394 research outputs found
Hardness of submodular cost allocation : lattice matching and a simplex coloring conjecture
We consider the Minimum Submodular Cost Allocation (MSCA) problem. In this problem, we are given k submodular cost functions f1, ... , fk: 2V -> R+ and the goal is to partition V into k sets A1, ..., Ak so as to minimize the total cost sumi = 1,k fi(Ai). We show that MSCA is inapproximable within any multiplicative factor even in very restricted settings; prior to our work, only Set Cover hardness was known. In light of this negative result, we turn our attention to special cases of the problem. We consider the setting in which each function fi satisfies fi = gi + h, where each gi is monotone submodular and h is (possibly non-monotone) submodular. We give an O(k log |V|) approximation for this problem. We provide some evidence that a factor of k may be necessary, even in the special case of HyperLabel. In particular, we formulate a simplex-coloring conjecture that implies a Unique-Games-hardness of (k - 1 - epsilon) for k-uniform HyperLabel and label set [k]. We provide a proof of the simplex-coloring conjecture for k=3
Random Coordinate Descent Methods for Minimizing Decomposable Submodular Functions
Submodular function minimization is a fundamental optimization problem that
arises in several applications in machine learning and computer vision. The
problem is known to be solvable in polynomial time, but general purpose
algorithms have high running times and are unsuitable for large-scale problems.
Recent work have used convex optimization techniques to obtain very practical
algorithms for minimizing functions that are sums of ``simple" functions. In
this paper, we use random coordinate descent methods to obtain algorithms with
faster linear convergence rates and cheaper iteration costs. Compared to
alternating projection methods, our algorithms do not rely on full-dimensional
vector operations and they converge in significantly fewer iterations
Constrained Submodular Maximization: Beyond 1/e
In this work, we present a new algorithm for maximizing a non-monotone
submodular function subject to a general constraint. Our algorithm finds an
approximate fractional solution for maximizing the multilinear extension of the
function over a down-closed polytope. The approximation guarantee is 0.372 and
it is the first improvement over the 1/e approximation achieved by the unified
Continuous Greedy algorithm [Feldman et al., FOCS 2011]
Hardness of Submodular Cost Allocation: Lattice Matching and a Simplex Coloring Conjecture
We consider the Minimum Submodular Cost Allocation (MSCA) problem.
In this problem, we are given k submodular cost functions f_1, ... ,
f_k: 2^V -> R_+ and the goal is to partition V into k sets A_1, ...,
A_k so as to minimize the total cost sum_{i = 1}^k f_i(A_i). We show
that MSCA is inapproximable within any multiplicative factor even in
very restricted settings; prior to our work, only Set Cover hardness
was known. In light of this negative result, we turn our attention
to special cases of the problem. We consider the setting in which
each function f_i satisfies f_i = g_i + h, where each g_i is monotone
submodular and h is (possibly non-monotone) submodular. We give an
O(k log |V|) approximation for this problem. We provide some evidence
that a factor of k may be necessary, even in the special case of
HyperLabel. In particular, we formulate a simplex-coloring
conjecture that implies a Unique-Games-hardness of (k - 1 - epsilon)
for k-uniform HyperLabel and label set [k]. We provide a proof of the
simplex-coloring conjecture for k=3
Fast Clustering with Lower Bounds: No Customer too Far, No Shop too Small
We study the \LowerBoundedCenter (\lbc) problem, which is a clustering
problem that can be viewed as a variant of the \kCenter problem. In the \lbc
problem, we are given a set of points P in a metric space and a lower bound
\lambda, and the goal is to select a set C \subseteq P of centers and an
assignment that maps each point in P to a center of C such that each center of
C is assigned at least \lambda points. The price of an assignment is the
maximum distance between a point and the center it is assigned to, and the goal
is to find a set of centers and an assignment of minimum price. We give a
constant factor approximation algorithm for the \lbc problem that runs in O(n
\log n) time when the input points lie in the d-dimensional Euclidean space
R^d, where d is a constant. We also prove that this problem cannot be
approximated within a factor of 1.8-\epsilon unless P = \NP even if the input
points are points in the Euclidean plane R^2.Comment: 14 page
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