201 research outputs found
Unified greedy approximability beyond submodular maximization
We consider classes of objective functions of cardinality constrained
maximization problems for which the greedy algorithm guarantees a constant
approximation. We propose the new class of --augmentable
functions and prove that it encompasses several important subclasses, such as
functions of bounded submodularity ratio, -augmentable functions, and
weighted rank functions of an independence system of bounded rank quotient - as
well as additional objective functions for which the greedy algorithm yields an
approximation. For this general class of functions, we show a tight bound of
on
the approximation ratio of the greedy algorithm that tightly interpolates
between bounds from the literature for functions of bounded submodularity ratio
and for -augmentable functions. In paritcular, as a by-product, we
close a gap left in [Math.Prog., 2020] by obtaining a tight lower bound for
-augmentable functions for all . For weighted rank
functions of independence systems, our tight bound becomes
, which recovers the known bound of for
independence systems of rank quotient at least
Wide partitions, Latin tableaux, and Rota's basis conjecture
Say that mu is a ``subpartition'' of an integer partition lambda if the
multiset of parts of mu is a submultiset of the parts of lambda, and define an
integer partition lambda to be ``wide'' if for every subpartition mu of lambda,
mu >= mu' in dominance order (where mu' denotes the conjugate or transpose of
mu). Then Brian Taylor and the first author have conjectured that an integer
partition lambda is wide if and only if there exists a tableau of shape lambda
such that (1) for all i, the entries in the ith row of the tableau are
precisely the integers from 1 to lambda_i inclusive, and (2) for all j, the
entries in the jth column of the tableau are pairwise distinct. This conjecture
was originally motivated by Rota's basis conjecture and, if true, yields a new
class of integer multiflow problems that satisfy max-flow min-cut and
integrality. Wide partitions also yield a class of graphs that satisfy
``delta-conjugacy'' (in the sense of Greene and Kleitman), and the above
conjecture implies that these graphs furthermore have a completely saturated
stable set partition. We present several partial results, but the conjecture
remains very much open.Comment: Joined forces with Goemans and Vondrak---several new partial results;
28 pages, submitted to Adv. Appl. Mat
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