An LP-based algorithm for the data association problem in multitarget tracking
Computers and Operations Research
Two-phase greedy algorithms for some classes of combinatorial linear programs
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Two-phase greedy algorithms for some classes of combinatorial linear programs
ACM Transactions on Algorithms (TALG)
Min CSP on four elements: moving beyond submodularity
CP'11 Proceedings of the 17th international conference on Principles and practice of constraint programming
Note on pseudolattices, lattices and submodular linear programs
Discrete Optimization
Dual greedy polyhedra, choice functions, and abstract convex geometries
Discrete Optimization
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A greedy algorithm solves a dual pair of linear programs where the primal variables are associated to the elements of a sublattice B of a finite product lattice, and the cost coefficients define a submodular function on B. This approach links and generalizes two well-known classes of greedily solvable linear programs. The primal problem generalizes the (ordinary and multi-index) transportation problems satisfying a Monge condition (Hoffman 1963; Bein et al. 1995) to the case of forbidden cells where the nonforbidden cells form a sublattice. The dual problem generalizes to an arbitrary finite product lattice the linear optimization problem over submodular polyhedra (Lovasz 1983; Fujishige and Tomizawa 1983), which stemmed from the work of Edmonds (1970) on polymatroids. Our model and results also encompass the dual pairs of linear programs and their greedy solutions defined by Lovasz (1983) for the special case of the Boolean algebra, and by Faigle and Kern (1996) for the case of so-called "rooted forests." We also discuss relationships between Monge properties and submodularity, and present a class of problems with submodular costs arising in production and logistics.