Network Formulations of Mixed-Integer Programs
Mathematics of Operations Research
New Hardness Results for Diophantine Approximation
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
The mixing set with divisible capacities
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
Scheduling periodic tasks in a hard real-time environment
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
A note on the continuous mixing set
Operations Research Letters
The mixing set with divisible capacities: A simple approach
Operations Research Letters
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SIAM Journal on Optimization
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We study the set $$S = \{(x, y) \in \Re_{+} \times Z^{n} : x + B_{j} y_{j} \geq b_{j}, j = 1, \ldots, n\}$$, where $$B_{j}, b_{j} \in \Re_{+} - \{0\}$$, j = 1, ..., n, and B 1 | ... | B n . The set S generalizes the mixed-integer rounding (MIR) set of Nemhauser and Wolsey and the mixing-MIR set of Günlük and Pochet. In addition, it arises as a substructure in general mixed-integer programming (MIP), such as in lot-sizing. Despite its importance, a number of basic questions about S remain unanswered, including the tractability of optimization over S and how to efficiently find a most violated cutting plane valid for P = conv (S). We address these questions by analyzing the extreme points and extreme rays of P. We give all extreme points and extreme rays of P. In the worst case, the number of extreme points grows exponentially with n. However, we show that, in some interesting cases, it is bounded by a polynomial of n. In such cases, it is possible to derive strong cutting planes for P efficiently. Finally, we use our results on the extreme points of P to give a polynomial-time algorithm for solving optimization over S.