7 research outputs found
Parameterized Complexity of Streaming Diameter and Connectivity Problems
We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size k allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is O(logn) for any fixed k. Underlying these algorithms is a method to execute a breadth-first search in O(k) passes and O(klogn) bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where Ω(n/p) bits of memory is needed for any p-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph H, for most H. For some cases, we can also show one-pass, Ω(nlogn) bits of memory lower bounds. We also prove a much stronger Ω(n2/p) lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size k. This yields a kernel of 2k vertices (with O(k2) edges) produced as a stream in poly(k) passes and only O(klogn) bits of memory
The Parameterised Complexity of Integer Multicommodity Flow
The Integer Multicommodity Flow problem has been studied extensively in the
literature. However, from a parameterised perspective, mostly special cases,
such as the Disjoint Paths problem, have been considered. Therefore, we
investigate the parameterised complexity of the general Integer Multicommodity
Flow problem. We show that the decision version of this problem on directed
graphs for a constant number of commodities, when the capacities are given in
unary, is XNLP-complete with pathwidth as parameter and XALP-complete with
treewidth as parameter. When the capacities are given in binary, the problem is
NP-complete even for graphs of pathwidth at most 13. We give related results
for undirected graphs. These results imply that the problem is unlikely to be
fixed-parameter tractable by these parameters.
In contrast, we show that the problem does become fixed-parameter tractable
when weighted tree partition width (a variant of tree partition width for edge
weighted graphs) is used as parameter
Parameterized Complexity of Streaming Diameter and Connectivity Problems
We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size k allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is O(log n) for any fixed k. Underlying these algorithms is a method to execute a breadth-first search in O(k) passes and O(klog n) bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where Ω(n/p) bits of memory is needed for any p-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph H, for most H. For some cases, we can also show one-pass, Ω(nlog n) bits of memory lower bounds. We also prove a much stronger Ω(n2/p) lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size k. This yields a kernel of 2k vertices (with O(k2) edges) produced as a stream in poly(k) passes and only O(klog n) bits of memory
Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem
We study Steiner Forest on -subgraph-free graphs, that is, graphs that do
not contain some fixed graph as a (not necessarily induced) subgraph. We
are motivated by a recent framework that completely characterizes the
complexity of many problems on -subgraph-free graphs. However, in contrast
to e.g. the related Steiner Tree problem, Steiner Forest falls outside this
framework. Hence, the complexity of Steiner Forest on -subgraph-free graphs
remained tantalizingly open. In this paper, we make significant progress
towards determining the complexity of Steiner Forest on -subgraph-free
graphs. Our main results are four novel polynomial-time algorithms for
different excluded graphs that are central to further understand its
complexity. Along the way, we study the complexity of Steiner Forest for graphs
with a small -deletion set, that is, a small set of vertices such that
each component of has size at most . Using this parameter, we give two
noteworthy algorithms that we later employ as subroutines. First, we prove
Steiner Forest is FPT parameterized by when (i.e. the vertex cover
number). Second, we prove Steiner Forest is polynomial-time solvable for graphs
with a 2-deletion set of size at most 2. The latter result is tight, as the
problem is NP-complete for graphs with a 3-deletion set of size 2
Having Fun in Learning Formal Specifications
There are many benefits in providing formal specifications for our software.
However, teaching students to do this is not always easy as courses on formal
methods are often experienced as dry by students. This paper presents a game
called FormalZ that teachers can use to introduce some variation in their
class. Students can have some fun in playing the game and, while doing so, also
learn the basics of writing formal specifications in the form of pre- and
post-conditions. Unlike existing software engineering themed education games
such as Pex and Code Defenders, FormalZ takes the deep gamification approach
where playing gets a more central role in order to generate more engagement.
This short paper presents our work in progress: the first implementation of
FormalZ along with the result of a preliminary users' evaluation. This
implementation is functionally complete and tested, but the polishing of its
user interface is still future work
Parameterized Complexity of Streaming Diameter and Connectivity Problems
We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size k allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is O(log n) for any fixed k. Underlying these algorithms is a method to execute a breadth-first search in O(k) passes and O(klog n) bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where Ω(n/p) bits of memory is needed for any p-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph H, for most H. For some cases, we can also show one-pass, Ω(nlog n) bits of memory lower bounds. We also prove a much stronger Ω(n2/p) lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size k. This yields a kernel of 2k vertices (with O(k2) edges) produced as a stream in poly(k) passes and only O(klog n) bits of memory