311 research outputs found
Parameterized Algorithms on Perfect Graphs for deletion to -graphs
For fixed integers , a graph is called an {\em
-graph} if the vertex set can be partitioned into
independent sets and cliques. The class of graphs
generalizes -colourable graphs (when and hence not surprisingly,
determining whether a given graph is an -graph is \NP-hard even when
or in general graphs.
When and are part of the input, then the recognition problem is
NP-hard even if the input graph is a perfect graph (where the {\sc Chromatic
Number} problem is solvable in polynomial time). It is also known to be
fixed-parameter tractable (FPT) on perfect graphs when parameterized by and
. I.e. there is an f(r+\ell) \cdot n^{\Oh(1)} algorithm on perfect
graphs on vertices where is some (exponential) function of and
.
In this paper, we consider the parameterized complexity of the following
problem, which we call {\sc Vertex Partization}. Given a perfect graph and
positive integers decide whether there exists a set of size at most such that the deletion of from results in an
-graph. We obtain the following results: \begin{enumerate} \item {\sc
Vertex Partization} on perfect graphs is FPT when parameterized by .
\item The problem does not admit any polynomial sized kernel when parameterized
by . In other words, in polynomial time, the input graph can not be
compressed to an equivalent instance of size polynomial in . In fact,
our result holds even when .
\item When are universal constants, then {\sc Vertex Partization} on
perfect graphs, parameterized by , has a polynomial sized kernel.
\end{enumerate
A practical fpt algorithm for Flow Decomposition and transcript assembly
The Flow Decomposition problem, which asks for the smallest set of weighted
paths that "covers" a flow on a DAG, has recently been used as an important
computational step in transcript assembly. We prove the problem is in FPT when
parameterized by the number of paths by giving a practical linear fpt
algorithm. Further, we implement and engineer a Flow Decomposition solver based
on this algorithm, and evaluate its performance on RNA-sequence data.
Crucially, our solver finds exact solutions while achieving runtimes
competitive with a state-of-the-art heuristic. Finally, we contextualize our
design choices with two hardness results related to preprocessing and weight
recovery. Specifically, -Flow Decomposition does not admit polynomial
kernels under standard complexity assumptions, and the related problem of
assigning (known) weights to a given set of paths is NP-hard.Comment: Introduces software package Toboggan: Version 1.0.
http://dx.doi.org/10.5281/zenodo.82163
Parameterized Algorithms on Perfect Graphs for Deletion to (r,l)-Graphs
For fixed integers r,l >= 0, a graph G is called an (r,l)-graph if the vertex set V(G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r+l. The class of (r,l) graphs generalizes r-colourable graphs (when l=0) and hence not surprisingly, determining whether a given graph is an (r,l)-graph is NP-hard even when r >= 3 or l >= 3 in general graphs.
When r and ell are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and l. I.e. there is an f(r+l) n^O(1) algorithm on perfect graphs on n vertices where f is a function of r and l. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and l.
In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r,l,k decide whether there exists a set S subset or equal to V(G) of size at most k such that the deletion of S from G results in an (r,l)-graph. This problem generalizes well studied problems such as Vertex Cover (when r=1 and l=0), Odd Cycle Transversal (when r=2, l=0) and Split Vertex Deletion (when r=1=l).
1. Vertex Partization on perfect graphs is FPT when parameterized by k+r+l.
2. The problem, when parameterized by k+r+l, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k+r+l. In fact, our result holds even when k=0.
3. When r,ell are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel
Human Performance on Hard Non-Euclidean Graph Problems: Vertex Cover
Recent studies on a computationally hard visual optimization problem, the Traveling Salesperson Problem (TSP), indicate that humans are capable of finding close to optimal solutions in near-linear time. The current study is a preliminary step in investigating human performance on another hard problem, the Minimum Vertex Cover Problem, in which solvers attempt to find a smallest set of vertices that ensures that every edge in an undirected graph is incident with at least one of the selected vertices. We identify appropriate measures of performance, examine features of problem instances that impact performance, and describe strategies typically employed by participants to solve instances of the Vertex Cover problem
Evaluating Datalog via Tree Automata and Cycluits
We investigate parameterizations of both database instances and queries that
make query evaluation fixed-parameter tractable in combined complexity. We show
that clique-frontier-guarded Datalog with stratified negation (CFG-Datalog)
enjoys bilinear-time evaluation on structures of bounded treewidth for programs
of bounded rule size. Such programs capture in particular conjunctive queries
with simplicial decompositions of bounded width, guarded negation fragment
queries of bounded CQ-rank, or two-way regular path queries. Our result is
shown by translating to alternating two-way automata, whose semantics is
defined via cyclic provenance circuits (cycluits) that can be tractably
evaluated.Comment: 56 pages, 63 references. Journal version of "Combined Tractability of
Query Evaluation via Tree Automata and Cycluits (Extended Version)" at
arXiv:1612.04203. Up to the stylesheet, page/environment numbering, and
possible minor publisher-induced changes, this is the exact content of the
journal paper that will appear in Theory of Computing Systems. Update wrt
version 1: latest reviewer feedbac
Grid Recognition: Classical and Parameterized Computational Perspectives
Grid graphs, and, more generally, grid graphs, form one of the
most basic classes of geometric graphs. Over the past few decades, a large body
of works studied the (in)tractability of various computational problems on grid
graphs, which often yield substantially faster algorithms than general graphs.
Unfortunately, the recognition of a grid graph is particularly hard -- it was
shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this
paper, we provide several positive results in this regard in the framework of
parameterized complexity (additionally, we present new and complementary
hardness results). Specifically, our contribution is threefold. First, we show
that the problem is fixed-parameter tractable (FPT) parameterized by where is the maximum size of a connected component of
. This also implies that the problem is FPT parameterized by
where is the treedepth of (to be compared with the hardness
for pathwidth 2 where ). Further, we derive as a corollary that strip
packing is FPT with respect to the height of the strip plus the maximum of the
dimensions of the packed rectangles, which was previously only known to be in
XP. Second, we present a new parameterization, denoted , relating graph
distance to geometric distance, which may be of independent interest. We show
that the problem is para-NP-hard parameterized by , but FPT parameterized
by on trees, as well as FPT parameterized by . Third, we show that
the recognition of grid graphs is NP-hard on graphs of pathwidth 2
where . Moreover, when and are unrestricted, we show that the
problem is NP-hard on trees of pathwidth 2, but trivially solvable in
polynomial time on graphs of pathwidth 1
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