80 research outputs found
Tight Kernel Bounds for Problems on Graphs with Small Degeneracy
In this paper we consider kernelization for problems on d-degenerate graphs,
i.e. graphs such that any subgraph contains a vertex of degree at most .
This graph class generalizes many classes of graphs for which effective
kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and
H-topological-minor free graphs. We show that for several natural problems on
d-degenerate graphs the best known kernelization upper bounds are essentially
tight.Comment: Full version of ESA 201
Lossy Kernelization
In this paper we propose a new framework for analyzing the performance of
preprocessing algorithms. Our framework builds on the notion of kernelization
from parameterized complexity. However, as opposed to the original notion of
kernelization, our definitions combine well with approximation algorithms and
heuristics. The key new definition is that of a polynomial size
-approximate kernel. Loosely speaking, a polynomial size
-approximate kernel is a polynomial time pre-processing algorithm that
takes as input an instance to a parameterized problem, and outputs
another instance to the same problem, such that . Additionally, for every , a -approximate solution
to the pre-processed instance can be turned in polynomial time into a
-approximate solution to the original instance .
Our main technical contribution are -approximate kernels of
polynomial size for three problems, namely Connected Vertex Cover, Disjoint
Cycle Packing and Disjoint Factors. These problems are known not to admit any
polynomial size kernels unless . Our approximate
kernels simultaneously beat both the lower bounds on the (normal) kernel size,
and the hardness of approximation lower bounds for all three problems. On the
negative side we prove that Longest Path parameterized by the length of the
path and Set Cover parameterized by the universe size do not admit even an
-approximate kernel of polynomial size, for any , unless
. In order to prove this lower bound we need to combine
in a non-trivial way the techniques used for showing kernelization lower bounds
with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and
approximate kernel lower bounds for Set Cover and Hitting Set parameterized
by universe siz
Essentially Tight Kernels For (Weakly) Closed Graphs
We study kernelization of classic hard graph problems when the input graphs
fulfill triadic closure properties. More precisely, we consider the recently
introduced parameters closure number and the weak closure number
[Fox et al., SICOMP 2020] in addition to the standard parameter solution size
. For Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching
we obtain the first kernels of size and , respectively, thus extending previous kernelization
results on degenerate graphs. The kernels are essentially tight, since these
problems are unlikely to admit kernels of size by previous
results on their kernelization complexity in degenerate graphs [Cygan et al.,
ACM TALG 2017]. In addition, we provide lower bounds for the kernelization of
Independent Set on graphs with constant closure number~ and kernels for
Dominating Set on weakly closed split graphs and weakly closed bipartite
graphs
Exponential Time Paradigms Through the Polynomial Time Lens
We propose a general approach to modelling algorithmic paradigms for the exact solution of NP-hard problems. Our approach is based on polynomial time reductions to succinct versions of problems solvable in polynomial time. We use this viewpoint to explore and compare the power of paradigms such as branching and dynamic programming, and to shed light on the true complexity of various problems.
As one instantiation, we model branching using the notion of witness compression, i.e., reducibility to the circuit satisfiability problem parameterized by the number of variables of the circuit. We show this is equivalent to the previously studied notion of `OPP-algorithms\u27, and provide a technique for proving conditional lower bounds for witness compressions via a constructive variant of AND-composition, which is a notion previously studied in theory of preprocessing. In the context of parameterized complexity we use this to show that problems such as Pathwidth and Treewidth and Independent Set parameterized by pathwidth do not have witness compression, assuming NP subseteq coNP/poly. Since these problems admit fast fixed parameter tractable algorithms via dynamic programming, this shows that dynamic programming can be stronger than branching, under a standard complexity hypothesis. Our approach has applications outside parameterized complexity as well: for example, we show if a polynomial time algorithm outputs a maximum independent set of a given planar graph on n vertices with probability exp(-n^{1-epsilon}) for some epsilon>0, then NP subseteq coNP/poly. This negative result dims the prospects for one very natural approach to sub-exponential time algorithms for problems on planar graphs.
