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)[11] and the Projection Games Conjecture (PGC)[18], 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 in 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 k^1-δ in 2^kF·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 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 OPT²). 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 [14] by showing the well-studied Strongly Connected Steiner Subgraph problem (which is W[1]-hard and does not have any polynomial time constant factor approximation) has a constant factor FPT-approximation.