110,062 research outputs found
Tropical complementarity problems and Nash equilibria
Linear complementarity programming is a generalization of linear programming
which encompasses the computation of Nash equilibria for bimatrix games. While
the latter problem is PPAD-complete, we show that the tropical analogue of the
complementarity problem associated with Nash equilibria can be solved in
polynomial time. Moreover, we prove that the Lemke--Howson algorithm carries
over the tropical setting and performs a linear number of pivots in the worst
case. A consequence of this result is a new class of (classical) bimatrix games
for which Nash equilibria computation can be done in polynomial time
Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures
The min-rank of a graph was introduced by Haemers (1978) to bound the Shannon
capacity of a graph. This parameter of a graph has recently gained much more
attention from the research community after the work of Bar-Yossef et al.
(2006). In their paper, it was shown that the min-rank of a graph G
characterizes the optimal scalar linear solution of an instance of the Index
Coding with Side Information (ICSI) problem described by the graph G. It was
shown by Peeters (1996) that computing the min-rank of a general graph is an
NP-hard problem. There are very few known families of graphs whose min-ranks
can be found in polynomial time. In this work, we introduce a new family of
graphs with efficiently computed min-ranks. Specifically, we establish a
polynomial time dynamic programming algorithm to compute the min-ranks of
graphs having simple tree structures. Intuitively, such graphs are obtained by
gluing together, in a tree-like structure, any set of graphs for which the
min-ranks can be determined in polynomial time. A polynomial time algorithm to
recognize such graphs is also proposed.Comment: Accepted by Algorithmica, 30 page
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
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