• Corpus ID: 7957655

On Making Directed Graphs Eulerian

  title={On Making Directed Graphs Eulerian},
  author={Manuel Sorge},
A directed graph is called Eulerian, if it contains a tour that traverses every arc in the graph exactly once. We study the problem of Eulerian extension (EE) where a directed multigraph G and a weight function is given and it is asked whether G can be made Eulerian by adding arcs whose total weight does not exceed a given threshold. This problem is motivated through applications in vehicle routing and flowshop scheduling. However, EE is NP-hard and thus we use the parameterized complexity… 
From Few Components to an Eulerian Graph by Adding Arcs
It is shown that EE is fixed-parameter tractable with respect to the combined parameter "number of connected components in the underlying undirected multigraph" and "sum of indeg - outdeg over all vertices v in the input multigraph where this value is positive".
Efficient Algorithms for Eulerian Extension and Rural Postman
This work shows that the NP-hard Weighted Multigraph Eulerian Extension problem is fixed-parameter tractable with respect to the number of extension arcs, and presents several polynomial-time algorithms for natural Euleria extension problems, including undirected variants which can be defined analogously to the directed ones.
Parameterized Complexity of Eulerian Deletion Problems
A randomized FPT algorithm for making an undirected graph Eulerian by deleting the minimum number of edges, based on a novel application of the color coding technique is shown, proving that this is not possible unless NP⊆coNP/poly.


Efficient Algorithms for Eulerian Extension
The main result is to show that the NP-hard WEIGHTED MULTIGRAPH EULERIAN EXTENSION is fixed-parameter tractable with respect to the number k of extension edges (arcs).
On general routing problems
It is shown that a proposed conversion of required nodes to required arcs is not allowed and that the problem remains polynomial complete if Q = θ, and the proposed transformations from M vehicle to single vehicle problems are shown to be incorrect.
The spanning subgraphs of eulerian graphs
It is shown that a connected graph G spans an eulerian graph if and only if G is not spanned by an odd complete bigraph K(2m + 1, 2n + 1). A disconnected graph spans an eulerian graph if and only if
Digraphs - theory, algorithms and applications
Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science, and it will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.
Algorithms for Enumerating All Spanning Trees of Undirected and Weighted Graphs
Algorithms for enumeration of spanning trees in undirected graphs, with and without weights, are presented, based on swapping edges in a fundamental cycle to construct a computation tree.
On Eulerian Extension Problems and their Application to Sequencing Problems
We introduce a new technique for solving several sequencing problems. We consider Gilmore and Gomory’s variant of the Traveling Salesman Problem and two variants of no-wait flowshop scheduling, the
Incompressibility through Colors and IDs
This paper shows how to combine results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems, and rules out the existence of compression algorithms for many of the problems in question.
On general routing problems: Comments
The RPP, TSP, and GRP a r e a l l polynomial complete rout ing problems, and why they requi re branch and bound (subtour e l imina t ion) a lgori thms, as well as the theorem presented.
Reducibility Among Combinatorial Problems
  • R. Karp
  • Computer Science
    50 Years of Integer Programming
  • 1972
Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.