1,487 research outputs found
Greed is Still Good: Maximizing Monotone Submodular+Supermodular Functions
We analyze the performance of the greedy algorithm, and also a discrete
semi-gradient based algorithm, for maximizing the sum of a suBmodular and
suPermodular (BP) function (both of which are non-negative monotone
non-decreasing) under two types of constraints, either a cardinality constraint
or matroid independence constraints. These problems occur naturally
in several real-world applications in data science, machine learning, and
artificial intelligence. The problems are ordinarily inapproximable to any
factor (as we show). Using the curvature of the submodular term, and
introducing for the supermodular term (a natural dual curvature for
supermodular functions), however, both of which are computable in linear time,
we show that BP maximization can be efficiently approximated by both the greedy
and the semi-gradient based algorithm. The algorithms yield multiplicative
guarantees of and
for the two types of constraints
respectively. For pure monotone supermodular constrained maximization, these
yield and for the two types of constraints
respectively. We also analyze the hardness of BP maximization and show that our
guarantees match hardness by a constant factor and by respectively.
Computational experiments are also provided supporting our analysis
Non-submodular Function Maximization subject to a Matroid Constraint, with Applications
The standard greedy algorithm has been recently shown to enjoy approximation
guarantees for constrained non-submodular nondecreasing set function
maximization. While these recent results allow to better characterize the
empirical success of the greedy algorithm, they are only applicable to simple
cardinality constraints. In this paper, we study the problem of maximizing a
non-submodular nondecreasing set function subject to a general matroid
constraint. We first show that the standard greedy algorithm offers an
approximation factor of , where
is the submodularity ratio of the function and is the rank of the
matroid. Then, we show that the same greedy algorithm offers a constant
approximation factor of , where is the
generalized curvature of the function. In addition, we demonstrate that these
approximation guarantees are applicable to several real-world applications in
which the submodularity ratio and the generalized curvature can be bounded.
Finally, we show that our greedy algorithm does achieve a competitive
performance in practice using a variety of experiments on synthetic and
real-world data.Comment: Added missing citations and changed strong submodularity ratio to
generalized curvatur
Submodular Optimization Problems and Greedy Strategies: A Survey
The greedy strategy is an approximation algorithm to solve optimization
problems arising in decision making with multiple actions. How good is the
greedy strategy compared to the optimal solution? In this survey, we mainly
consider two classes of optimization problems where the objective function is
submodular. The first is set submodular optimization, which is to choose a set
of actions to optimize a set submodular objective function, and the second is
string submodular optimization, which is to choose an ordered set of actions to
optimize a string submodular function. Our emphasis here is on performance
bounds for the greedy strategy in submodular optimization problems.
Specifically, we review performance bounds for the greedy strategy, more
general and improved bounds in terms of curvature, performance bounds for the
batched greedy strategy, and performance bounds for Nash equilibria
Polyhedral aspects of Submodularity, Convexity and Concavity
Seminal work by Edmonds and Lovasz shows the strong connection between
submodularity and convexity. Submodular functions have tight modular lower
bounds, and subdifferentials in a manner akin to convex functions. They also
admit poly-time algorithms for minimization and satisfy the Fenchel duality
theorem and the Discrete Seperation Theorem, both of which are fundamental
characteristics of convex functions. Submodular functions also show signs
similar to concavity. Submodular maximization, though NP hard, admits constant
factor approximation guarantees. Concave functions composed with modular
functions are submodular, and they also satisfy diminishing returns property.
This manuscript provides a more complete picture on the relationship between
submodularity with convexity and concavity, by extending many of the results
connecting submodularity with convexity to the concave aspects of
submodularity. We first show the existence of superdifferentials, and
efficiently computable tight modular upper bounds of a submodular function.
While we show that it is hard to characterize this polyhedron, we obtain inner
and outer bounds on the superdifferential along with certain specific and
useful supergradients. We then investigate forms of concave extensions of
submodular functions and show interesting relationships to submodular
maximization. We next show connections between optimality conditions over the
superdifferentials and submodular maximization, and show how forms of
approximate optimality conditions translate into approximation factors for
maximization. We end this paper by studying versions of the discrete seperation
theorem and the Fenchel duality theorem when seen from the concave point of
view. In every case, we relate our results to the existing results from the
convex point of view, thereby improving the analysis of the relationship
between submodularity, convexity, and concavity.Comment: 38 pages, 10 figure
Dependent Randomized Rounding for Matroid Polytopes and Applications
Motivated by several applications, we consider the problem of randomly
rounding a fractional solution in a matroid (base) polytope to an integral one.
We consider the pipage rounding technique and also present a new technique,
randomized swap rounding. Our main technical results are concentration bounds
for functions of random variables arising from these rounding techniques. We
prove Chernoff-type concentration bounds for linear functions of random
variables arising from both techniques, and also a lower-tail exponential bound
for monotone submodular functions of variables arising from randomized swap
rounding.
