Exact Algorithm

Analysis of different combinatorial search algorithms has shown that they have a set of distinctive features in common. The paper suggests a number of reusable blocks that support these features and provide high-level design of combinatorial accelerators.

An edge dominating set in a graph G = (V, E) is a subset of the edges D ⊆ E such that every edge in E is adjacent or equal to some edge in D. The problem of finding an edge dominating set of minimum cardinality is NP-hard. We present a faster exact exponential time algorithm for this problem. Our algorithm uses O(1.3226 n ) time and polynomial space. The algorithm combines an enumeration approach of minimal vertex covers in the input graph with the branch and reduce paradigm. Its time bound is obtained using the measure and conquer technique. The algorithm is obtained by starting with a slower algorithm which is refined stepwise. In each of these refinement steps, the worst cases in the measure and conquer analysis of the current algorithm are reconsidered and a new branching strategy is proposed on one of these worst cases. In this way a series of algorithms appears, each one slightly faster than the previous, ending in the O(1.3226 n ) time algorithm. For each algorithm in the series, we also give a lower bound on its running time. We also show that the related problems: minimum weight edge dominating set, minimum maximal matching and minimum weight maximal matching can be solved in O(1.3226 n ) time and polynomial space using modifications of the algorithm for edge dominating set. In addition we consider the matrix dominating set problem which we solve in O(1.3226 n+m ) time and polynomial space, and the parametrised minimum weight maximal matching problem for which we obtain an O * (2.4178 k ) time algorithm.

Many application problems in networks can be modeled as optimization problems in a graph G. For example, the maximum path coloring (Max-PC) problem, described next is an abstract model for many routing problems: Given a set P of paths in G and k colors, find a maximum subset of P and assign a color to each path of the subset such that the paths with the same color are edge-disjoint. One approach for solving an optimization problem in G is to decompose G into subgraphs, find partial solutions in each subgraph and combine the partial solutions into a solution of the problem. A carving-decomposition (branch-decomposition) of G is a system of edge-cuts (vertex-cuts) which decomposes G into subgraphs with each edge (vertex) a minimal subgraph. We give a carving-decomposition based exact algorithm and 1.58-approximation algorithm for the Max-PC problem. Let L be the maximum number of paths in P on any edge of G and let γ be the maximum cardinality of any edge-cut in a given carving-decomposition. Our exact algorithm and approximation algorithm run in O((L + 1) 1.5kγ n 2 ) and O((L + 1) 1.5γ kn 2 ) time, respectively. Our computational study shows that the exact algorithm can solve the Max-PC problem for small k and γ in a practical time and the approximation algorithm gives solutions close to optimal ones for practical values of k and L, and small γ. We also conduct a computational study on a branch-decomposition based exact algorithm for the maximum edge degreebounded connected subgraph (MEDBCS) problem. Our result shows that the MEDBCS problem of vertex-degree bounded by 3 can be solved for graphs with small branchwidth in a practical time.

Facility location problems are often encountered in many areas such as distribution, transportation and telecommunication. We describe a new solution approach for the capacitated facility location problem in which each customer is served by a single facility. An important class of heuristic solution methods for these problems are Lagrangian heuristics which have been shown to produce high quality solutions and at the same time be quite robust. A primal heuristic, based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satis®ed, is incorporated into the Lagrangian heuristic. Finally, a branch-and-bound method, based on the Lagrangian heuristic is developed, and compared computationally to the commercial code CPLEX. The computational results indicate that the proposed method is very ecient.

Una entre las metaheurísticas más exitosas que aparecieron en los últimos aos del siglo pasado es GRASP, un método multi-arranque diseñado para resolver problemas difíciles en optimización combinatoria. En su versión básica cada iteración consiste en dos fases: una fase constructiva cuyo producto es una solución factible buena, aunque no necesariamente un óptimo local, y una búsqueda local, durante la cual se examinan vecindades de la solución, al llegar a un óptimo local la iteración termina. La iteraciones continúan, guardando la mejor solución encontrada en cada una de ellas, hasta que se alcanza un criterio de terminación. En este capítulo se revisan las componentes básicas de GRASP, junto con algunas estrategias para el ajuste de parámetros como es GRASP reactivo. De aquí sigue una discusión acerca del uso de reencadenamiento de trayectorias como una forma de introducir aspectos de memoria dentro de la búsqueda local. El capítulo concluye con una descripción de GRASP en paralelo, con una extensa experimentación para comparar implementaciones independientes contra cooperativas.

