On the Parameterized Complexity of Computing st-Orientations with Few Transitive Edges

Orienting the edges of an undirected graph such that the resulting digraph satisfies some given constraints is a classical problem in graph theory, with multiple algorithmic applications. In particular, an $st$-orientation orients each edge of the input graph such that the resulting digraph is acycl...

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Vydáno v:Journal of graph algorithms and applications Ročník 29; číslo 1; s. 247 - 266
Hlavní autoři: Binucci, Carla, Liotta, Giuseppe, Montecchiani, Fabrizio, Ortali, Giacomo, Piselli, Tommaso
Médium: Journal Article
Jazyk:angličtina
Vydáno: 12.08.2025
ISSN:1526-1719, 1526-1719
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Abstract Orienting the edges of an undirected graph such that the resulting digraph satisfies some given constraints is a classical problem in graph theory, with multiple algorithmic applications. In particular, an $st$-orientation orients each edge of the input graph such that the resulting digraph is acyclic, and it contains a single source $s$ and a single sink $t$. Computing an $st$-orientation of a graph can be done efficiently, and it finds notable applications in graph algorithms and in particular in graph drawing. On the other hand, finding an $st$-orientation with at most $k$ transitive edges is more challenging and it was recently proven to be \NP-hard already when $k=0$. We strengthen this result for graphs of bounded diameter, and for graphs of bounded vertex degree. These computational lower bounds naturally raise the question about which structural parameters can lead to tractable parameterizations of the problem. Our main result is a fixed-parameter tractable algorithm parameterized by treewidth.
AbstractList Orienting the edges of an undirected graph such that the resulting digraph satisfies some given constraints is a classical problem in graph theory, with multiple algorithmic applications. In particular, an $st$-orientation orients each edge of the input graph such that the resulting digraph is acyclic, and it contains a single source $s$ and a single sink $t$. Computing an $st$-orientation of a graph can be done efficiently, and it finds notable applications in graph algorithms and in particular in graph drawing. On the other hand, finding an $st$-orientation with at most $k$ transitive edges is more challenging and it was recently proven to be \NP-hard already when $k=0$. We strengthen this result for graphs of bounded diameter, and for graphs of bounded vertex degree. These computational lower bounds naturally raise the question about which structural parameters can lead to tractable parameterizations of the problem. Our main result is a fixed-parameter tractable algorithm parameterized by treewidth.
Author Binucci, Carla
Ortali, Giacomo
Piselli, Tommaso
Liotta, Giuseppe
Montecchiani, Fabrizio
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