Guillermo De Ita Luna

Computing #2SAT and #2UNSAT by binary patterns

We present some results about the parametric complexity of #2SAT and #2UNSAT, which consist on counting the number of models and falsifying assignments, respectively, for two Conjunctive Forms (2-CFs) . Firstly, we show some cases where given a formula F, #2SAT(F) can be bounded above by considering a binary pattern analysis over its set of clauses. Secondly, since #2SAT(F)=2n-#2UNSAT(F) we show that, by considering the constrained graph GF of F, if GF represents an acyclic graph then, #UNSAT(F) can be computed in polynomial time. To the best of our knowledge, this is the first time where #2UNSAT is computed through its constrained graph, since the inclusion-exclusion formula has been commonly used for computing #UNSAT(F).

The Incremental Satisfiability Problem for a Two Conjunctive Normal Form

Abstract We propose a novel method to review K ⊢ ϕ when K and ϕ are both in Conjunctive Normal Forms (CF). We extend our method to solve the incremental satisfiablity problem (ISAT), and we present different cases where ISAT can be solved in polynomial time. Especially, we present an algorithm for 2-ISAT. Our last algorithm allow us to establish an upper bound for the time-complexity of 2-ISAT, as well as to establish some tractable cases for the 2-ISAT problem.

Using Binary Patterns for Counting Falsifying Assignments of Conjunctive Forms

The representation of the set of falsifying assignments of clauses via binary patterns has been useful in the design of algorithms for solving #FAL (counting the number of falsifying assignments of conjunctive forms (CF)). Given as input a CF formula F expressed by m clauses defined over n variables, we present a deterministic algorithm for computing #FAL(F). Principally, our algorithm computes non-intersecting subsets of falsifying assignments of F until the space of falsifying assignments defined by F is covered. Due to #SAT(F) = 2n-#FAL(F), results about #FAL can be established dually for #SAT. The time complexity of our proposals for computing #FAL(F) is established according to the number of clauses and the number of variables of F.

An Approximate Algorithm for the Chromatic Number of Graphs

Abstract We have designed a novel polynomial-time approximate algorithm for the graph vertex colouring problem. Contrary to the common top-down strategy for solving the colouring graph problem, we propose a bottom-up algorithm for colouring graphs. Given an input graph G, we establish an upper bound to approximate the colouring of the input grap given by ⌈ δ ( G ) / 2 ⌉ + 2 where δ ( G ) is the average degree of G.

An Enumerative Algorithm for #2SAT

Abstract Counting models for two conjunctive forms (2-CF), problem known as #2SAT, is a classic #P-complete problem. We determine different discrete structures on the constrained graph of the 2-CF formula allowing the efficient computation of #2SAT. We show that if the constrained graph of a 2-CF F is acyclic or it has only cycles, which are independent each other, then #2SAT( F ) is computed efficiently. On the other hand, we design a bottom-up procedure to compute #2SAT( F ) in an incremental way. Given a formula F , our procedure begins with the maximum subformula which does not have intersecting cycles, let say F 0 . In each iteration of the procedure, a new clause C i ∈ ( F − F 0 ) is considered in order to form F i = ( F i − 1 ∧ C i ) and then to compute #2SAT( F i ) based on the computation of #2SAT( F i − 1 ).

A New Optimization Strategy for Solving the Fall-Off Boundary Value Problem in Pixel-Value Differencing Steganography

In Digital Image Steganography, Pixel-Value Differencing (PVD) methods use the difference between neighboring pixel values to determine the amount of data bits to be inserted. The main advantage of these methods is the size of input data that an image can hold. However, the fall-off boundary problem and the fall in error problem are persistent in many PVD steganographic methods. This results in an incorrect output image. To fix these issues, usually the pixel values are either somehow adjusted or simply not considered to carry part of the input data. In this paper, we enhance the Tri-way Pixel-Value Differencing method by finding an optimal pixel value for each pixel pair such that it carries the maximum input data possible without ignoring any pair and without yielding incorrect pixel values.

Modelling 3-Coloring of Outerplanar Graphs via Incremental Satisfiability

Abstract A novel method to model the problem of 3-coloring on outerplanar graphs is presented. The proposal is based on the specification of the logical constraints for the 3-coloring of an outerplanar graph in a dinaymic way, resulting in a polynomial-time instance of the incremental satisfiability problem. This proposal can be extended to consider other polynomial-time instances of the 3-coloring problem.

Extending Extremal Polygonal Arrays for the Merrifield-Simmons Index

Polygonal array graphs have been widely investigated, and they represent a relevant area of interest in mathematical chemistry because they have been used to study intrinsic properties of molecular graphs. For example, to determine the Merrifield-Simmons index of a polygonal array \(A_n\) that is the number of independent sets of that graph, denoted as \(i(A_n)\).

A Parametric Polynomial Deterministic Algorithm for #2SAT

Counting models for two Conjunctive Normal Form formulae (2-CFs), known as the #2SAT problem, is a classic #P complete problem. It is known that if the constraint graph of a 2-CF F is acyclic or contains loops and parallel edges, \(\#2SAT(F)\) can be computed efficiently. In this paper we address the cyclic case different from loops and parallel edges.

Applying Max-2SAT to Efficient Belief Revision

Belief revision is a NP-hard problem even when the Knowledge Base (Ó) is formed by Horn clauses. In this paper, we present a new belief revision operator running efficiently if the initial knowledge base is a two conjunctive form(conjunction of unit or binary clauses, denoted as 2-CF).Such revision operator *I on a new formula F, denoted as (Ó *IF), relies heavily on the selected model I of F. If the model Ican be computed in polynomial time (e.g. if F is Horn or a 2-CF), then the complete belief revision process has a polynomial time complexity. However, if F has not restrictions, our proposal request to apply only one NP oracle call (that is, the necessary call to compute a model of F) that involves an exponential time over the length of F. Afterwards, to compute (Ó *I F) is done in a polynomial time over the length of Ó. It is common to consider the length of Ó much longer than the length of F. Thus, our proposal of belief revision is in the complexity class PNP[1].