14,346 research outputs found
Random Weighting, Asymptotic Counting, and Inverse Isoperimetry
For a family X of k-subsets of the set 1,...,n, let |X| be the cardinality of
X and let Gamma(X,mu) be the expected maximum weight of a subset from X when
the weights of 1,...,n are chosen independently at random from a symmetric
probability distribution mu on R. We consider the inverse isoperimetric problem
of finding mu for which Gamma(X,mu) gives the best estimate of ln|X|. We prove
that the optimal choice of mu is the logistic distribution, in which case
Gamma(X,mu) provides an asymptotically tight estimate of ln|X| as k^{-1}ln|X|
grows. Since in many important cases Gamma(X,mu) can be easily computed, we
obtain computationally efficient approximation algorithms for a variety of
counting problems. Given mu, we describe families X of a given cardinality with
the minimum value of Gamma(X,mu), thus extending and sharpening various
isoperimetric inequalities in the Boolean cube.Comment: The revision contains a new isoperimetric theorem, some other
improvements and extensions; 29 pages, 1 figur
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
Meta-Kernelization with Structural Parameters
Meta-kernelization theorems are general results that provide polynomial
kernels for large classes of parameterized problems. The known
meta-kernelization theorems, in particular the results of Bodlaender et al.
(FOCS'09) and of Fomin et al. (FOCS'10), apply to optimization problems
parameterized by solution size. We present the first meta-kernelization
theorems that use a structural parameters of the input and not the solution
size. Let C be a graph class. We define the C-cover number of a graph to be a
the smallest number of modules the vertex set can be partitioned into, such
that each module induces a subgraph that belongs to the class C. We show that
each graph problem that can be expressed in Monadic Second Order (MSO) logic
has a polynomial kernel with a linear number of vertices when parameterized by
the C-cover number for any fixed class C of bounded rank-width (or
equivalently, of bounded clique-width, or bounded Boolean width). Many graph
problems such as Independent Dominating Set, c-Coloring, and c-Domatic Number
are covered by this meta-kernelization result. Our second result applies to MSO
expressible optimization problems, such as Minimum Vertex Cover, Minimum
Dominating Set, and Maximum Clique. We show that these problems admit a
polynomial annotated kernel with a linear number of vertices
Real root finding for equivariant semi-algebraic systems
Let be a real closed field. We consider basic semi-algebraic sets defined
by -variate equations/inequalities of symmetric polynomials and an
equivariant family of polynomials, all of them of degree bounded by .
Such a semi-algebraic set is invariant by the action of the symmetric group. We
show that such a set is either empty or it contains a point with at most
distinct coordinates. Combining this geometric result with efficient algorithms
for real root finding (based on the critical point method), one can decide the
emptiness of basic semi-algebraic sets defined by polynomials of degree
in time . This improves the state-of-the-art which is exponential
in . When the variables are quantified and the
coefficients of the input system depend on parameters , one
also demonstrates that the corresponding one-block quantifier elimination
problem can be solved in time
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
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