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, , the problem does not admit a (d + 2 − )^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·cw⁾, showing that the problem is FPT by d + cw, and improving the previously best know algorithm for this case.