43 research outputs found
A Duality Based 2-Approximation Algorithm for Maximum Agreement Forest
We give a 2-approximation algorithm for the Maximum Agreement Forest problem
on two rooted binary trees. This NP-hard problem has been studied extensively
in the past two decades, since it can be used to compute the Subtree
Prune-and-Regraft (SPR) distance between two phylogenetic trees. Our result
improves on the very recent 2.5-approximation algorithm due to Shi, Feng, You
and Wang (2015). Our algorithm is the first approximation algorithm for this
problem that uses LP duality in its analysis
Layers and Matroids for the Traveling Salesman's Paths
Gottschalk and Vygen proved that every solution of the subtour elimination
linear program for traveling salesman paths is a convex combination of more and
more restrictive "generalized Gao-trees". We give a short proof of this fact,
as a layered convex combination of bases of a sequence of increasingly
restrictive matroids. A strongly polynomial, combinatorial algorithm follows
for finding this convex combination, which is a new tool offering polyhedral
insight, already instrumental in recent results for the path TSP
A Duality Based 2-Approximation Algorithm for Maximum Agreement Forest
We give a 2-approximation algorithm for the Maximum Agreement Forest problem
on two rooted binary trees. This NP-hard problem has been studied extensively
in the past two decades, since it can be used to compute the rooted Subtree
Prune-and-Regraft (rSPR) distance between two phylogenetic trees. Our algorithm
is combinatorial and its running time is quadratic in the input size. To prove
the approximation guarantee, we construct a feasible dual solution for a novel
linear programming formulation. In addition, we show this linear program is
stronger than previously known formulations, and we give a compact formulation,
showing that it can be solved in polynomial tim
Split Scheduling with Uniform Setup Times
We study a scheduling problem in which jobs may be split into parts, where
the parts of a split job may be processed simultaneously on more than one
machine. Each part of a job requires a setup time, however, on the machine
where the job part is processed. During setup a machine cannot process or set
up any other job. We concentrate on the basic case in which setup times are
job-, machine-, and sequence-independent. Problems of this kind were
encountered when modelling practical problems in planning disaster relief
operations. Our main algorithmic result is a polynomial-time algorithm for
minimising total completion time on two parallel identical machines. We argue
why the same problem with three machines is not an easy extension of the
two-machine case, leaving the complexity of this case as a tantalising open
problem. We give a constant-factor approximation algorithm for the general case
with any number of machines and a polynomial-time approximation scheme for a
fixed number of machines. For the version with objective minimising weighted
total completion time we prove NP-hardness. Finally, we conclude with an
overview of the state of the art for other split scheduling problems with job-,
machine-, and sequence-independent setup times
A Duality Based 2-Approximation Algorithm for Maximum Agreement Forest
We give a 2-approximation algorithm for the Maximum Agreement Forest problem on two rooted binary trees. This NP-hard problem has been studied extensively in the past two decades, since it can be used to compute the rooted Subtree Prune-and-Regraft (rSPR) distance between two phylogenetic trees. Our algorithm is combinatorial and its running time is quadratic in the input size. To prove the approximation guarantee, we construct a feasible dual solution for a novel linear programming formulation. In addition, we show this linear program is stronger than previously known formulations, and we give a compact formulation, showing that it can be solved in polynomial tim
Split scheduling with uniform setup times
We study a scheduling problem in which jobs
may be split into parts, where the parts of a split job may be
processed simultaneously on more than one machine. Each
part of a job requires a setup time, however, on the machine
where the job part is processed. During setup, a machine
cannot process or set up any other job. We concentrate on
the basic case in which setup times are job-, machine- and
sequence-independent. Problems of this kind were encountered
when modelling practical problems in planning dis-
aster relief operations. Our main algorithmic result is a
polynomial-time algorithm for minimising total completion
time on two parallel identical machines. We argue, why the
same problem with threemachines is not an easy extension of
the two-machine case, leaving the complexity of this case as a
tantalising open problem. We give a constant-factor approximation
algorithm for the general case with any number of
machines and a polynomial-time approximation scheme for
a fixed number of machines. For the version with the objective
to minimise total weighted completion time, we prove
NP-hardness. Finally, we conclude with an overview of the
state of the art for other split scheduling problems with job-,
machine- and sequence-independent setup times