Yijia Chen

On parameterized approximability

Combining classical approximability questions with parameterized complexity, we introduce a theory of parameterized approximability. The main intention of this theory is to deal with the efficient approximation of small cost solutions for optimisation problems.

Understanding the Complexity of Induced Subgraph Isomorphisms

We study left-hand side restrictions of the induced subgraph isomorphismproblem: Fixing a class , for given graphs G and arbitrary Hwe ask for induced subgraphs of Hisomorphic to G. For the homomorphism problem this kind of restriction has been studied by Grohe and Dalmau, Kolaitis and Vardi for the decision problem and by Dalmau and Jonsson for its counting variant. We give a dichotomy result for both variants of the induced subgraph isomorphism problem. Under some assumption from parameterized complexity theory, these problems are solvable in polynomial time if and only if contains no arbitrarily large graphs. All classifications are given by means of parameterized complexity. The results are presented for arbitrary structures of bounded arity which implies, for example, analogous results for directed graphs. Furthermore, we show that no such dichotomy is possible in the sense of classical complexity. That is, if there are classes such that the induced subgraph isomorphism problem on is neither in nor -complete. This argument may be of independent interest, because it is applicable to various parameterized problems.

On Parameterized Path and Chordless Path Problems

We study the parameterized complexity of various path (and cycle) problems, the parameter being the length of the path. For example, we show that the problem of the existence of a maximal path of length k in a graph G is fixed-parameter tractable, while its counting version is #W[1]- complete. The corresponding problems for chordless (or induced) paths are W[2]-complete and #W[2]-complete respectively. With the tools developed in this paper we derive the NP-completeness of a related classical problem, thereby solving a problem due to Hedetniemi.

Machine Characterizations of the Classes of the W-Hierarchy

We give machine characterizations of the complexity classes of the W-hierarchy. Moreover, for every class of this hierarchy, we present a parameterized halting problem complete for this class.

An Isomorphism Between Subexponential and Parameterized Complexity Theory

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.

On p-optimal proof systems and logics for PTIME

We prove that TAUT has a p-optimal proof system if and only if a logic related to least fixed-point logic captures polynomial time on all finite structures. Furthermore, we show that TAUT has no effective p-optimal proof system if NTIME(hO(1)) ⊈ DTIME(hO(log h)) for every time constructible and increasing function h.

On miniaturized problems in parameterized complexity theory

We introduce a general notion of miniaturization of a problem that comprises the different miniaturizations of concrete problems considered so far. We develop parts of the basic theory of miniaturizations. Using the appropriate logical formalism, we show that the miniaturization of a definable problem in W[t] lies in W[t], too. In particular, the miniaturization of the dominating set problem is in W[2]. Furthermore, we investigate the relation between f(k)ċno(k) time and subexponential time algorithms for the dominating set problem and for the clique problem.

The Constant Inapproximability of the Parameterized Dominating Set Problem

We prove that there is no fpt-algorithm that can approximate the dominating set problem with any constant ratio, unless FPT = W[1]. Our hardness reduction is built on the second authors recent W[1]-hardness proof of the biclique problem [25]. This yields, among other things, a proof without the PCP machinery that the classical dominating set problem has no polynomial time constant approximation under the exponential time hypothesis.

Subexponential time and fixed-parameter tractability: exploiting the miniaturization mapping

Recently it has been shown that the miniaturization mapping ℳ faithfully translates bexponential parameterized complexity into (unbounded) parameterized complexity. We determine the pre-images under ℳ of various (classes of) problems. For many parameterized problems whose underlying classical problem is in NP we show that the pre-images coincide with natural reparameterizations that take into account the amount of non-determinism needed to solve them.

On slicewise monotone parameterized problems and optimal proof systems for TAUT

For a reasonable sound and complete proof calculus for firstorder logic consider the problem to decide, given a sentence ϕ of firstorder logic and a natural number n, whether ϕ has no proof of length ≤ n. We show that there is a nondeterministic algorithm accepting this problem which, for fixed ϕ, has running time bounded by a polynomial in n if and only if there is an optimal proof system for the set TAUT of tautologies of propositional logic. This equivalence is an instance of a general result linking the complexity of so-called slicewise monotone parameterized problems with the existence of an optimal proof system for TAUT.