34 research outputs found
Streaming Kernelization
Kernelization is a formalization of preprocessing for combinatorially hard
problems. We modify the standard definition for kernelization, which allows any
polynomial-time algorithm for the preprocessing, by requiring instead that the
preprocessing runs in a streaming setting and uses
bits of memory on instances . We obtain
several results in this new setting, depending on the number of passes over the
input that such a streaming kernelization is allowed to make. Edge Dominating
Set turns out as an interesting example because it has no single-pass
kernelization but two passes over the input suffice to match the bounds of the
best standard kernelization
A shortcut to (sun)flowers: Kernels in logarithmic space or linear time
We investigate whether kernelization results can be obtained if we restrict
kernelization algorithms to run in logarithmic space. This restriction for
kernelization is motivated by the question of what results are attainable for
preprocessing via simple and/or local reduction rules. We find kernelizations
for d-Hitting Set(k), d-Set Packing(k), Edge Dominating Set(k) and a number of
hitting and packing problems in graphs, each running in logspace. Additionally,
we return to the question of linear-time kernelization. For d-Hitting Set(k) a
linear-time kernelization was given by van Bevern [Algorithmica (2014)]. We
give a simpler procedure and save a large constant factor in the size bound.
Furthermore, we show that we can obtain a linear-time kernel for d-Set
Packing(k) as well.Comment: 18 page
Towards Optimal and Expressive Kernelization for d-Hitting Set
d-Hitting Set is the NP-hard problem of selecting at most k vertices of a
hypergraph so that each hyperedge, all of which have cardinality at most d,
contains at least one selected vertex. The applications of d-Hitting Set are,
for example, fault diagnosis, automatic program verification, and the
noise-minimizing assignment of frequencies to radio transmitters.
We show a linear-time algorithm that transforms an instance of d-Hitting Set
into an equivalent instance comprising at most O(k^d) hyperedges and vertices.
In terms of parameterized complexity, this is a problem kernel. Our
kernelization algorithm is based on speeding up the well-known approach of
finding and shrinking sunflowers in hypergraphs, which yields problem kernels
with structural properties that we condense into the concept of expressive
kernelization.
We conduct experiments to show that our kernelization algorithm can kernelize
instances with more than 10^7 hyperedges in less than five minutes.
Finally, we show that the number of vertices in the problem kernel can be
further reduced to O(k^{d-1}) with additional O(k^{1.5 d}) processing time by
nontrivially combining the sunflower technique with d-Hitting Set problem
kernels due to Abu-Khzam and Moser.Comment: This version gives corrected experimental results, adds additional
figures, and more formally defines "expressive kernelization
Unit Interval Editing is Fixed-Parameter Tractable
Given a graph~ and integers , , and~, the unit interval
editing problem asks whether can be transformed into a unit interval graph
by at most vertex deletions, edge deletions, and edge
additions. We give an algorithm solving this problem in time , where , and denote respectively
the numbers of vertices and edges of . Therefore, it is fixed-parameter
tractable parameterized by the total number of allowed operations.
Our algorithm implies the fixed-parameter tractability of the unit interval
edge deletion problem, for which we also present a more efficient algorithm
running in time . Another result is an -time algorithm for the unit interval vertex deletion problem,
significantly improving the algorithm of van 't Hof and Villanger, which runs
in time .Comment: An extended abstract of this paper has appeared in the proceedings of
ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an
appendix is provided for a brief overview of related graph classe
Precedence-constrained scheduling problems parameterized by partial order width
Negatively answering a question posed by Mnich and Wiese (Math. Program.
154(1-2):533-562), we show that P2|prec,|, the
problem of finding a non-preemptive minimum-makespan schedule for
precedence-constrained jobs of lengths 1 and 2 on two parallel identical
machines, is W[2]-hard parameterized by the width of the partial order giving
the precedence constraints. To this end, we show that Shuffle Product, the
problem of deciding whether a given word can be obtained by interleaving the
letters of other given words, is W[2]-hard parameterized by , thus
additionally answering a question posed by Rizzi and Vialette (CSR 2013).
Finally, refining a geometric algorithm due to Servakh (Diskretn. Anal. Issled.
Oper. 7(1):75-82), we show that the more general Resource-Constrained Project
Scheduling problem is fixed-parameter tractable parameterized by the partial
order width combined with the maximum allowed difference between the earliest
possible and factual starting time of a job.Comment: 14 pages plus appendi
Network-based dissolution
We introduce a novel graph-theoretic dissolution model which applies to a number of redistribution scenarios such as gerrymandering or work economization. The central aspect of our model is to delete some vertices and redistribute their "load" to neighboring vertices in a completely balanced way. We investigate how the underlying graph structure, the pre-knowledge about which vertices to delete, and the relation between old and new "vertex load" influence the computational complexity of the underlying easy-to-describe graph problems, thereby identifying both tractable and intractable cases
On the complexity of computing the -restricted edge-connectivity of a graph
The \emph{-restricted edge-connectivity} of a graph , denoted by
, is defined as the minimum size of an edge set whose removal
leaves exactly two connected components each containing at least vertices.
This graph invariant, which can be seen as a generalization of a minimum
edge-cut, has been extensively studied from a combinatorial point of view.
However, very little is known about the complexity of computing .
Very recently, in the parameterized complexity community the notion of
\emph{good edge separation} of a graph has been defined, which happens to be
essentially the same as the -restricted edge-connectivity. Motivated by the
relevance of this invariant from both combinatorial and algorithmic points of
view, in this article we initiate a systematic study of its computational
complexity, with special emphasis on its parameterized complexity for several
choices of the parameters. We provide a number of NP-hardness and W[1]-hardness
results, as well as FPT-algorithms.Comment: 16 pages, 4 figure
Enumerating Isolated Cliques in Temporal Networks
Isolation is a concept from the world of clique enumeration that is mostly
used to model communities that do not have much contact to the outside world.
Herein, a clique is considered isolated if it has few edges connecting it to
the rest of the graph. Motivated by recent work on enumerating cliques in
temporal networks, we lift the isolation concept to this setting. We discover
that the addition of the time dimension leads to six distinct natural isolation
concepts. Our main contribution is the development of fixed-parameter
enumeration algorithms for five of these six clique types employing the
parameter "degree of isolation". On the empirical side, we implement and test
these algorithms on (temporal) social network data, obtaining encouraging
preliminary results