Parameterized Orientable Deletion

A graph is $d$-orientable if its edges can be oriented so that the maximum in-degree of the resulting digraph is at most $d$. $d$-orientability is a well-studied concept with close connections to fundamental graph-theoretic notions and applications as a load balancing problem. In this paper we consider the d-ORIENTABLE DELETION problem: given a graph $G=(V,E)$, delete the minimum number of vertices to make $G$ $d$-orientable. We contribute a number of results that improve the state of the art on this problem. Specifically:

• We show that the problem is W[2]-hard and $\log n$-inapproximable with respect to $k$, the number of deleted vertices. This closes the gap in the problem’s approximability.
• We completely characterize the parameterized complexity of the problem on chordal graphs: it is FPT parameterized by $d+k$, but W-hard for each of the parameters $d,k$ separately.
• We show that, under the SETH, for all $d,ϵ$, the problem does not admit a $(d+2-ϵ{)}^{tw}$, algorithm where $tw$ is the graph’s treewidth, resolving as a special case an open problem on the complexity of PSEUDOFOREST DELETION.
• We show that the problem is W-hard parameterized by the input graph’s clique-width. Complementing this, we provide an algorithm running in time $d^{O(d\cdot cw)}$, showing that the problem is FPT by $d+cw$, and improving the previously best known algorithm for this case.