14,393 research outputs found
Parameterized Approximation Schemes using Graph Widths
Combining the techniques of approximation algorithms and parameterized
complexity has long been considered a promising research area, but relatively
few results are currently known. In this paper we study the parameterized
approximability of a number of problems which are known to be hard to solve
exactly when parameterized by treewidth or clique-width. Our main contribution
is to present a natural randomized rounding technique that extends well-known
ideas and can be used for both of these widths. Applying this very generic
technique we obtain approximation schemes for a number of problems, evading
both polynomial-time inapproximability and parameterized intractability bounds
Parameterized Streaming Algorithms for Vertex Cover
As graphs continue to grow in size, we seek ways to effectively process such
data at scale. The model of streaming graph processing, in which a compact
summary is maintained as each edge insertion/deletion is observed, is an
attractive one. However, few results are known for optimization problems over
such dynamic graph streams.
In this paper, we introduce a new approach to handling graph streams, by
instead seeking solutions for the parameterized versions of these problems
where we are given a parameter and the objective is to decide whether there
is a solution bounded by . By combining kernelization techniques with
randomized sketch structures, we obtain the first streaming algorithms for the
parameterized versions of the Vertex Cover problem. We consider the following
three models for a graph stream on nodes:
1. The insertion-only model where the edges can only be added.
2. The dynamic model where edges can be both inserted and deleted.
3. The \emph{promised} dynamic model where we are guaranteed that at each
timestamp there is a solution of size at most .
In each of these three models we are able to design parameterized streaming
algorithms for the Vertex Cover problem. We are also able to show matching
lower bound for the space complexity of our algorithms.
(Due to the arXiv limit of 1920 characters for abstract field, please see the
abstract in the paper for detailed description of our results)Comment: Fixed some typo
Parameterized Complexity Analysis of Randomized Search Heuristics
This chapter compiles a number of results that apply the theory of
parameterized algorithmics to the running-time analysis of randomized search
heuristics such as evolutionary algorithms. The parameterized approach
articulates the running time of algorithms solving combinatorial problems in
finer detail than traditional approaches from classical complexity theory. We
outline the main results and proof techniques for a collection of randomized
search heuristics tasked to solve NP-hard combinatorial optimization problems
such as finding a minimum vertex cover in a graph, finding a maximum leaf
spanning tree in a graph, and the traveling salesperson problem.Comment: This is a preliminary version of a chapter in the book "Theory of
Evolutionary Computation: Recent Developments in Discrete Optimization",
edited by Benjamin Doerr and Frank Neumann, published by Springe
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be
made bipartite by deleting at most of its vertices. In a breakthrough
result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a
\BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial
runtime of uniform degree for every fixed . It is known that this implies a
polynomial-time compression algorithm that turns OCT instances into equivalent
instances of size at most \BigOh(4^k), a so-called kernelization. Since then
the existence of a polynomial kernel for OCT, i.e., a kernelization with size
bounded polynomially in , has turned into one of the main open questions in
the study of kernelization.
This work provides the first (randomized) polynomial kernelization for OCT.
We introduce a novel kernelization approach based on matroid theory, where we
encode all relevant information about a problem instance into a matroid with a
representation of size polynomial in . For OCT, the matroid is built to
allow us to simulate the computation of the iterative compression step of the
algorithm of Reed, Smith, and Vetta, applied (for only one round) to an
approximate odd cycle transversal which it is aiming to shrink to size . The
process is randomized with one-sided error exponentially small in , where
the result can contain false positives but no false negatives, and the size
guarantee is cubic in the size of the approximate solution. Combined with an
\BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a
reduction of the instance to size \BigOh(k^{4.5}), implying a randomized
polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape
Parameterized Algorithms for Graph Partitioning Problems
We study a broad class of graph partitioning problems, where each problem is
specified by a graph , and parameters and . We seek a subset
of size , such that is at most
(or at least) , where are constants
defining the problem, and are the cardinalities of the edge sets
having both endpoints, and exactly one endpoint, in , respectively. This
class of fixed cardinality graph partitioning problems (FGPP) encompasses Max
-Cut, Min -Vertex Cover, -Densest Subgraph, and -Sparsest
Subgraph.
Our main result is an algorithm for any problem in
this class, where is the maximum degree in the input graph.
This resolves an open question posed by Bonnet et al. [IPEC 2013]. We obtain
faster algorithms for certain subclasses of FGPPs, parameterized by , or by
. In particular, we give an time algorithm for Max
-Cut, thus improving significantly the best known time
algorithm
Fast Witness Extraction Using a Decision Oracle
The gist of many (NP-)hard combinatorial problems is to decide whether a
universe of elements contains a witness consisting of elements that
match some prescribed pattern. For some of these problems there are known
advanced algebra-based FPT algorithms which solve the decision problem but do
not return the witness. We investigate techniques for turning such a
YES/NO-decision oracle into an algorithm for extracting a single witness, with
an objective to obtain practical scalability for large values of . By
relying on techniques from combinatorial group testing, we demonstrate that a
witness may be extracted with queries to either a deterministic or
a randomized set inclusion oracle with one-sided probability of error.
Furthermore, we demonstrate through implementation and experiments that the
algebra-based FPT algorithms are practical, in particular in the setting of the
-path problem. Also discussed are engineering issues such as optimizing
finite field arithmetic.Comment: Journal version, 16 pages. Extended abstract presented at ESA'1
Narrow sieves for parameterized paths and packings
We present randomized algorithms for some well-studied, hard combinatorial
problems: the k-path problem, the p-packing of q-sets problem, and the
q-dimensional p-matching problem. Our algorithms solve these problems with high
probability in time exponential only in the parameter (k, p, q) and using
polynomial space; the constant bases of the exponentials are significantly
smaller than in previous works. For example, for the k-path problem the
improvement is from 2 to 1.66. We also show how to detect if a d-regular graph
admits an edge coloring with colors in time within a polynomial factor of
O(2^{(d-1)n/2}).
Our techniques build upon and generalize some recently published ideas by I.
Koutis (ICALP 2009), R. Williams (IPL 2009), and A. Bj\"orklund (STACS 2010,
FOCS 2010)
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