Computational Complexity of Simultaneous Elementary Matching Problems

The purpose of this paper is not to give an overview of the state of art in unification theory. It is intended to be a short introduction into the area of equational unification which should give the reader a feeling for what unification theory might be about. The basic notions such as complete and minimal complete sets of unifiers, and unification types of equational theories are introduced and illustrated by examples. Then we shall describe the original motivations for considering unification (in the empty theory) in resolution theorem proving and term rewriting. Starting with Robinson's first unification algorithm it will be sketched how more efficient unification algorithms can be derived.We shall then explain the reasons which lead to the introduction of unification in non-empty theories into the above mentioned areas theorem proving and term rewriting. For theory unification it makes a difference whether single equations or systems of equations are considered. In addition, one has to be careful with regard to the signature over which the terms of the unification problems can be built. This leads to the distinction between elementary unification, unification with constants, and general unification (where arbitrary free function symbols may occur). Going from elementary unification to general unification is an instance of the so-called combination problem for equational theories which can be formulated as follows: Let E, F be equational theories over disjoint signatures. How can unification algorithms for E, F be combined to a unification algorithm for the theory E ∪ F.KeywordsFunction SymbolEquational TheoryVariable NodeUnification AlgorithmCritical PairThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Haplotype inference is a challenging problem in bioinformatics that consists in inferring the basic genetic constitution of diploid organisms on the basis of their genotype. This piece of information allows researchers to perform association studies for the genetic variants involved in diseases and the individual responses to therapeutic agents. A notable approach to the problem is to encode it as a combinatorial problem (under certain hypotheses, such as the pure parsimony) and to solve it using off-the-shelf combinatorial optimization techniques. In this paper, we present and discuss an approach based on hybridization of two meta-heuristics, one being a population based learning algorithm and the other a local search. We test our approach by solving instances from common Haplotype inference benchmarks. Results show that this approach achieves an improvement on solution quality with respect to the application of a single 'pure' algorithm.

In recent years, the introduction of electric vehicles (EVs) has become increasingly desirable from an environmental perspective. However, it is important to develop suitable charging equipment and infrastructure, as EVs tend to have shorter cruising distances than gasoline-powered vehicles. Various data-driven methods have been proposed for the placement of charging stations (CSs). To date, however, there have been few simulation-driven approaches that can take account of the microscale interactions among vehicles. Therefore, in this paper, we propose an efficient placement method based on multi-agent traffic simulations with an improved scoring function. In addition, as the number of CSs increase in proportion to the spread of EVs, we solve a multiple location design problem formulated as a combinatorial optimization technique. Numerical experiments were conducted on a realistic network in the central part of Wakayama Prefecture in Japan, and it was shown that the proposed method could improve the performance indicator of CS placement compared with the conventional method. Furthermore, the scenario of multiple CS installations at one time demonstrated that good combinations could be obtained.

Quantization, defined as the act of attributing a finite number of grey-levels to an image, is an essential task in image acquisition and coding. It is also intricately linked to various image analysis tasks, such as denoising and segmentation. In this paper, we investigate quantization combined with regularity constraints, a little-studied area which is of interest, in particular, when quantizing in the presence of noise or other acquisition artifacts. We present an optimization approach to the problem involving a novel two-step, iterative, flexible, joint quantizing-regularization method featuring both convex and combinatorial optimization techniques. We show that when using a small number of grey-levels, our approach can yield better quality images in terms of SNR, with lower entropy, than conventional optimal quantization methods.

Central to the functioning of any living cell, the metabolic network is a complex network of biochemical reactions. It may also be viewed as an elaborate production system, integrating a diversity of internal and external signals in order to efficiently produce the energy and the biochemical precursors to ensure all cellular functions. Even in simple organisms like bacteria, it shows a striking level of coordination, adapting to very different growth media. Constraint-based models constitute an efficient mathematical framework to compute optimal metabolic configurations, at the scale of a whole genome. Combining the constraint-based approach "Resource Balance Analysis" with combinatorial optimization techniques, we propose a general method to explore these configurations, based on the inference of logical rules governing the activation of metabolic fluxes in response to diverse extracellular media. Using the concept of partial Boolean functions, we notably introduce a novel tractable algorithm to infer monotone Boolean functions on a minimal support. Monotonicity seems particularly relevant in this context, since the orderliness exhibited by the metabolic network's dynamical behavior is expected to give rise to relatively simple rules. First results are promising, as the application of the method on Bacillus subtilis central carbon metabolism allows to recover known regulations as well as to investigate lesser known parts of the global regulatory network.

Database query optimization is a hard research problem. Exhaustive techniques are adequate for trivial instances only, while combinatorial optimization techniques are vulnerable to the peculiarities of specific instances. We propose a model based on genetic programming to address this problem, motivated by its robustness and efficiency in a wide area of search problems. We adapt the genetic programming paradigm to the requirements of the query optimization problem, showing that the nature of the problem makes genetic programming a particularly attractive approach to it.

