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Theoretical Computer Science
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We prove that approximating the Max. Acyclic Subgraphproblem within a factor better than 1/2 is Unique Gameshard. Specifically, for every constant $\eps$ 0 thefollowing holds: given a directed graph $G$ that has anacyclic subgraph consisting of a fraction $(1-\eps)$ of itsedges, if one can efficiently find an acyclic subgraph of$G$ with more than $(1/2+\eps)$ of its edges, then the UGCis false. Note that it is trivial to find an acyclicsubgraph with $1/2$ the edges, by taking either the forwardor backward edges in an arbitrary ordering of the verticesof $G$. The existence of a $\rho$-approximation algorithmfor \rho 1/2$ has been a basic open problem for a while.Our result is the first tight inapproximability result foran ordering problem. The starting point of our reduction isa directed acyclic subgraph (DAG) in which every cut isnearly-balanced in the sense that the number of forward andbackward edges crossing the cut are nearly equal; such DAGswere constructed by Charikar et al. Using this, we are ableto study Max. Acyclic Subgraph, which is a constraintsatisfaction problem (CSP) over an unbounded domain, byrelating it to a proxy CSP over a bounded domain. The latteris then amenable to powerful techniques based on theinvariance principle.Our results also give a super-constant factorinapproximability result for the Feedback Arc Set problem.Using our reductions, we also obtain SDP integrality gapsfor both the problems.