Fixed-Parameter and Approximation Algorithms: A New Look

A Fixed-Parameter Tractable (FPT) ρ-approximation algorithm for a minimization (resp. maximization) parameterized problem P is an FPT-algorithm that, given an instance (x, k) ∈ P computes a solution of cost at most k · ρ(k) (resp. k/ρ(k)) if a solution of cost at most (resp. at least) k exists; otherwise the output can be arbitrary. For well-known intractable problems such as the W[1]-hard Clique and W[2]-hard Set Cover problems, the natural question is whether we can get any FPT-approximation. It is widely believed that both Clique and Set-Cover admit no FPT ρ-approximation algorithm, for any increasing function ρ. However, to the best of our knowledge, there has been no progress towards proving this conjecture. Assuming standard conjectures such as the Exponential Time Hypothesis (ETH) [18] and the Projection Games Conjecture (PGC) [27], we make the first progress towards proving this conjecture by showing that – Under the ETH and PGC, there exist constants F1, F2 > 0 such that the Set Cover problem does not admit a FPT approximation algorithm with ratio k1 in 2 F2 · poly(N,M) time, where N is the size of the universe and M is the number of sets. – Unless NP ⊆ SUBEXP, for every 1 > δ > 0 there exists a constant F (δ) > 0 such that Clique has no FPT cost approximation with ratio k1−δ in 2 F · poly(n) time, where n is the number of vertices in the graph. In the second part of the paper we consider various W[1]-hard problems such as Directed Multicut, Directed Steiner Tree, Directed Steiner Forest, Directed Steiner Network and Minimum Size Edge Cover. For all these problem we give polynomial time f(OPT)-approximation algorithms for some small function f (the largest approximation ratio we give is OPT). Our results indicate a potential separation between the classes W[1] and W[2]; since no W[2]-hard problem is known to have a polynomial time f(OPT)-approximation for any function f . Finally, we answer a question by Marx [23] by showing the well-studied Strongly Connected Steiner ? Supported in part by NSF CAREER award 1053605, NSF grant CCF-1161626, ONR YIP award N000141110662, DARPA/AFOSR grant FA9550-12-1-0423, and a University of Maryland Research and Scholarship Award (RASA). The first author was also supported by a Summer International Research Fellowship from University of Maryland. The second author is also with AT&T Labs. ?? Supported in part by NSF grant number 434923. Subgraph problem (which is W[1]-hard and does not have any polynomial time constant factor approximation) has a constant factor FPTapproximation.