3,502 research outputs found
Maximizing Symmetric Submodular Functions
Symmetric submodular functions are an important family of submodular
functions capturing many interesting cases including cut functions of graphs
and hypergraphs. Maximization of such functions subject to various constraints
receives little attention by current research, unlike similar minimization
problems which have been widely studied. In this work, we identify a few
submodular maximization problems for which one can get a better approximation
for symmetric objectives than the state of the art approximation for general
submodular functions.
We first consider the problem of maximizing a non-negative symmetric
submodular function subject to a
down-monotone solvable polytope . For
this problem we describe an algorithm producing a fractional solution of value
at least , where is the optimal integral solution.
Our second result considers the problem for a
non-negative symmetric submodular function . For this problem, we give an approximation ratio that depends on
the value and is always at least . Our method can
also be applied to non-negative non-symmetric submodular functions, in which
case it produces approximation, improving over the best known
result for this problem. For unconstrained maximization of a non-negative
symmetric submodular function we describe a deterministic linear-time
-approximation algorithm. Finally, we give a -approximation algorithm for Submodular Welfare with players having
identical non-negative submodular utility functions, and show that this is the
best possible approximation ratio for the problem.Comment: 31 pages, an extended abstract appeared in ESA 201
Randomized Composable Core-sets for Distributed Submodular Maximization
An effective technique for solving optimization problems over massive data
sets is to partition the data into smaller pieces, solve the problem on each
piece and compute a representative solution from it, and finally obtain a
solution inside the union of the representative solutions for all pieces. This
technique can be captured via the concept of {\em composable core-sets}, and
has been recently applied to solve diversity maximization problems as well as
several clustering problems. However, for coverage and submodular maximization
problems, impossibility bounds are known for this technique \cite{IMMM14}. In
this paper, we focus on efficient construction of a randomized variant of
composable core-sets where the above idea is applied on a {\em random
clustering} of the data. We employ this technique for the coverage, monotone
and non-monotone submodular maximization problems. Our results significantly
improve upon the hardness results for non-randomized core-sets, and imply
improved results for submodular maximization in a distributed and streaming
settings.
In summary, we show that a simple greedy algorithm results in a
-approximate randomized composable core-set for submodular maximization
under a cardinality constraint. This is in contrast to a known impossibility result for (non-randomized) composable core-set. Our
result also extends to non-monotone submodular functions, and leads to the
first 2-round MapReduce-based constant-factor approximation algorithm with
total communication complexity for either monotone or non-monotone
functions. Finally, using an improved analysis technique and a new algorithm
, we present an improved -approximation algorithm
for monotone submodular maximization, which is in turn the first
MapReduce-based algorithm beating factor in a constant number of rounds
Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms
Constrained submodular maximization problems have long been studied, with
near-optimal results known under a variety of constraints when the submodular
function is monotone. The case of non-monotone submodular maximization is less
understood: the first approximation algorithms even for the unconstrainted
setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC
'09, APPROX '09) show how to approximately maximize non-monotone submodular
functions when the constraints are given by the intersection of p matroid
constraints; their algorithm is based on local-search procedures that consider
p-swaps, and hence the running time may be n^Omega(p), implying their algorithm
is polynomial-time only for constantly many matroids. In this paper, we give
algorithms that work for p-independence systems (which generalize constraints
given by the intersection of p matroids), where the running time is poly(n,p).
Our algorithm essentially reduces the non-monotone maximization problem to
multiple runs of the greedy algorithm previously used in the monotone case.
Our idea of using existing algorithms for monotone functions to solve the
non-monotone case also works for maximizing a submodular function with respect
to a knapsack constraint: we get a simple greedy-based constant-factor
approximation for this problem.
With these simpler algorithms, we are able to adapt our approach to
constrained non-monotone submodular maximization to the (online) secretary
setting, where elements arrive one at a time in random order, and the algorithm
must make irrevocable decisions about whether or not to select each element as
it arrives. We give constant approximations in this secretary setting when the
algorithm is constrained subject to a uniform matroid or a partition matroid,
and give an O(log k) approximation when it is constrained by a general matroid
of rank k.Comment: In the Proceedings of WINE 201
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