The following are examples of our applications: (1) We give a
(1-1/e-epsilon)-approximation algorithm for the problem of maximizing a
monotone submodular function subject to 1 matroid and k linear constraints, for
any constant k and epsilon>0. (2) We present a result on minimax packing
problems that involve a matroid base constraint. We give an O(log m / log log
m)-approximation for the general problem Min {lambda: x \in {0,1}^N, x \in
B(M), Ax <= lambda b}, where m is the number of packing constraints. (3) We
generalize the continuous greedy algorithm to problems involving multiple
submodular functions, and use it to find a (1-1/e-epsilon)-approximate pareto
set for the problem of maximizing a constant number of monotone submodular
functions subject to a matroid constraint. An example is the Submodular Welfare
Problem where we are looking for an approximate pareto set with respect to
individual players' utilities.Comment: Rico Zenklusen joined as an author; paper substantially expanded
compared to previous version; note a slight change in the titl
Performance Bounds for the -Batch Greedy Strategy in Optimization Problems with Curvature
The -batch greedy strategy is an approximate algorithm to solve
optimization problems where the optimal solution is hard to obtain. Starting
with the empty set, the -batch greedy strategy adds a batch of elements
to the current solution set with the largest gain in the objective function
while satisfying the constraints. In this paper, we bound the performance of
the -batch greedy strategy with respect to the optimal strategy by defining
the total curvature . We show that when the objective function is
nondecreasing and submodular, the -batch greedy strategy satisfies a
harmonic bound for a general matroid constraint and an
exponential bound for a
uniform matroid constraint, where divides the cardinality of the maximal
set in the general matroid, is an integer, and is the rank of the
uniform matroid. We also compare the performance of the -batch greedy
strategy with that of the -batch greedy strategy when divides .
Specifically, we prove that when the objective function is nondecreasing and
submodular, the -batch greedy strategy has better harmonic and exponential
bounds in terms of the total curvature. Finally, we illustrate our results by
considering a task-assignment problem.Comment: This paper has been accepted by 2016 AC
Complete enumeration of small realizable oriented matroids
Enumeration of all combinatorial types of point configurations and polytopes
is a fundamental problem in combinatorial geometry. Although many studies have
been done, most of them are for 2-dimensional and non-degenerate cases.
Finschi and Fukuda (2001) published the first database of oriented matroids
including degenerate (i.e. non-uniform) ones and of higher ranks. In this
paper, we investigate algorithmic ways to classify them in terms of
realizability, although the underlying decision problem of realizability
checking is NP-hard. As an application, we determine all possible combinatorial
types (including degenerate ones) of 3-dimensional configurations of 8 points,
2-dimensional configurations of 9 points and 5-dimensional configurations of 9
points. We could also determine all possible combinatorial types of 5-polytopes
with 9 vertices.Comment: 19 pages, 2 figure
Optimization with More than One Budget
A natural way to deal with multiple, partially conflicting objectives is
turning all the objectives but one into budget constraints. Some classical
polynomial-time optimization problems, such as spanning tree and forest,
shortest path, (perfect) matching, independent set (basis) in a matroid or in
the intersection of two matroids, become NP-hard even with one budget
constraint. Still, for most of these problems deterministic and randomized
polynomial-time approximation schemes are known. In the case of two or more
budgets, typically only multi-criteria approximation schemes are available,
which return slightly infeasible solutions. Not much is known however for the
case of strict budget constraints: filling this gap is the main goal of this
paper.
We show that shortest path, perfect matching, and spanning tree (and hence
matroid basis and matroid intersection basis) are inapproximable already with
two budget constraints. For the remaining problems, whose set of solutions
forms an independence system, we present deterministic and randomized
polynomial-time approximation schemes for a constant number k of budget
constraints. Our results are based on a variety of techniques:
1. We present a simple and powerful mechanism to transform multi-criteria
approximation schemes into pure approximation schemes.
2. We show that points in low dimensional faces of any matroid polytope are
almost integral, an interesting result on its own. This gives a deterministic
approximation scheme for k-budgeted matroid independent set.
3. We present a deterministic approximation scheme for 2-budgeted matching.
The backbone of this result is a purely topological property of curves in R^2
Submodular Function Maximization over Distributive and Integer Lattices
The problem of maximizing non-negative submodular functions has been studied
extensively in the last few years. However, most papers consider submodular set
functions. Recently, several advances have been made for the more general case
of submodular functions on the integer lattice. In this paper, we present a
deterministic -approximation for maximizing a submodular function
on a bounded integer lattice using a Double Greedy
framework. Moreover, we show that the analysis is tight and that other ideas
used for approximating set functions cannot easily be extended. In contrast to
set functions, submodularity on the integer lattice does not imply the
so-called diminishing returns property. Assuming this property, it was shown
that many results for set functions can also be obtained for the integer
lattice. In this paper, we consider a further generalization. Instead of the
integer lattice, we consider a distributive lattice as the function domain and
assume the diminishing returns (DR) property. On the one hand, we show that
some approximation algorithms match the set functions setting. In particular,
we can obtain a -approximation for unconstrained maximization, a
-approximation for monotone functions under a cardinality
constraint and a -approximation for a poset matroid constraint. On
the other hand, for a knapsack constraint, the problem becomes significantly
harder: even for monotone DR-submodular functions, we show that there is no
-approximation for every
under the assumption that cannot be solved in time
Combinatorial Optimization Problems with Interaction Costs: Complexity and Solvable Cases
We introduce and study the combinatorial optimization problem with
interaction costs (COPIC). COPIC is the problem of finding two combinatorial
structures, one from each of two given families, such that the sum of their
independent linear costs and the interaction costs between elements of the two
selected structures is minimized. COPIC generalizes the quadratic assignment
problem and many other well studied combinatorial optimization problems, and
hence covers many real world applications. We show how various topics from
different areas in the literature can be formulated as special cases of COPIC.
The main contributions of this paper are results on the computational
complexity and approximability of COPIC for different families of combinatorial
structures (e.g. spanning trees, paths, matroids), and special structures of
the interaction costs. More specifically, we analyze the complexity if the
interaction cost matrix is parameterized by its rank and if it is a diagonal
matrix. Also, we determine the structure of the intersection cost matrix, such
that COPIC is equivalent to independently solving linear optimization problems
for the two given families of combinatorial structures
- …