3 research outputs found
Symmetric Submodular Function Minimization Under Hereditary Family Constraints
We present an efficient algorithm to find non-empty minimizers of a symmetric
submodular function over any family of sets closed under inclusion. This for
example includes families defined by a cardinality constraint, a knapsack
constraint, a matroid independence constraint, or any combination of such
constraints. Our algorithm make oracle calls to the submodular
function where is the cardinality of the ground set. In contrast, the
problem of minimizing a general submodular function under a cardinality
constraint is known to be inapproximable within (Svitkina
and Fleischer [2008]).
The algorithm is similar to an algorithm of Nagamochi and Ibaraki [1998] to
find all nontrivial inclusionwise minimal minimizers of a symmetric submodular
function over a set of cardinality using oracle calls. Their
procedure in turn is based on Queyranne's algorithm [1998] to minimize a
symmetric submodularComment: 13 pages, Submitted to SODA 201
Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions
We investigate three related and important problems connected to machine
learning: approximating a submodular function everywhere, learning a submodular
function (in a PAC-like setting [53]), and constrained minimization of
submodular functions. We show that the complexity of all three problems depends
on the 'curvature' of the submodular function, and provide lower and upper
bounds that refine and improve previous results [3, 16, 18, 52]. Our proof
techniques are fairly generic. We either use a black-box transformation of the
function (for approximation and learning), or a transformation of algorithms to
use an appropriate surrogate function (for minimization). Curiously, curvature
has been known to influence approximations for submodular maximization [7, 55],
but its effect on minimization, approximation and learning has hitherto been
open. We complete this picture, and also support our theoretical claims by
empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201
Density Functions subject to a Co-Matroid Constraint
In this paper we consider the problem of finding the {\em densest} subset
subject to {\em co-matroid constraints}. We are given a {\em monotone
supermodular} set function defined over a universe , and the density of
a subset is defined to be f(S)/\crd{S}. This generalizes the concept of
graph density. Co-matroid constraints are the following: given matroid \calM
a set is feasible, iff the complement of is {\em independent} in the
matroid. Under such constraints, the problem becomes \np-hard. The specific
case of graph density has been considered in literature under specific
co-matroid constraints, for example, the cardinality matroid and the partition
matroid. We show a 2-approximation for finding the densest subset subject to
co-matroid constraints. Thus, for instance, we improve the approximation
guarantees for the result for partition matroids in the literature