Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time n o(k) m O(1) , where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t − 1)-st level W [t − 1] of the W-hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted sat, hitting set, set cover, and feature set, cannot be solved in time n o(k) m O(1) , where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W [1] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted q-sat (for any fixed q ≥ 2), clique, independent set, and dominating set, cannot be solved in time n o(k) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n k m O(1) or O(n k). 1

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/ɛ)n o(1/ɛ) for any function f unless an unlikely collapse occurs in parameterized complexity theory.

Given a graph and pairs s i t i of terminals, the edge-disjoint paths problem is to determine whether there exist s i t i paths that do not share any edges. We consider this problem on acyclic digraphs. It is known to be NP-complete and solvable in time n where k is the number of paths. It has been a long-standing open question whether it is fixed-parameter tractable in k. We resolve this question in the negative: we show that the problem is W [1]-hard. In fact it remains W [1]-hard even if the demand graph consists of two sets of parallel edges. On a

An FPT algorithm with a running time of O(n is described for the Set Splitting problem, parameterized by the number k of sets to be split. It is also shown that there can be no FPT unless the satisfiability of n-variable 3SAT instances can be decided in time .

Abstract. The notion of fixed-parameter approximation is introduced to investigate the approximability of optimization problems within the framework of fixed-parameter computation. This work partially aims at enhancing the world of fixed-parameter computation in parallel with the conventional theory of computation that includes both exact and approximate computations. In particular, it is proved that fixed-parameter approximability is closely related to the approximation of small-cost solutions in polynomial time. It is also demonstrated that many fixedparameter intractable problems are not fixed-parameter approximable. On the other hand, fixed-parameter approximation appears to be a viable approach to solving some inapproximable yet important optimization problems. For instance, all problems in the class MAX SNP admit fixed-parameter approximation schemes in time O(2 O((1−ɛ/O(1))k) p(n)) for any small ɛ>0. 1

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Abstract. In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number ml(G) of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring, Hamilton Path, Minimum Dominating Set, Minimum Bandwidth or many other problems, for graphs of bounded max leaf number? What optimization problems are W [1]-hard under this parameterization? We do two things: (1) We describe much improved FPT algorithms for a large number of graph problems, for input graphs G for which ml(G) ≤ k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (FPT) runtimes O ∗ (f(k)). (2) The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions. 1

Abstract. We establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories. Key words. parameterized complexity, exponential time complexity, exponential time hypothesis, subexponential time

For an edge-weighted connected undirected graph, the minimum k-way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into k connected components. The problem is NP-hard when k is part of the input and W[1]-hard when k is taken as a parameter. A simple algorithm for approximating a minimum k-way cut is to iteratively increase the number of components of the graph by h − 1, where 2 ≤ h ≤ k, until the graph has k components. The approximation ratio of this algorithm is known for h ≤ 3 but is open for h ≥ 4. In this paper, we consider a general algorithm that iteratively increases the number of components of the graph by hi − 1, where h1 ≤ h2 ≤ · · · ≤ hq and �q i=1 (hi − 1) = k − 1. We prove that the approximation ratio of this general algorithm is 2 − ( �q � � � � hi k) / , which is tight. Our result implies i=1 2 that the approximation ratio of the simple algorithm is 2 − h/k + O(h 2 /k 2) in general and 2 − h/k if k − 1 is a multiple of h − 1. Key words approximation algorithm, k-way cut, k-way split. 1 2 Approximation ratio for k-way cuts 2

We investigate and classify fixed parameter complexity of several infinite duration games, including Rabin, Streett, Muller, parity, mean payoff, and simple stochastic, using different natural parameterizations. Most known fixed parameter intractable games are PSPACE- or EXP-complete classically, AW [∗] or XP-hard parametrically, and are all finite duration games. In contrast, the games we consider are infinite duration, solvable in positional or finite memory strategies, and belong to “lower ” complexity classes, like NP and/or coNP. However, the best known algorithms they possess are of complexity n f(k) , i.e., XP is the only upper bound, with no known parametric lower bounds. We demonstrate that under different parameterizations these games may have different or equivalent FPT-statuses, and present several tractable and intractable cases. 1