Model checking is the process of verifying whether a model of a concurrent system satisfies a specified temporal property. Symbolic algorithms based on Binary Decision Diagrams (BDDs) have significantly increased the size of the models that can be verified. The main problem in symbolic model checking is the image computation problem, i.e., efficiently computing the successors or predecessors of a set of states. This paper is an in-depth study of the image computation problem. We analyze and evaluate several newheuristics, metrics, and algorithms for this problem. The algorithms use combinatorial optimization techniques such as hill climbing, simulated annealing, and ordering by recursive partitioning to obtain better results than was previously the case. Theoretical analysis and systematic experimentation are used to evaluate the algorithms.

A combinatorial optimization technique has been developed and applied to the scheduling problem in data path synthesis. The cost function is minimized using a gradient-like method, and constraints are satisfied by a new technique analogous to the combination of the penalty method and the feasible direction method used for nonlinear optimization. To overcome the drawbacks of assigning operations to control steps one at a time, this technique assigns all the operations to control steps simultaneously. Experimental results show that this method is as good as or better than other published methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Possibilities of the Cross-Entropy Method Usage in the Control Theory

An algebra is said to be polynomially n−dense if all equational theories extending the equational theory of the algebra with constants have a relative base consisting of equations in no more than n variables. In this paper, we investigate polynomial density of commutative semigroups. In particular, we prove that, for n &gt; 1, a commutative semigroup is (n − 1)-dense if and only if its subsemigroup consisting of all n−factor-products is either a monoid or a union of groups of a bounded order. Moreover, a commutative semigroup is 0-dense if and only if it is a bounded semilattice. For semilattices, we give a full description of the corresponding lattices of equational theories.

This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-T composition, where T is a continuous triangular norm. When such a system is consistent, although we know that the solution set can be characterized by a maximum solution and finitely many minimal solutions, it is still a challenging task to find all minimal solutions in an efficient manner. Using the representation theorem of continuous triangular norms, we show that the systems of sup-T equations can be divided into two categories depending on the involved triangular norm. When the triangular norm is Archimedean, the minimal solutions correspond one-to-one to the irredundant coverings of a set covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of a set covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-T equations can be reduced into a 0---1 integer programming problem in polynomial time. This work generalizes most, if not all, known results and provides a unified framework to deal with the problem of resolution and optimization of a system of sup-T equations. Further generalizations and related issues are also included for discussion.

An efficient multilayer general area router, named V4, for MCM and dense PCB designs is presented. The unique feature of the V4 router is that it uses no more than four vias to route every net and yet produces high quality routing solutions. A number of combinatorial optimization techniques are used in the router to produce high quality routing solutions in polynomial time. As a result, it is independent of net ordering, runs much faster, and has far less memory requirement compared to other multilayer general area routers. The router was tested on several examples, including two industrial MCM designs. Compared with the 3-D maze router, on average the V4 router uses 2% less wire length, 31% fewer vias, and runs 26 times faster. Compared with the SLICE router, on average the V4 router uses 4% less wire length and runs 4.6 times faster. >

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p ∈ P is seen by at least one guard g∈ G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃ ℝ, which has been studied earlier by other communities. The class ∃ ℝ consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP ⊆ ∃ ℝ. We prove that the art gallery problem is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃ ℝ. As an illustration of our techniques, we can show that for every compact semi-algebraic set S⊂ [0,1]2 there exists a polygon with rational coordinates that enforces one of the guards to be at any position p∈ S, in any optimal guarding. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. In the ∃ ℝ-hardness proof for the art gallery problem we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it can be used to obtain ∃ ℝ-hardness proofs for other problems. In particular, ETR-INV has been very recently used to prove ∃ ℝ-hardness of other geometric problems.

In this paper, we study the efficiency (both theoretically and computationally) of a class of valid inequalities for the minimum weighted elementary directed cycle problem (MWEDCP) in planar digraphs with negative weight elementary directed cycles. These valid inequalities are called cycle valid inequalities and are parametrized by an integer called inequality’s order. From a theoretical point of view, we prove that separating cycle valid inequalities of order 1 in planar digraph can be done in polynomial time. From a computational point of view, we present a cutting plane algorithm featuring the efficiency of a lifted form of the cycle valid inequalities of order 1. In addition to these lifted valid inequalities, our algorithm is also based on a mixed integer linear formulation of the MWEDCP. The computational results are carried out on randomly generated planar digraph instances of the MWEDCP. For all 29 instances considered, we obtain in average 26.47% gap improvement. Moreover, for some of our instances the strengthening process directly displays the optimal integer elementary directed cycle.

Deciding knowledge in security protocols under equational theories