13 research outputs found
Posimodular Function Optimization
Given a posimodular function on a finite set , we
consider the problem of finding a nonempty subset of that minimizes
. Posimodular functions often arise in combinatorial optimization such as
undirected cut functions. In this paper, we show that any algorithm for the
problem requires oracle calls to , where
. It contrasts to the fact that the submodular function minimization,
which is another generalization of cut functions, is polynomially solvable.
When the range of a given posimodular function is restricted to be
for some nonnegative integer , we show that
oracle calls are necessary, while we propose an
-time algorithm for the problem. Here, denotes the
time needed to evaluate the function value for a given .
We also consider the problem of maximizing a given posimodular function. We
show that oracle calls are necessary for solving the problem,
and that the problem has time complexity when
is the range of for some constant .Comment: 18 page
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
Convex Analysis and Optimization with Submodular Functions: a Tutorial
Set-functions appear in many areas of computer science and applied
mathematics, such as machine learning, computer vision, operations research or
electrical networks. Among these set-functions, submodular functions play an
important role, similar to convex functions on vector spaces. In this tutorial,
the theory of submodular functions is presented, in a self-contained way, with
all results shown from first principles. A good knowledge of convex analysis is
assumed
Approximating submodular -partition via principal partition sequence
In submodular -partition, the input is a non-negative submodular function
defined over a finite ground set (given by an evaluation oracle) along
with a positive integer and the goal is to find a partition of the ground
set into non-empty parts in order to minimize
. Narayanan, Roy, and Patkar (Journal of Algorithms, 1996)
designed an algorithm for submodular -partition based on the principal
partition sequence and showed that the approximation factor of their algorithm
is for the special case of graph cut functions (subsequently rediscovered
by Ravi and Sinha (Journal of Operational Research, 2008)). In this work, we
study the approximation factor of their algorithm for three subfamilies of
submodular functions -- monotone, symmetric, and posimodular, and show the
following results:
1. The approximation factor of their algorithm for monotone submodular
-partition is . This result improves on the -factor achievable via
other algorithms. Moreover, our upper bound of matches the recently shown
lower bound under polynomial number of function evaluation queries (Santiago,
IWOCA 2021). Our upper bound of is also the first improvement beyond
for a certain graph partitioning problem that is a special case of monotone
submodular -partition.
2. The approximation factor of their algorithm for symmetric submodular
-partition is . This result generalizes their approximation factor
analysis beyond graph cut functions.
3. The approximation factor of their algorithm for posimodular submodular
-partition is .
We also construct an example to show that the approximation factor of their
algorithm for arbitrary submodular functions is .Comment: Accepted to APPROX'2
The complexity of Boolean surjective general-valued CSPs
Valued constraint satisfaction problems (VCSPs) are discrete optimisation
problems with a -valued objective function given as
a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on
labels from and an optimal assignment is required to use both
labels from . Examples include the classical global Min-Cut problem in
graphs and the Minimum Distance problem studied in coding theory.
We establish a dichotomy theorem and thus give a complete complexity
classification of Boolean surjective VCSPs with respect to exact solvability.
Our work generalises the dichotomy for -valued constraint
languages (corresponding to surjective decision CSPs) obtained by Creignou and
H\'ebrard. For the maximisation problem of -valued
surjective VCSPs, we also establish a dichotomy theorem with respect to
approximability.
Unlike in the case of Boolean surjective (decision) CSPs, there appears a
novel tractable class of languages that is trivial in the non-surjective
setting. This newly discovered tractable class has an interesting mathematical
structure related to downsets and upsets. Our main contribution is identifying
this class and proving that it lies on the borderline of tractability. A
crucial part of our proof is a polynomial-time algorithm for enumerating all
near-optimal solutions to a generalised Min-Cut problem, which might be of
independent interest.Comment: v5: small corrections and improved presentatio
Contributions on secretary problems, independent sets of rectangles and related problems
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D