2 research outputs found
Exponential lower bounds for history-based simplex pivot rules on abstract cubes
The behavior of the simplex algorithm is a widely studied subject. Specifically, the question of the existence of a polynomial pivot rule for the simplex algorithm is of major importance. Here, we give exponential lower bounds for three history-based pivot rules. Those rules decide their next step based on memory of the past steps. In particular, we study Zadeh's least entered rule, Johnson's least-recently basic rule and Cunningham's least-recently considered (or round-robin) rule. We give exponential lower bounds on Acyclic Unique Sink Orientations of the abstract cube, for all of these pivot rules. For Johnson's rule our bound is the first superpolynomial one in any context; for Zadeh's it is the first one for AUSO. Those two are our main results.ISSN:1868-896
Unique End of Potential Line
This paper studies the complexity of problems in PPAD PLS that have
unique solutions. Three well-known examples of such problems are the problem of
finding a fixpoint of a contraction map, finding the unique sink of a Unique
Sink Orientation (USO), and solving the P-matrix Linear Complementarity Problem
(P-LCP). Each of these are promise-problems, and when the promise holds, they
always possess unique solutions.
We define the complexity class UEOPL to capture problems of this type. We
first define a class that we call EOPL, which consists of all problems that can
be reduced to End-of-Potential-Line. This problem merges the canonical
PPAD-complete problem End-of-Line, with the canonical PLS-complete problem
Sink-of-Dag, and so EOPL captures problems that can be solved by a
line-following algorithm that also simultaneously decreases a potential
function.
Promise-UEOPL is a promise-subclass of EOPL in which the line in the
End-of-Potential-Line instance is guaranteed to be unique via a promise. We
turn this into a non-promise class UEOPL, by adding an extra solution type to
EOPL that captures any pair of points that are provably on two different lines.
We show that UEOPL EOPL CLS, and that all of our
motivating problems are contained in UEOPL: specifically USO, P-LCP, and
finding a fixpoint of a Piecewise-Linear Contraction under an -norm all
lie in UEOPL. Our results also imply that parity games, mean-payoff games,
discounted games, and simple-stochastic games lie in UEOPL.
All of our containment results are proved via a reduction to a problem that
we call One-Permutation Discrete Contraction (OPDC). This problem is motivated
by a discretized version of contraction, but it is also closely related to the
USO problem. We show that OPDC lies in UEOPL, and we are also able to show that
OPDC is UEOPL-complete.Comment: This paper substantially revises and extends the work described in
our previous preprint "End of Potential Line'' (arXiv:1804.03450). The
abstract has been shortened to meet the arXiv character limi