Abstract
The partially disjoint paths problem is: given: a directed graph, vertices \(r_1,s_1,\ldots ,r_k,s_k\), and a set F of pairs \(\{i,j\}\) from \(\{1,\ldots ,k\}\), find: for each \(i=1,\ldots ,k\) a directed \(r_i-s_i\) path \(P_i\) such that if \(\{i,j\}\in F\) then \(P_i\) and \(P_j\) are disjoint. We show that for fixed k, this problem is solvable in polynomial time if the directed graph is planar. More generally, the problem is solvable in polynomial time for directed graphs embedded on a fixed compact surface. Moreover, one may specify for each edge a subset of \(\{1,\ldots ,k\}\) prescribing which of the \(r_i-s_i\) paths are allowed to traverse this edge.
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement n\({}^{\circ }\) 339109.
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Notes
- 1.
If a and b are periods of \(x=(x_1,\ldots ,x_n)\) with \(a<b\) and \(a+b\le n\), then \(b-a\) is a period of x. For let \(1\le i\le n-(b-a)\). We show \(x_{i+(b-a)}=x_i\). If \(i\le n-b\) then \(x_{i+(b-a)}=x_{(i+b)-a}=x_{i+b}=x_i\). If \(i>n-b\) then \(i>a\) (as \(i>n-b\ge a\), since \(a+b\le n\) by assumption), hence \(x_{i+(b-a)}=x_{(i-a)+b}=x_{i-a}=x_i\).
- 2.
Let A a polynomial-time algorithm that finds a solution for feasible instances. When we apply A to any instance, then if feasible, we find a feasible solution, and if infeasible, A gets stuck or has not delivered a solution in polynomial time.
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The author thanks the referee for helpful corrective remarks and suggestions.
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© 2019 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature
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Schrijver, A. (2019). Finding k Partially Disjoint Paths in a Directed Planar Graph. In: Bárány, I., Katona, G., Sali, A. (eds) Building Bridges II. Bolyai Society Mathematical Studies, vol 28. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59204-5_13
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