Frameworks for logically classifying polynomial-time optimisation problems.

We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠ NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems

Polynomial Kernelizations for MIN F^+Pi_1 and MAX NP

The relation of constant-factor approximability to fixed-parameter tractability and kernelization is a long-standing open question. We prove that two large classes of constant-factor approximable problems, namely~$textsc{MIN F}^+Pi_1$ and~$textsc{MAX NP}$, including the well-known subclass~$textsc{MAX SNP}$, admit polynomial kernelizations for their natural decision versions. This extends results of Cai and Chen (JCSS 1997), stating that the standard parameterizations of problems in~$textsc{MAX SNP}$ and~$textsc{MIN F}^+Pi_1$ are fixed-parameter tractable, and complements recent research on problems that do not admit polynomial kernelizations (Bodlaender et al. ICALP 2008)

On input read-modes of alternating Turing machines

AbstractA number of input read-modes of Turing machines have appeared in the literature. To investigate the differences among these input read-modes, we study log-time alternating Turing machines of constant alternations. For each fixed integer k ⩾ 1 and for each read-mode, a precise circuit characterization is established for log-time alternating Turing machines of k alternations, which is a nontrivial refinement of Ruzzo's circuit characterization of alternating Turing machines. These circuit characterizations indicate clearly the differences among the input read-modes. Complete languages in strong sense for each level of the log-time hierarchy are presented, refining a result by Buss. An application of these results to computational optimization problems is described

Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems

Descriptive Complexity for Counting Complexity Classes

Descriptive Complexity has been very successful in characterizing complexity classes of decision problems in terms of the properties definable in some logics. However, descriptive complexity for counting complexity classes, such as FP and #P, has not been systematically studied, and it is not as developed as its decision counterpart. In this paper, we propose a framework based on Weighted Logics to address this issue. Specifically, by focusing on the natural numbers we obtain a logic called Quantitative Second Order Logics (QSO), and show how some of its fragments can be used to capture fundamental counting complexity classes such as FP, #P and FPSPACE, among others. We also use QSO to define a hierarchy inside #P, identifying counting complexity classes with good closure and approximation properties, and which admit natural complete problems. Finally, we add recursion to QSO, and show how this extension naturally captures lower counting complexity classes such as #L