6 research outputs found
Parameterized Rural Postman Problem
The Directed Rural Postman Problem (DRPP) can be formulated as follows: given
a strongly connected directed multigraph with nonnegative integral
weights on the arcs, a subset of and a nonnegative integer ,
decide whether has a closed directed walk containing every arc of and
of total weight at most . Let be the number of weakly connected
components in the the subgraph of induced by . Sorge et al. (2012) ask
whether the DRPP is fixed-parameter tractable (FPT) when parameterized by ,
i.e., whether there is an algorithm of running time where is a
function of only and the notation suppresses polynomial factors.
Sorge et al. (2012) note that this question is of significant practical
relevance and has been open for more than thirty years. Using an algebraic
approach, we prove that DRPP has a randomized algorithm of running time
when is bounded by a polynomial in the number of vertices in
. We also show that the same result holds for the undirected version of
DRPP, where is a connected undirected multigraph
Editing to Eulerian Graphs
We investigate the problem of modifying a graph into a connected graph in
which the degree of each vertex satisfies a prescribed parity constraint. Let
, and denote the operations edge addition, edge deletion and
vertex deletion respectively. For any , we define
Connected Degree Parity Editing (CDPE()) to be the problem that takes
as input a graph , an integer and a function , and asks whether can be modified into a connected
graph with for each , using
at most operations from . We prove that
1. if or , then CDPE() can be solved in polynomial
time;
2. if , then CDPE() is
NP-complete and W[1]-hard when parameterized by , even if .
Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk,
Pilipczuk and Schlotter, our results completely classify the classical and
parameterized complexity of the CDPE() problem for all . We obtain the same classification for a natural variant of the
CDPE() problem on directed graphs, where the target is a weakly connected
digraph in which the difference between the in- and out-degree of every vertex
equals a prescribed value. As an important implication of our results, we
obtain polynomial-time algorithms for the Eulerian Editing problem and its
directed variant.Comment: 33 pages. An extended abstract of this paper will appear in the
proceedings of FSTTCS 201
On Eulerian extensions and their application to no-wait flowshop scheduling
We consider a variant of no-wait flowshop scheduling that is motivated by continuous casting in the multi-stage production process in steel manufacturing. The task is to find a feasible schedule with a minimum number of interruptions, i.e., continuous idle time intervals on the last production stage. Based on an interpretation as Eulerian Extension Problems, we fully settle the complexity status of any particular problem case: We give a very intuitive optimal algorithm for scheduling on two processing stages with one machine in the first stage, and we show that all other problem variants are strongly NP-hard. We also discuss alternative idle time related scheduling models and their justification in the considered steel manufacturing environment. Here, we derive constant factor approximations