Data reductions, fixed parameter tractability, and random weighted d-CNF satisfiability

Data reduction is a key technique in the study of fixed parameter algorithms. In the AI literature, pruning techniques based on simple and efficient-to-implement reduction rules also play a crucial role in the success of many industrial-strength solvers. Understanding the effectiveness and the applicability of data reduction as a technique for designing heuristics for intractable problems has been one of the main motivations in studying the phase transition of randomly-generated instances of NP-complete problems.In this paper, we take the initiative to study the power of data reductions in the context of random instances of a generic intractable parameterized problem, the weighted d-CNF satisfiability problem. We propose a non-trivial random model for the problem and study the probabilistic behavior of the random instances from the model. We design an algorithm based on data reduction and other algorithmic techniques and prove that the algorithm solves the random instances with high probability and in fixed-parameter polynomial time O(dknm) where n is the number of variables, m is the number of clauses, and k is the fixed parameter. We establish the exact threshold of the phase transition of the solution probability and show that in some region of the problem space, unsatisfiable random instances of the problem have parametric resolution proof of fixed-parameter polynomial size. Also discussed is a more general random model and the generalization of the results to the model.     

The computational complexity of ideal semantics

We analyse the computational complexity of the recently proposed ideal semantics within both abstract argumentation frameworks (afs) and assumption-based argumentation frameworks (abfs). It is shown that while typically less tractable than credulous admissibi-lity semantics, the natural decision problems arising with this extension-based model can, perhaps surprisingly, be decided more efficiently than sceptical preferred semantics. In particular the task of finding the unique ideal extension is easier than that of deciding if a given argument is accepted under the sceptical semantics. We provide efficient algorithmic approaches for the class of bipartite argumentation frameworks and, finally, present a number of technical results which offer strong indications that typical problems in ideal argumentation are complete for the class of languages decidable by polynomial time algorithms allowed to make non-adaptive queries to a C oracle, where C is an upper bound on the computational complexity of deciding credulous acceptance: C=np for afs and logic programming (lp) instantiations of abfs; for abfs modelling default theories.     

Range mode and range median queries in constant time and sub-quadratic space

Given a list of n items and a function defined over sub-lists, we study the space required for computing the function for arbitrary sub-lists in constant time.For the function mode we improve the previously known space bound O(n²/logn) to O(n²loglogn/log²n) words.For median the space bound is improved to O(n²loglog²n/log²n) words from O(n²⋅log(k)n/logn), where k is an arbitrary constant and log(k) is the iterated logarithm.     

On the distribution function of the complexity of finite sequences

Investigations of complexity of sequences lead to important applications such as effective data compression, testing of randomness, discriminating between information sources and many others. In this paper we establish formulae describing the distribution functions of random variables representing the complexity of finite sequences introduced by Lempel and Ziv in 1976. It is known that this quantity can be used as an estimator of entropy. We show that the distribution functions depend affinely on the probabilities of the so-called “exact” sequences.     

The quantum query complexity of the determinant

In this paper we give tight quantum query complexity bounds of some important linear algebra problems. We prove Θ(n²) quantum query bounds for verify the determinant, rank, matrix inverse and the matrix power problem.     

Semisimple algebras of almost minimal rank over the reals

A famous lower bound for the bilinear complexity of the multiplication in associative algebras is the Alder–Strassen bound. Algebras for which this bound is tight are called algebras of minimal rank. After 25 years of research, these algebras are now well understood. Here we start the investigation of the algebras for which the Alder–Strassen bound is off by one. As a first result, we completely characterize the semisimple algebras over R$\mathbb{R}$ whose bilinear complexity is by one larger than the Alder–Strassen bound. Furthermore, we characterize all algebras A$A$ (with radical) of minimal rank plus one over R$\mathbb{R}$ for which A/radA$A/radA$ has minimal rank plus one. The other possibility is that A/radA$A/radA$ has minimal rank. For this case, we only present a partial result.     

The subsequence composition of a string

Words that appear as constrained subsequences in a text-string are considered as possible indicators of the host string structure, hence also as a possible means of sequence comparison and classification. The constraint consists of imposing a bound on the number ω$\omega$ of positions in the text that may intervene between any two consecutive characters of a subsequence. A subset of such ω$\omega$-sequences is then characterized that consists, in intuitive terms, of sequences that could not be enriched with more characters without losing some occurrence in the text. A compact spatial representation is then proposed for these representative sequences, within which a number of parameters can be defined and measured. In the final part of the paper, such parameters are empirically analyzed on a small collection of text-strings endowed with various degrees of structure.     

Limiting negations in non-deterministic circuits

The minimum number of NOT gates in a Boolean circuit computing a Boolean function f is called the inversion complexity of f. In 1958, Markov determined the inversion complexity of every Boolean function and, in particular, proved that log₂(n+1) NOT gates are sufficient to compute any Boolean function on n variables. In this paper, we consider circuits computing non-deterministically and determine the inversion complexity of every Boolean function. In particular, we prove that one NOT gate is sufficient to compute any Boolean function in non-deterministic circuits if we can use an arbitrary number of guess inputs.     

Adaptive binary tree for fast SVM multiclass classification

This paper presents an adaptive binary tree (ABT) to reduce the test computational complexity of multiclass support vector machine (SVM). It achieves a fast classification by: (1) reducing the number of binary SVMs for one classification by using separating planes of some binary SVMs to discriminate other binary problems; (2) selecting the binary SVMs with the fewest average number of support vectors (SVs). The average number of SVs is proposed to denote the computational complexity to exclude one class. Compared with five well-known methods, experiments on many benchmark data sets demonstrate our method can speed up the test phase while remain the high accuracy of SVMs.