Parameter Compilation

In resolving instances of a computational problem, if multiple instances of interest share a feature in common, it may be fruitful to compile this feature into a format that allows for more efficient resolution, even if the compilation is relatively expensive. In this article, we introduce a formal framework for classifying problems according to their compilability. The basic object in our framework is that of a parameterized problem, which here is a language along with a parameterization---a map which provides, for each instance, a so-called parameter on which compilation may be performed. Our framework is positioned within the paradigm of parameterized complexity, and our notions are relatable to established concepts in the theory of parameterized complexity. Indeed, we view our framework as playing a unifying role, integrating together parameterized complexity and compilability theory

Parameterized complexity of machine scheduling: 15 open problems

Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc

The Global Implications of the Hard X-ray Excess in Type 1 AGN

Recent evidence for a strong 'hard excess' of flux at energies > 20 keV in some Suzaku observations of type 1 Active Galactic Nuclei (AGN) has motivated an exploratory study of the phenomenon in the local type 1 AGN population. We have selected all type 1 AGN in the Swift Burst Alert Telescope (BAT) 58-month catalog and cross-correlated them with the holdings of the Suzaku public archive. We find the hard excess phenomenon to be a ubiquitous property of type 1 AGN. Taken together, the spectral hardness and equivalent width of Fe K alpha emission are consistent with reprocessing by an ensemble of Compton-thick clouds that partially cover the continuum source. In the context of such a model, ~ 80 % of the sample has a hardness ratio consistent with > 50% covering of the continuum by low-ionization, Compton-thick gas. More detailed study of the three hardest X-ray spectra in our sample reveal a sharp Fe K absorption edge at ~ 7 keV in each of them, indicating that blurred reflection is not responsible for the very hard spectral forms. Simple considerations place the distribution of Compton-thick clouds at or within the optical broad line region.Comment: Accepted for publication in Ap

Automating Resolution is NP-Hard

We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatizable unless P = NP. Indeed, we show it is NP-hard to distinguish between formulas that have Resolution refutations of polynomial length and those that do not have subexponential length refutations. This also implies that Resolution is not automatizable in subexponential time or quasi-polynomial time unless NP is included in SUBEXP or QP, respectively

Parameterized complexity of DPLL search procedures

We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas

Parameterized bounded-depth Frege is not optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's

A Peculiar Flaring Episode of Cygnus X-1

Recent monitoring of Cyg X-1 with {\em RXTE} revealed a period of intense flaring, which started in October of 2000 and lasted until March of 2001. The source exhibited some quite unusual behaviors during this period. The soft X-ray flux of the source went up and down three times on a timescale of about one month, as discovered by the ASM aboard RXTE, before finally returning to the normal level (of the hard state). The observed spectral and temporal X-ray properties of Cyg X-1 are mostly intermediate between the canonical hard and soft states. This is known previously for strong X-ray flares, however, we show that the source did enter a period that resembles, in many ways, a sustained soft state during the last of the three flares. We make detailed comparisons between this flare and the 1996 state transition, in terms of the observed X-ray properties, such as flux--hardness correlation, X-ray spectrum, and power density spectrum. We point out the similarities and differences, and discuss possible implications of the results on our understanding of the phenomena of flares and state transitions associated with Cyg X-1.Comment: 4 pages, 3 figures, accepted for publication in ApJ Letter