19,360 research outputs found
Induced Matching below Guarantees: Average Paves the Way for Fixed-Parameter Tractability
In this work, we study the Induced Matching problem: Given an undirected
graph and an integer , is there an induced matching of size at
least ? An edge subset is an induced matching in if is a
matching such that there is no edge between two distinct edges of . Our work
looks into the parameterized complexity of Induced Matching with respect to
"below guarantee" parameterizations. We consider the parameterization for an upper bound on the size of any induced matching. For instance,
any induced matching is of size at most where is the number of
vertices, which gives us a parameter . In fact, there is a
straightforward -time algorithm for Induced
Matching [Moser and Thilikos, J. Discrete Algorithms]. Motivated by this, we
ask: Is Induced Matching FPT for a parameter smaller than ? In
search for such parameters, we consider and ,
where is the maximum matching size and is the maximum
independent set size of . We find that Induced Matching is presumably not
FPT when parameterized by or . In contrast to
these intractability results, we find that taking the average of the two helps
-- our main result is a branching algorithm that solves Induced Matching in
time. Our algorithm makes use
of the Gallai-Edmonds decomposition to find a structure to branch on
On the Parameterized Complexity of the Acyclic Matching Problem
A matching is a set of edges in a graph with no common endpoint. A matching M
is called acyclic if the induced subgraph on the endpoints of the edges in M is
acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for
an acyclic matching of size k in G. The problem is known to be NP-complete. In
this paper, we investigate the complexity of the problem in different aspects.
First, we prove that the problem remains NP-complete for the class of planar
bipartite graphs of maximum degree three and arbitrarily large girth. Also, the
problem remains NP-complete for the class of planar line graphs with maximum
degree four. Moreover, we study the parameterized complexity of the problem. In
particular, we prove that the problem is W[1]-hard on bipartite graphs with
respect to the parameter k. On the other hand, the problem is fixed parameter
tractable with respect to the parameters tw and (k, c4), where tw and c4 are
the treewidth and the number of cycles with length 4 of the input graph. We
also prove that the problem is fixed parameter tractable with respect to the
parameter k for the line graphs and every proper minor-closed class of graphs
(including planar graphs)
Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses
We present a series of almost settled inapproximability results for three
fundamental problems. The first in our series is the subexponential-time
inapproximability of the maximum independent set problem, a question studied in
the area of parameterized complexity. The second is the hardness of
approximating the maximum induced matching problem on bounded-degree bipartite
graphs. The last in our series is the tight hardness of approximating the
k-hypergraph pricing problem, a fundamental problem arising from the area of
algorithmic game theory. In particular, assuming the Exponential Time
Hypothesis, our two main results are:
- For any r larger than some constant, any r-approximation algorithm for the
maximum independent set problem must run in at least
2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of
2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the
domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et
al., 2013)
- For any k larger than some constant, there is no polynomial time min
(k^{1-\epsilon}, n^{1/2-\epsilon})-approximation algorithm for the k-hypergraph
pricing problem, where n is the number of vertices in an input graph. This
almost matches the upper bound of min (O(k), \tilde O(\sqrt{n})) (by Balcan and
Blum, 2007 and an algorithm in this paper).
We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness
for polynomial-time algorithms, the k-hypergraph pricing problem admits
n^{\delta} approximation for any \delta >0 in quasi-polynomial time. This puts
this problem in a rare approximability class in which approximability
thresholds can be improved significantly by allowing algorithms to run in
quasi-polynomial time.Comment: The full version of FOCS 201
Exploiting -Closure in Kernelization Algorithms for Graph Problems
A graph is c-closed if every pair of vertices with at least c common
neighbors is adjacent. The c-closure of a graph G is the smallest number such
that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated
it in the context of clique enumeration. We show that c-closure can be applied
in kernelization algorithms for several classic graph problems. We show that
Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a
kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with
O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed
graphs have polynomially-bounded Ramsey numbers, as we show
Assessing the Computational Complexity of Multi-Layer Subgraph Detection
Multi-layer graphs consist of several graphs (layers) over the same vertex
set. They are motivated by real-world problems where entities (vertices) are
associated via multiple types of relationships (edges in different layers). We
chart the border of computational (in)tractability for the class of subgraph
detection problems on multi-layer graphs, including fundamental problems such
as maximum matching, finding certain clique relaxations (motivated by community
detection), or path problems. Mostly encountering hardness results, sometimes
even for two or three layers, we can also spot some islands of tractability
Fast Parallel Fixed-Parameter Algorithms via Color Coding
Fixed-parameter algorithms have been successfully applied to solve numerous
difficult problems within acceptable time bounds on large inputs. However, most
fixed-parameter algorithms are inherently \emph{sequential} and, thus, make no
use of the parallel hardware present in modern computers. We show that parallel
fixed-parameter algorithms do not only exist for numerous parameterized
problems from the literature -- including vertex cover, packing problems,
cluster editing, cutting vertices, finding embeddings, or finding matchings --
but that there are parallel algorithms working in \emph{constant} time or at
least in time \emph{depending only on the parameter} (and not on the size of
the input) for these problems. Phrased in terms of complexity classes, we place
numerous natural parameterized problems in parameterized versions of AC. On
a more technical level, we show how the \emph{color coding} method can be
implemented in constant time and apply it to embedding problems for graphs of
bounded tree-width or tree-depth and to model checking first-order formulas in
graphs of bounded degree
- …