Unimodular Integer Caratheodory is Fixed Parameter Tractable

The monoid problem, also called the integer Carath\'eodory problem, is to decide if a given integer vector is a finite nonnegative integer combination of a given set of integer vectors, and find such a decomposition if one exists. This problem arises, for instance, in the context of bin packing and in the context of three-way tables. It was very recently shown that for fixed dimension the problem can be solved in polynomial time which is doubly exponential in the dimension. Here we show that when the set of integer points is defined by a totally unimodular matrix, the problem in fixed dimension can be solved much faster, in quadratic time, and with the dimension as a parameter, the problem is fixed parameter tractable. As a consequence of this, we also conclude that the huge three-way table problem is fixed parameter tractable as well.

[1]  Jesús A. De Loera,et al.  All Linear and Integer Programs Are Slim 3-Way Transportation Programs , 2006, SIAM J. Optim..

[2]  David S. Johnson.  The NP-Completeness Column: An Ongoing Guide , 1986, J. Algorithms.

[3]  R. Gomory,et al.  A Linear Programming Approach to the Cutting-Stock Problem , 1961 .

[4]  Shmuel Onn.  Huge multiway table problems , 2014, Discret. Optim..

[5]  Nimrod Megiddo,et al.  Linear Programming in Linear Time When the Dimension Is Fixed , 1984, JACM.

[6]  Stephen E. Fienberg,et al.  Three centuries of categorical data analysis: Log-linear models and maximum likelihood estimation , 2007 .

[7]  L. Trotter,et al.  Integer Rounding and Polyhedral Decomposition for Totally Unimodular Systems , 1978 .

[8]  Alexander Martin,et al.  A counterexample to an integer analogue of Carathéodory's theorem , 1999 .

[9]  Raymond Hemmecke,et al.  A polynomial oracle-time algorithm for convex integer minimization , 2007, Math. Program..

[10]  Frits C. R. Spieksma,et al.  A Polynomial Algorithm for Multiprocessor Scheduling with Two Job Lengths , 2001, Math. Oper. Res..

[11]  Thomas Rothvoß,et al.  Polynomiality for Bin Packing with a Constant Number of Item Types , 2013, SODA.

[12]  Raymond Hemmecke,et al.  n-Fold integer programming in cubic time , 2011, Math. Program..

[13]  Friedrich Eisenbrand,et al.  Carathéodory bounds for integer cones , 2006, Oper. Res. Lett..

[14]  Shmuel Onn,et al.  Huge Unimodular n-Fold Programs , 2015, SIAM J. Discret. Math..

[15]  S. Onn.  Nonlinear Discrete Optimization , 2010 .

[16]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[17]  Alan J. Hoffman,et al.  Integral Boundary Points of Convex Polyhedra , 2010, 50 Years of Integer Programming.

[18]  Jesús A. De Loera,et al.  The Complexity of Three-Way Statistical Tables , 2002, SIAM J. Comput..

[19]  András Sebö,et al.  Hilbert Bases, Caratheodory's Theorem and Combinatorial Optimization , 1990, IPCO.