1,941 research outputs found
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
Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts
A skew-symmetric graph is a directed graph with an
involution on the set of vertices and arcs. In this paper, we
introduce a separation problem, -Skew-Symmetric Multicut, where we are given
a skew-symmetric graph , a family of of -sized subsets of
vertices and an integer . The objective is to decide if there is a set
of arcs such that every set in the family has a vertex
such that and are in different connected components of
. In this paper, we give an algorithm for
this problem which runs in time , where is the
number of arcs in the graph, the number of vertices and the length
of the family given in the input.
Using our algorithm, we show that Almost 2-SAT has an algorithm with running
time and we obtain algorithms for {\sc Odd Cycle Transversal}
and {\sc Edge Bipartization} which run in time and
respectively. This resolves an open problem posed by Reed,
Smith and Vetta [Operations Research Letters, 2003] and improves upon the
earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010].
We also show that Deletion q-Horn Backdoor Set Detection is a special case of
3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor
Set Detection which runs in time . This gives the first
fixed-parameter tractable algorithm for this problem answering a question posed
in a paper by a superset of the authors [STACS, 2013]. Using this result, we
get an algorithm for Satisfiability which runs in time where
is the size of the smallest q-Horn deletion backdoor set, with being
the length of the input formula
Continued Fractions with Multiple Limits
For integers , we study divergent continued fractions whose
numerators and denominators in each of the arithmetic progressions modulo
converge. Special cases give, among other things, an infinite sequence of
divergence theorems, the first of which is the classical Stern-Stolz theorem.
We give a theorem on a general class of Poincar{\'e} type recurrences which
shows that they tend to limits when the limits are taken in residue classes and
the roots of their characteristic polynomials are distinct roots of unity.
We also generalize a curious -continued fraction of Ramanujan's with three
limits to a continued fraction with distinct limit points, . The
limits are evaluated in terms of ratios of certain unusual series.
Finally, we show how to use Daniel Bernoulli's continued fraction in an
elementary way to create analytic continued fractions with limit points,
for any positive integer .Comment: 29 pages. Updated/new conten
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