subject to Z Σ jEJ: (j,m) EE Pjxjm Z Σ mEM (j,m) EE Ijm 20 Ijm = 1 Vm € M Vje J V(j, m) € E Show that if r is a verter solution to the LP, then the graph (J, M, E*) is a forest, where E* CE contains exactly the edges (j, m) € E such that Ijm > 0. (c) Root each tree in the forest in some job vertex, and assign each job to one of its children (arbitrarily) in the tree, or to its parent if it is a leaf. Show that this assignment has cost at most 20PT. Given n jobs, where job j has size pj, and m machines, we want to assign the jobs to the machines and minimize the maximum load of any machine (the sum of the p, for jobs j assigned to it). Furthermore, the jobs are "picky": there is a bipartite graph (J, M, E) with J the jobs, M the machines, and a job j can be assigned to machine m iff (j, m) € E. Denote by OPT the cost of the optimal assignment (the maximum load of a machine). (a) Show that the greedy algorithm, which considers the jobs in an arbitrary order and assignes the job to the least loaded machine (so far) that can take it, can output a solution of cost log n OPT (for some input graph, job order and and job sizes). (b) Consider the following linear program (with variables Z€R and TRE): min subject to Z Σ jEJ: (j,m) EE Pjxjm Z Σ mEM (j,m) EE Ijm 20 Ijm = 1 Vm € M Vje J V(j, m) € E Show that if r is a verter solution to the LP, then the graph (J, M, E*) is a forest, where E* CE contains exactly the edges (j, m) € E such that Ijm > 0. (c) Root each tree in the forest in some job vertex, and assign each job to one of its children (arbitrarily) in the tree, or to its parent if it is a leaf. Show that this assignment has cost at most 20PT.
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a The greedy algorithm will always assign the current job to the least loaded machine that can take ... View the full answer
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