Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph

We prove that approximating the Max Acyclic Subgraph problem within a factor better than 1/2 is Unique-Games hard. Specifically, for every constant ε > 0 the following holds: given a directed graph G that has an acyclic subgraph consisting of a fraction (1 − ε) of its edges, if one can efficiently find an acyclic subgraph of G with more than (1/2 + ε) of its edges, then the UGC is false. Note that it is trivial to find an acyclic subgraph with 1/2 the edges, by taking either the forward or backward edges in an arbitrary ordering of the vertices of G. The existence of a ρ-approximation algorithm for ρ > 1/2 has been a basic open problem for a while. Our result is the first tight inapproximability result for an ordering problem. The starting point of our reduction is a directed acyclic subgraph (DAG) in which every cut is nearly-balanced in the sense that the number of forward and backward edges crossing the cut are nearly equal; such DAGs were constructed in [3]. Using...

• Type: Conference
• Year: 2008
• Where: FOCS
• Authors: Venkatesan Guruswami, Rajsekar Manokaran, Prasad Raghavendra
• Keywords: Acyclic Subgraph, Acyclic Subgraph Problem, FOCS 2008, Max Acyclic Subgraph, Theoretical Computer Science