Patrick Chervet , Roland Grappe , Mathieu Lacroix , Francesco Pisanu , Roberto Wolfler Calvo
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引用次数: 0
Abstract
In this paper, we study the complexity of some fundamental questions regarding box-totally dual integral (box-TDI) polyhedra. First, although box-TDI polyhedra have strong integrality properties, we prove that Integer Programming over box-TDI polyhedra is NP-complete, that is, finding an integer point optimizing a linear function over a box-TDI polyhedron is hard. Second, we complement the result of Ding et al. (2008) who proved that deciding whether a given system is box-TDI is co-NP-complete: we prove that recognizing whether a polyhedron is box-TDI is co-NP-complete.
To derive these complexity results, we exhibit new classes of totally equimodular matrices – a generalization of totally unimodular matrices – by characterizing the total equimodularity of incidence matrices of graphs.
本文研究了盒-完全对偶积分多面体的一些基本问题的复杂性。首先，尽管盒- tdi多面体具有很强的整体性，但我们证明了盒- tdi多面体上的整数规划是np完全的，即很难找到一个整数点来优化一个线性函数。其次，我们补充了Ding et al.(2008)证明判定给定系统是否为box-TDI是共np完全的结果:我们证明了识别多面体是否为box-TDI是共np完全的。为了得到这些复杂性结果，我们通过刻画图关联矩阵的全等模性，展示了一类新的全等模矩阵——全等模矩阵的推广。
期刊介绍：
Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.
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