0-1 Integer Linear Programming with a Linear Number of Constraints

We give an exact algorithm for the 0-1 Integer Linear Programming problem with a linear number of constraints that improves over exhaustive search by an exponential factor. Specifically, our algorithm runs in time $2^{(1-\text{poly}(1/c))n}$ where n is the number of variables and cn is the number of constraints. The key idea for the algorithm is a reduction to the Vector Domination problem and a new algorithm for that subproblem.

[1]  Russell Impagliazzo,et al.  Which problems have strongly exponential complexity? , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[2]  Timothy M. Chan.  All-Pairs Shortest Paths with Real Weights in O(n3/log n) Time , 2008, Algorithmica.

[3]  Ryan Williams,et al.  A new algorithm for optimal 2-constraint satisfaction and its implications , 2005, Theor. Comput. Sci..

[4]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[5]  Russell Impagliazzo,et al.  A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits , 2012, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[6]  Ryan Williams,et al.  New algorithms and lower bounds for circuits with linear threshold gates , 2014, STOC.

[7]  Jon Louis Bentley,et al.  Multidimensional divide-and-conquer , 1980, CACM.

[8]  Vassil Guliashki,et al.  LINEAR INTEGER PROGRAMMING METHODS AND APPROACHES-A SURVEY , 2011 .

[9]  Rainer Schuler,et al.  An algorithm for the satisfiability problem of formulas in conjunctive normal form , 2005, J. Algorithms.