English, Article, Journal or magazine article edition: Network formulations of mixed-integer programs CONFORTI, Michele; DI SUMMA, Marco; EISENBRAND, Fritz; ...

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Bookmark: http://trove.nla.gov.au/version/178224
Physical Description
  • preprint
Language
  • English

Edition details

Title
  • Network formulations of mixed-integer programs
Author
  • CONFORTI, Michele
  • DI SUMMA, Marco
  • EISENBRAND, Fritz
  • WOLSEY, Laurence A.
Physical Description
  • preprint
Notes
  • We consider mixed-integer sets of the type M IX T U =​ {x : Ax b; xi integer, i I}, where A is a totally unimodular matrix, b is an arbitrary vector and I is a nonempty subset of the column indices of A. We show that the problem of checking nonemptiness of a set M IX T U is NP-complete when A contains at most two nonzeros per column. This is in contrast to the case when A is TU and contains at most two nonzeros per row. Denoting the set by M IX 2T U , we provide an extended formulation for the convex hull of M IX 2T U whose constraint matrix is the dual of a network matrix, and with integer right hand side vector. The size of this formulation depends on the number |F | of distinct fractional parts taken by the continuous variables in the extreme points of conv(M IX 2T U ). When this number is polynomial in the dimension of the matrix A, the formulation is of polynomial size and the optimization problem over M IX 2T U lies in P. We show that there are instances for which |F | is of exponential size, and we also give conditions under which |F | is of polynomial size. Finally we show that these results for the set M IX 2T U 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.
  • mixed-integer set, totally unimodular matrix, extended formulation, convex hull, dual of network matrix.
  • RePEc:ctl:louvec:2006117
Language
  • English
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