As two other illustrations (more exploratory) of our approach, we model algorithms based on inclusion-exclusion or group algebras via the notion of "parity compression", and we model a subclass of dynamic programming algorithms with the notion of "disjunctive dynamic programming". These models give us a way to naturally classify various parameterized problems with FPT algorithms. In the case of the dynamic programming model, we show that Independent Set parameterized by pathwidth is complete for this model
An Approximate Kernel for Connected Feedback Vertex Set
The Feedback Vertex Set problem is a fundamental computational problem which has been the subject of intensive study in various domains of algorithmics. In this problem, one is given an undirected graph G and an integer k as input. The objective is to determine whether at most k vertices can be deleted from G such that the resulting graph is acyclic. The study of preprocessing algorithms for this problem has a long and rich history, culminating in the quadratic kernelization of Thomasse [SODA 2010].
However, it is known that when the solution is required to induce a connected subgraph (such a set is called a connected feedback vertex set), a polynomial kernelization is unlikely to exist and the problem is NP-hard to approximate below a factor of 2 (assuming the Unique Games Conjecture).
In this paper, we show that if one is interested in only preserving approximate solutions (even of quality arbitrarily close to the optimum), then there is a drastic improvement in our ability to preprocess this problem. Specifically, we prove that for every fixed 0<epsilon<1, graph G, and k in N, the following holds:
There is a polynomial time computable graph G\u27 of size k^O(1) such that for every c >= 1, any c-approximate connected feedback vertex set of G\u27 of size at most k is a c * (1+epsilon)-approximate connected feedback vertex set of G.
Our result adds to the set of approximate kernelization algorithms introduced by Lokshtanov et al. [STOC 2017]. As a consequence of our main result, we show that Connected Feedback Vertex Set can be approximated within a factor min{OPT^O(1),n^(1-delta)} in polynomial time for some delta>0
Point Line Cover: The Easy Kernel is Essentially Tight
The input to the NP-hard Point Line Cover problem (PLC) consists of a set
of points on the plane and a positive integer , and the question is
whether there exists a set of at most lines which pass through all points
in . A simple polynomial-time reduction reduces any input to one with at
most points. We show that this is essentially tight under standard
assumptions. More precisely, unless the polynomial hierarchy collapses to its
third level, there is no polynomial-time algorithm that reduces every instance
of PLC to an equivalent instance with points, for
any . This answers, in the negative, an open problem posed by
Lokshtanov (PhD Thesis, 2009).
Our proof uses the machinery for deriving lower bounds on the size of kernels
developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients:
We first show, by reduction from Vertex Cover, that PLC---conditionally---has
no kernel of total size bits. This does not directly imply
the claimed lower bound on the number of points, since the best known
polynomial-time encoding of a PLC instance with points requires
bits. To get around this we build on work of Goodman et al.
(STOC 1989) and devise an oracle communication protocol of cost
for PLC; its main building block is a bound of for the order
types of points that are not necessarily in general position, and an
explicit algorithm that enumerates all possible order types of n points. This
protocol and the lower bound on total size together yield the stated lower
bound on the number of points.
While a number of essentially tight polynomial lower bounds on total sizes of
kernels are known, our result is---to the best of our knowledge---the first to
show a nontrivial lower bound for structural/secondary parameters
On Sparsification for Computing Treewidth
We investigate whether an n-vertex instance (G,k) of Treewidth, asking
whether the graph G has treewidth at most k, can efficiently be made sparse
without changing its answer. By giving a special form of OR-cross-composition,
we prove that this is unlikely: if there is an e > 0 and a polynomial-time
algorithm that reduces n-vertex Treewidth instances to equivalent instances, of
an arbitrary problem, with O(n^{2-e}) bits, then NP is in coNP/poly and the
polynomial hierarchy collapses to its third level.
Our sparsification lower bound has implications for structural
parameterizations of Treewidth: parameterizations by measures that do not
exceed the vertex count, cannot have kernels with O(k^{2-e}) bits for any e >
0, unless NP is in coNP/poly. Motivated by the question of determining the
optimal kernel size for Treewidth parameterized by vertex cover, we improve the
O(k^3)-vertex kernel from Bodlaender et al. (STACS 2011) to a kernel with
O(k^2) vertices. Our improved kernel is based on a novel form of
treewidth-invariant set. We use the q-expansion lemma of Fomin et al. (STACS
2011) to find such sets efficiently in graphs whose vertex count is
superquadratic in their vertex cover number.Comment: 21 pages. Full version of the extended abstract presented at IPEC
201
- …