Abstract
Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer $k\geq 2$ and arbitrary small $\epsilon > 0$, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor $k-\epsilon$ even on graphs where the vertices can be almost partitioned into $k$ solutions. This gives a more structured and yet simpler (albeit using the “It Ain't Over Till It's Over” theorem) UG-hardness result for the Feedback Vertex Set problem than the previous hardness result. In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is UG-hard to approximate within any constant. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations. An extended abstract of this paper appeared in the Proceedings of the 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, 2012 (APPROX'12).
Highlights
Many interesting discrete optimization problems can be formulated as the problem of finding, in a given graph, a large induced subgraph with a desired property
One of the most studied such problems is the Feedback Vertex Set (FVS) problem where the property is acyclicity, i. e., given a directed graph G = (V, E) we wish to delete the minimum number of vertices so that the resulting graph is acyclic
Another example is the DAG Vertex Deletion (DVD) problem, where we are given an integer k and a directed acyclic graph and we wish to delete the minimum number of vertices so that the resulting graph has no path of length1 k
Summary
Many interesting discrete optimization problems can be formulated as the problem of finding, in a given (directed) graph, a large induced subgraph with a desired property. Even though the initial motivation of this work was to better understand the approximability of DTCT (and DVD), the techniques that we develop lead to a more structured UGC-based hardness result for FVS: similar to the recent results for Vertex Cover on k-uniform hypergraphs by Bansal and Khot [1, 2], we show that, for any integer k ≥ 2 and arbitrarily small ε > 0, there is no (k −ε)-approximation algorithm for FVS even on graphs where the vertices can be almost partitioned into k feedback vertex sets. When proving UGC-based inapproximability results, the main task is usually to design “gadgets” for the problems considered that simulate a so-called dictatorship test Once we have such “dictatorship gadgets,” the process of obtaining UGC-based hardness results often follows from () fairly standard arguments. This explains the difficulty in obtaining good approximation algorithms for DTCT and further motivates alternative approaches such as the bicriteria approach by Skutella [18] that approximates the DTCT within a constant factor if the deadline is allowed to be violated by a constant factor
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