12,519 research outputs found
Bi-Criteria and Approximation Algorithms for Restricted Matchings
In this work we study approximation algorithms for the \textit{Bounded Color
Matching} problem (a.k.a. Restricted Matching problem) which is defined as
follows: given a graph in which each edge has a color and a profit
, we want to compute a maximum (cardinality or profit)
matching in which no more than edges of color are
present. This kind of problems, beside the theoretical interest on its own
right, emerges in multi-fiber optical networking systems, where we interpret
each unique wavelength that can travel through the fiber as a color class and
we would like to establish communication between pairs of systems. We study
approximation and bi-criteria algorithms for this problem which are based on
linear programming techniques and, in particular, on polyhedral
characterizations of the natural linear formulation of the problem. In our
setting, we allow violations of the bounds and we model our problem as a
bi-criteria problem: we have two objectives to optimize namely (a) to maximize
the profit (maximum matching) while (b) minimizing the violation of the color
bounds. We prove how we can "beat" the integrality gap of the natural linear
programming formulation of the problem by allowing only a slight violation of
the color bounds. In particular, our main result is \textit{constant}
approximation bounds for both criteria of the corresponding bi-criteria
optimization problem
A Deterministic Algorithm for the Vertex Connectivity Survivable Network Design Problem
In the vertex connectivity survivable network design problem we are given an
undirected graph G = (V,E) and connectivity requirement r(u,v) for each pair of
vertices u,v. We are also given a cost function on the set of edges. Our goal
is to find the minimum cost subset of edges such that for every pair (u,v) of
vertices we have r(u,v) vertex disjoint paths in the graph induced by the
chosen edges. Recently, Chuzhoy and Khanna presented a randomized algorithm
that achieves a factor of O(k^3 log n) for this problem where k is the maximum
connectivity requirement. In this paper we derandomize their algorithm to get a
deterministic O(k^3 log n) factor algorithm. Another problem of interest is the
single source version of the problem, where there is a special vertex s and all
non-zero connectivity requirements must involve s. We also give a deterministic
O(k^2 log n) algorithm for this problem
Generalised Pattern Matching Revisited
In the problem of
[STOC'94, Muthukrishnan and Palem], we are given a text of length over
an alphabet , a pattern of length over an alphabet
, and a matching relationship ,
and must return all substrings of that match (reporting) or the number
of mismatches between each substring of of length and (counting).
In this work, we improve over all previously known algorithms for this problem
for various parameters describing the input instance:
* being the maximum number of characters that match a fixed
character,
* being the number of pairs of matching characters,
* being the total number of disjoint intervals of characters
that match the characters of the pattern .
At the heart of our new deterministic upper bounds for and
lies a faster construction of superimposed codes, which solves
an open problem posed in [FOCS'97, Indyk] and can be of independent interest.
To conclude, we demonstrate first lower bounds for . We start by
showing that any deterministic or Monte Carlo algorithm for must
use time, and then proceed to show higher lower bounds
for combinatorial algorithms. These bounds show that our algorithms are almost
optimal, unless a radically new approach is developed
Decomposition-Based Method for Sparse Semidefinite Relaxations of Polynomial Optimization Problems
We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in [1, 2] that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decompositionbased method, which partitions the relaxations according to their sparsity pattern. The decomposition-based method that we propose is an extension to semidefinite programming of the Benders decomposition for linear programs [3] .Polynomial optimization, Semidefinite programming, Sparse SDP relaxations, Benders decomposition
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