12 research outputs found
FPT-algorithms for some problems related to integer programming
In this paper, we present FPT-algorithms for special cases of the shortest
lattice vector, integer linear programming, and simplex width computation
problems, when matrices included in the problems' formulations are near square.
The parameter is the maximum absolute value of rank minors of the corresponding
matrices. Additionally, we present FPT-algorithms with respect to the same
parameter for the problems, when the matrices have no singular rank
sub-matrices.Comment: arXiv admin note: text overlap with arXiv:1710.00321 From author:
some minor corrections has been don
Submodular Minimization Under Congruency Constraints
Submodular function minimization (SFM) is a fundamental and efficiently
solvable problem class in combinatorial optimization with a multitude of
applications in various fields. Surprisingly, there is only very little known
about constraint types under which SFM remains efficiently solvable. The
arguably most relevant non-trivial constraint class for which polynomial SFM
algorithms are known are parity constraints, i.e., optimizing only over sets of
odd (or even) cardinality. Parity constraints capture classical combinatorial
optimization problems like the odd-cut problem, and they are a key tool in a
recent technique to efficiently solve integer programs with a constraint matrix
whose subdeterminants are bounded by two in absolute value.
We show that efficient SFM is possible even for a significantly larger class
than parity constraints, by introducing a new approach that combines techniques
from Combinatorial Optimization, Combinatorics, and Number Theory. In
particular, we can show that efficient SFM is possible over all sets (of any
given lattice) of cardinality r mod m, as long as m is a constant prime power.
This covers generalizations of the odd-cut problem with open complexity status,
and with relevance in the context of integer programming with higher
subdeterminants. To obtain our results, we establish a connection between the
correctness of a natural algorithm, and the inexistence of set systems with
specific combinatorial properties. We introduce a general technique to disprove
the existence of such set systems, which allows for obtaining extensions of our
results beyond the above-mentioned setting. These extensions settle two open
questions raised by Geelen and Kapadia [Combinatorica, 2017] in the context of
computing the girth and cogirth of certain types of binary matroids
Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting
Mixed-integer mathematical programs are among the most commonly used models
for a wide set of problems in Operations Research and related fields. However,
there is still very little known about what can be expressed by small
mixed-integer programs. In particular, prior to this work, it was open whether
some classical problems, like the minimum odd-cut problem, can be expressed by
a compact mixed-integer program with few (even constantly many) integer
variables. This is in stark contrast to linear formulations, where recent
breakthroughs in the field of extended formulations have shown that many
polytopes associated to classical combinatorial optimization problems do not
even admit approximate extended formulations of sub-exponential size.
We provide a general framework for lifting inapproximability results of
extended formulations to the setting of mixed-integer extended formulations,
and obtain almost tight lower bounds on the number of integer variables needed
to describe a variety of classical combinatorial optimization problems. Among
the implications we obtain, we show that any mixed-integer extended formulation
of sub-exponential size for the matching polytope, cut polytope, traveling
salesman polytope or dominant of the odd-cut polytope, needs many integer variables, where is the number of vertices of the
underlying graph. Conversely, the above-mentioned polyhedra admit
polynomial-size mixed-integer formulations with only or (for the traveling salesman polytope) many integer variables.
Our results build upon a new decomposition technique that, for any convex set
, allows for approximating any mixed-integer description of by the
intersection of with the union of a small number of affine subspaces.Comment: A conference version of this paper will be presented at SODA 201
An Algorithmic Theory of Integer Programming
We study the general integer programming problem where the number of
variables is a variable part of the input. We consider two natural
parameters of the constraint matrix : its numeric measure and its
sparsity measure . We show that integer programming can be solved in time
, where is some computable function of the
parameters and , and is the binary encoding length of the input. In
particular, integer programming is fixed-parameter tractable parameterized by
and , and is solvable in polynomial time for every fixed and .
Our results also extend to nonlinear separable convex objective functions.
Moreover, for linear objectives, we derive a strongly-polynomial algorithm,
that is, with running time , independent of the rest of
the input data.
We obtain these results by developing an algorithmic framework based on the
idea of iterative augmentation: starting from an initial feasible solution, we
show how to quickly find augmenting steps which rapidly converge to an optimum.
A central notion in this framework is the Graver basis of the matrix , which
constitutes a set of fundamental augmenting steps. The iterative augmentation
idea is then enhanced via the use of other techniques such as new and improved
bounds on the Graver basis, rapid solution of integer programs with bounded
variables, proximity theorems and a new proximity-scaling algorithm, the notion
of a reduced objective function, and others.
As a consequence of our work, we advance the state of the art of solving
block-structured integer programs. In particular, we develop near-linear time
algorithms for -fold, tree-fold, and -stage stochastic integer programs.
We also discuss some of the many applications of these classes.Comment: Revision 2: - strengthened dual treedepth lower bound - simplified
proximity-scaling algorith
Integer programs with bounded subdeterminants and two nonzeros per row
We give a strongly polynomial-time algorithm for integer linear programs
defined by integer coefficient matrices whose subdeterminants are bounded by a
constant and that contain at most two nonzero entries in each row. The core of
our approach is the first polynomial-time algorithm for the weighted stable set
problem on graphs that do not contain more than vertex-disjoint odd cycles,
where is any constant. Previously, polynomial-time algorithms were only
known for (bipartite graphs) and for .
We observe that integer linear programs defined by coefficient matrices with
bounded subdeterminants and two nonzeros per column can be also solved in
strongly polynomial-time, using a reduction to -matching