213 research outputs found
On Kernelization and Approximation for the Vector Connectivity Problem
In the Vector Connectivity problem we are given an undirected graph
, a demand function , and an integer
. The question is whether there exists a set of at most vertices
such that every vertex has at least
vertex-disjoint paths to ; this abstractly captures questions about placing
servers or warehouses relative to demands. The problem is \NP-hard already for
instances with (Cicalese et al., arXiv '14), admits a log-factor
approximation (Boros et al., Networks '14), and is fixed-parameter tractable in
terms of~ (Lokshtanov, unpublished '14). We prove several results regarding
kernelization and approximation for Vector Connectivity and the variant Vector
-Connectivity where the upper bound on demands is a fixed constant. For
Vector -Connectivity we give a factor -approximation algorithm and
construct a vertex-linear kernelization, i.e., an efficient reduction to an
equivalent instance with vertices. For Vector Connectivity we have
a factor -approximation and we can show that it has no
kernelization to size polynomial in or even unless
, making optimal
for Vector -Connectivity. Finally, we provide a write-up for fixed-parameter
tractability of Vector Connectivity() by giving an alternative FPT algorithm
based on matroid intersection.Comment: Non-constructive Kernelization argument, improved technical details
of signature
Polynomial Kernels for Weighted Problems
Kernelization is a formalization of efficient preprocessing for NP-hard
problems using the framework of parameterized complexity. Among open problems
in kernelization it has been asked many times whether there are deterministic
polynomial kernelizations for Subset Sum and Knapsack when parameterized by the
number of items.
We answer both questions affirmatively by using an algorithm for compressing
numbers due to Frank and Tardos (Combinatorica 1987). This result had been
first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We
further illustrate its applicability by giving polynomial kernels also for
weighted versions of several well-studied parameterized problems. Furthermore,
when parameterized by the different item sizes we obtain a polynomial
kernelization for Subset Sum and an exponential kernelization for Knapsack.
Finally, we also obtain kernelization results for polynomial integer programs
Tree Deletion Set has a Polynomial Kernel (but no OPT^O(1) approximation)
In the Tree Deletion Set problem the input is a graph G together with an
integer k. The objective is to determine whether there exists a set S of at
most k vertices such that G-S is a tree. The problem is NP-complete and even
NP-hard to approximate within any factor of OPT^c for any constant c. In this
paper we give a O(k^4) size kernel for the Tree Deletion Set problem. To the
best of our knowledge our result is the first counterexample to the
"conventional wisdom" that kernelization algorithms automatically provide
approximation algorithms with approximation ratio close to the size of the
kernel. An appealing feature of our kernelization algorithm is a new algebraic
reduction rule that we use to handle the instances on which Tree Deletion Set
is hard to approximate
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
A Fast Parameterized Algorithm for Co-Path Set
The k-CO-PATH SET problem asks, given a graph G and a positive integer k,
whether one can delete k edges from G so that the remainder is a collection of
disjoint paths. We give a linear-time fpt algorithm with complexity
O^*(1.588^k) for deciding k-CO-PATH SET, significantly improving the previously
best known O^*(2.17^k) of Feng, Zhou, and Wang (2015). Our main tool is a new
O^*(4^{tw(G)}) algorithm for CO-PATH SET using the Cut&Count framework, where
tw(G) denotes treewidth. In general graphs, we combine this with a branching
algorithm which refines a 6k-kernel into reduced instances, which we prove have
bounded treewidth
The Graph Motif problem parameterized by the structure of the input graph
The Graph Motif problem was introduced in 2006 in the context of biological
networks. It consists of deciding whether or not a multiset of colors occurs in
a connected subgraph of a vertex-colored graph. Graph Motif has been mostly
analyzed from the standpoint of parameterized complexity. The main parameters
which came into consideration were the size of the multiset and the number of
colors. Though, in the many applications of Graph Motif, the input graph
originates from real-life and has structure. Motivated by this prosaic
observation, we systematically study its complexity relatively to graph
structural parameters. For a wide range of parameters, we give new or improved
FPT algorithms, or show that the problem remains intractable. For the FPT
cases, we also give some kernelization lower bounds as well as some ETH-based
lower bounds on the worst case running time. Interestingly, we establish that
Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which
is, to the best of our knowledge, the first problem to behave this way.Comment: 24 pages, accepted in DAM, conference version in IPEC 201
Fixed-parameter algorithms for minimum-cost edge-connectivity augmentation
We consider connectivity-augmentation problems in a setting where each potential new edge has a non-negative cost associated with it, and the task is to achieve a certain connectivity target with at most p new edges of minimum total cost. The main result is that the minimum cost augmentation of edge-connectivity from k − 1 to k with at most p new edges is fixed-parameter tractable parameterized by p and admits a polynomial kernel. We also prove the fixed-parameter tractability of increasing edge connectivity from 0 to 2 and increasing node connectivity from 1 to 2
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