FPT-algorithms for The Shortest Lattice Vector and Integer Linear Programming Problems

In this paper, we present FPT-algorithms for special cases of the shortest vector problem (SVP) and the integer linear programming problem (ILP), when matrices included to the problems' formulations are near square. The main parameter is the maximal absolute value of rank minors of matrices included to the problem formulation. Additionally, we present FPT-algorithms with respect to the same main parameter for the problems, when the matrices have no singular rank sub-matrices.

[1]  Miklós Ajtai,et al.  Generating hard instances of lattice problems (extended abstract) , 1996, STOC '96.

[2]  D. Gribanov The Flatness Theorem for Some Class of Polytopes and Searching an Integer Point , 2014 .

[3]  Dmitry V. Gribanov,et al.  On integer programming with bounded determinants , 2015, Optim. Lett..

[4]  Ravi Kannan,et al.  Improved algorithms for integer programming and related lattice problems , 1983, STOC.

[5]  C. Siegel,et al.  Lectures on the Geometry of Numbers , 1989 .

[6]  Valery Shevchenko Qualitative Topics in Integer Linear Programming , 1996 .

[7]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[8]  George Labahn,et al.  Asymptotically fast computation of Hermite normal forms of integer matrices , 1996, ISSAC '96.

[9]  Johannes Blömer,et al.  Sampling Methods for Shortest Vectors, Closest Vectors and Successive Minima , 2007, ICALP.

[10]  Ravi Kumar,et al.  Sampling short lattice vectors and the closest lattice vector problem , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[11]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[12]  Arne Storjohann,et al.  Near optimal algorithms for computing Smith normal forms of integer matrices , 1996, ISSAC '96.

[13]  D. Malyshev,et al.  Classes of graphs critical for the edge list-ranking problem , 2014, Journal of Applied and Industrial Mathematics.

[14]  V. E. Alekseev,et al.  On easy and hard hereditary classes of graphs with respect to the independent set problem , 2003, Discret. Appl. Math..

[15]  Vadim V. Lozin,et al.  NP-hard graph problems and boundary classes of graphs , 2007, Theor. Comput. Sci..

[16]  Dmitriy S. Malyshev,et al.  The computational complexity of three graph problems for instances with bounded minors of constraint matrices , 2017, Discret. Appl. Math..

[17]  Michael J. Todd,et al.  Polynomial Algorithms for Linear Programming , 1988 .

[18]  Vadim V. Lozin,et al.  Boundary properties of graphs for algorithmic graph problems , 2011, Theor. Comput. Sci..

[19]  Manfred W. Padberg,et al.  The boolean quadric polytope: Some characteristics, facets and relatives , 1989, Math. Program..

[20]  U. Fincke,et al.  Improved methods for calculating vectors of short length in a lattice , 1985 .

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

[22]  Friedrich Eisenbrand,et al.  On Sub-determinants and the Diameter of Polyhedra , 2011, SoCG '12.

[23]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[24]  Ravi Kumar,et al.  A sieve algorithm for the shortest lattice vector problem , 2001, STOC '01.

[25]  Christos H. Papadimitriou,et al.  On the complexity of integer programming , 1981, JACM.

[26]  D. Malyshev A study of the boundary graph classes for colorability problems , 2013, Journal of Applied and Industrial Mathematics.

[27]  Rico Zenklusen,et al.  A strongly polynomial algorithm for bimodular integer linear programming , 2017, STOC.

[28]  Santosh S. Vempala,et al.  Enumerative Lattice Algorithms in any Norm Via M-ellipsoid Coverings , 2010, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[29]  Ravi Kannan,et al.  Minkowski's Convex Body Theorem and Integer Programming , 1987, Math. Oper. Res..

[30]  S. Vajda,et al.  Integer Programming and Network Flows , 1970 .

[31]  Vadim V. Lozin,et al.  Boundary classes of graphs for the dominating set problem , 2004, Discrete Mathematics.

[32]  R E Gomory,et al.  ON THE RELATION BETWEEN INTEGER AND NONINTEGER SOLUTIONS TO LINEAR PROGRAMS. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[33]  Sergey I. Veselov,et al.  Integer program with bimodular matrix , 2008, Discret. Optim..

[34]  Santosh S. Vempala,et al.  A note on non-degenerate integer programs with small sub-determinants , 2016, Oper. Res. Lett..

[35]  Santosh S. Vempala,et al.  Enumerative Algorithms for the Shortest and Closest Lattice Vector Problems in Any Norm via M-Ellipsoid Coverings , 2010, ArXiv.

[36]  Dmitry V. Gribanov,et al.  The width and integer optimization on simplices with bounded minors of the constraint matrices , 2016, Optimization Letters.

[37]  Friedrich Eisenbrand,et al.  Covering cubes and the closest vector problem , 2011, SoCG '11.

[38]  J. Cheon,et al.  Approximate Algorithms on Lattices with Small Determinant ( Extended Abstract ) , 2016 .

[39]  Friedrich Eisenbrand,et al.  On Sub-determinants and the Diameter of Polyhedra , 2014, Discret. Comput. Geom..

[40]  B. David Saunders,et al.  Computing the smith forms of integer matrices and solving related problems , 2005 .

[41]  D. Malyshev Critical elements in combinatorially closed families of graph classes , 2017 .

[42]  D. V. Sirotkin,et al.  Polynomial-time solvability of the independent set problem in a certain class of subcubic planar graphs , 2017 .

[43]  F. Thorne,et al.  Geometry of Numbers , 2017, Algebraic Number Theory.

[44]  Michael E. Pohst,et al.  A procedure for determining algebraic integers of given norm , 1983, EUROCAL.

[45]  C. A. Rogers,et al.  An Introduction to the Geometry of Numbers , 1959 .

[46]  David K. Smith Theory of Linear and Integer Programming , 1987 .

[47]  Panos M. Pardalos,et al.  Critical hereditary graph classes: a survey , 2016, Optim. Lett..

[48]  Daniele Micciancio,et al.  A Deterministic Single Exponential Time Algorithm for Most Lattice Problems based on Voronoi Cell Computations ( Extended Abstract ) , 2009 .

[49]  Santosh S. Vempala,et al.  Geometric random edge , 2014, Math. Program..

[50]  Carsten Moldenhauer,et al.  Solving the Stable Set Problem in Terms of the Odd Cycle Packing Number , 2014, FSTTCS.

[51]  Éva Tardos,et al.  A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs , 1986, Oper. Res..

[52]  Damien Stehlé,et al.  Algorithms for the Shortest and Closest Lattice Vector Problems , 2011, IWCC.

[53]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.