Strong computational lower bounds via parameterized complexity

Abstract

We develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes WEIGHTED SATISFIABILITY, DOMINATING SET, HITTING SET, SET COVER, CLIQUE, and INDEPENDENT SET. For example, although a trivial enumeration can easily test in time O(n k ) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time f (k)n o(k) for any function f , even if we restrict the parameter values to be bounded by an arbitrarily small function of n. Under the same assumption, we prove that even if we restrict the parameter values k to be of the order Θ(μ(n)) for any reasonable function μ, no algorithm of running time n o(k) can test if a graph of n vertices has a clique of size k. Similar strong lower bounds on the computational complexity are also derived for other NP-hard problems in the above class. Our techniques can be further extended to derive computational lower bounds on polynomial time approximation schemes for NP-hard optimization problems. For example, we prove that the NP-hard DISTINGUISHING SUBSTRING SELECTION problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time f (1/ )n o(1/ ) for any function f unless an unlikely collapse occurs in parameterized complexity theory. ✩ A preliminary version of this paper "Linear FPT reductions and computational lower bounds" was presented at

Computing Vertex-Surjective Homomorphisms to Partially Reflexive Trees

“…In G we did not denote any vertex labels but they are all the same as in Figure 4. Instead, we show that all displayed vertices in the top dotted area including g (1) s and g (2) s are mapped to p by h, whereas all vertices in the bottom dotted area including g (1) t and g (2) t are mapped to q. Also note that the two matching edges in G are in 1-to-1 correspondence with the two paths (of length 1) in G , the ends of which are mapped to x1 and x2.…”

“…This follows from combining the aforementioned observation that for all k ≥ 1 a connected graph G on at least two vertices allows a surjective homomorphism to S k if and only if G has an independent set of size at least k with the result of Chen et al [2] who showed that there is no algorithm that solves Independent Set on n-vertex graphs in time f (k) · n o(k) , unless the Exponential Time Hypothesis fails.…”

“…, e d that are incident with u in G. Because G has minimum degree at least two, C u has at most one other vertex if u / ∈ {s, t}; otherwise C u has two other vertices. If C u has one other vertex then we denote this vertex by g (1) u , and if C u has two other vertices then we denote these vertices by g (1) u and g (2) u , respectively. 2.…”

“…This does not violate the definition of a homomorphism either, because the only vertices of the subgraphs F 1 and F 2 of G that have neighbors outside F 1 and F 2 are g (1) s , g (2) s , and g (1)there is at least one edge uv ∈ V G with u ∈ V 1 and v ∈ V 2 . Hence, the vertices x 1 , .…”

Refining complexity analyses in planning by exploiting the exponential time hypothesis

“…= NP: our understanding of this kind of complexity questions is currently far too weak. We content ourselves by saying that the ETH has deep connections with many topics in computer science, such as the existence of subexponential algorithms for NP-complete problems [16,18,25], the complexity and approximability of optimisation problems [9,23] and parameterised complexity theory [10,11], and the failure of ETH would have far-reaching consequences.…”