Integer and combinatorial optimization
Integer and combinatorial optimization
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
The Continuous Mixing Polyhedron
Mathematics of Operations Research
Mathematical Programming: Series A and B
SIAM Journal on Discrete Mathematics
Compact formulations as a union of polyhedra
Mathematical Programming: Series A and B
The mixing-MIR set with divisible capacities
Mathematical Programming: Series A and B
Extended Formulations for Packing and Partitioning Orbitopes
Mathematics of Operations Research
The mixing set with divisible capacities
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
Lot-Sizing with Stock Upper Bounds and Fixed Charges
SIAM Journal on Discrete Mathematics
Fixed-charge transportation on a path: linear programming formulations
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
On a class of mixed-integer sets with a single integer variable
Operations Research Letters
Mixing Sets Linked by Bidirected Paths
SIAM Journal on Optimization
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We consider mixed-integer sets described by system of linear inequalities in which the constraint matrix A is totally unimodular; the right-hand side is arbitrary vector; and a subset of the variables is required to be integer. We show that the problem of checking nonemptiness of a set of this type is NP-complete, even in the case in which the linear system describes mixed-integer network flows with half-integral requirement on the nodes. This is in contrast to the case in which A is totally unimodular and contains at most two nonzeros per row. In this case, we provide an extended formulation for the convex hull of solutions whose constraint matrix is a dual-network matrix with an integral right-hand-side vector. The size of this formulation depends on the number of distinct fractional parts taken by the continuous variables in the extreme points of the convex hull of the given set. When this number is polynomial in the dimension of A, the extended formulation is of polynomial size. If, in addition, the corresponding list of fractional parts can be computed efficiently, then our result provides a polynomial algorithm for the optimization problem over these sets. We show that there are instances for which this list is of exponential size, and we also give conditions under which it is short and can be efficiently computed. Finally, we show that these results provide a unified framework leading to polynomial-size extended formulations for several generalizations of mixing sets and lot-sizing sets studied in the last few years.