Facility location problems are often encountered in many areas such as distribution, transportation and telecommunication. We describe a new solution approach for the capacitated facility location problem in which each customer is served by a single facility. An important class of heuristic solution methods for these problems are Lagrangian heuristics which have been shown to produce high quality solutions and at the same time be quite robust. A primal heuristic, based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satis®ed, is incorporated into the Lagrangian heuristic. Finally, a branch-and-bound method, based on the Lagrangian heuristic is developed, and compared computationally to the commercial code CPLEX. The computational results indicate that the proposed method is very ecient.

Consider a website containing a collection of webpages with data such as in Yahoo or the Open Directory project. Each page is associated with a weight representing the frequency with which that page is accessed by users. In the tree hierarchy representation, accessing each page requires the user to travel along the path leading to it from the root. By enhancing the index tree with additional edges (hotlinks) one may reduce the access cost of the system. In other words, the hotlinks reduce the expected number of steps needed to reach a leaf page from the tree root, assuming that the user knows which hotlinks to take. The hotlink enhancement problem involves finding a set of hotlinks minimizing this cost. This article proposes the first exact algorithm for the hotlink enhancement problem. This algorithm runs in polynomial time for trees with logarithmic depth. Experiments conducted with real data show that significant improvement in the expected number of accesses per search can be ac...

Given a set of graphs, the median graph is defined as the graph which has the smallest sum of distances (SOD) to all the graphs in the set. It has been proposed as a tool to obtain the representative of such a set. In spite of its potential applications, the existing algorithms are computationally complex and have a very limited applicability. In this paper we propose a new approach for the exact computation of the median graph based on graph embedding in vector spaces. Graphs are embedded into a vector space and the median is computed in the vector domain. After that, the median graph is recovered from this median vector. Experiments on a synthetic database show that our approach outperforms the previous existing exact algorithms both on computation time and number of SOD computations.

Median graphs have been presented as a useful tool for capturing the essential information of a set of graphs. Nevertheless, computation of optimal solutions is a very hard problem. In this work we present a new and more efficient optimal algorithm for the median graph computation. With the use of a particular cost function that permits the definition of the graph edit distance in terms of the maximum common subgraph, and a prediction function in the backtracking algorithm, we reduce the size of the search space, avoiding the evaluation of a great amount of states and still obtaining the exact median. We present a set of experiments comparing our new algorithm against the previous existing exact algorithm using synthetic data. In addition, we present the first application of the exact median graph computation to real data and we compare the results against an approximate algorithm based on genetic search. These experimental results show that our algorithm outperforms the previous existing exact algorithm and in addition show the potential applicability of the exact solutions to real problems.

This paper investigates the budget variant of the discrete time/cost trade-off problem (DTCTP). This multi-mode project scheduling problem requires assigning modes to the activities of a project so that the total completion time is minimized and the budget and the precedence constraints are satisfied. This problem is often encountered in practice as timely completion of the projects without exceeding the budget is crucial. The contribution of this paper to the literatures is to describe an effective Benders Decomposition-based exact algorithm to solve the DTCTP instances of realistic sizes. Although Benders Decomposition often exhibits a very slow convergence, we have included several algorithmic features to enhance the performance of the proposed tailored approach. Computational results attest to the efficacy of the proposed algorithm, which can solve large-scale instances to optimality.

In this paper we consider the Project Scheduling Problem with resource constraints, where the objective is to minimize the project makespan. We present a new 0-1 linear programming formulation of the problem that requires an exponential number of variables, corresponding to all feasible subsets of activities that can be simultaneously executed without violating resource or precedence constraints. Different relaxations of the above formulation are used to derive new lower bounds, which dominate the value of the longest path on the precedence graph and are tighter than the bound proposed by Stinson et al. (1978). A tree search algorithm, based on the above formulation, that uses new lower bounds and dominance criteria is also presented. Computational results indicate that the exact algorithm can solve hard instances that cannot be solved by the best algorithms reported in the literature.

A matrix A = [aij] of nonnegative integers must be partitioned into p blocks (submatrices) corresponding to a set of vertical cuts parallel to the columns and a set of horizontal cuts parallel to the rows. With each block is associated a cost equal to the sum of its elements. We consider the problem of finding a matrix partitioning that minimizes the cost of the block of maximum cost. In this paper a mathematical formulation of the problem is given and used to derive both exact and heuristic algorithms. Lower bounds and dominance criteria are exploited in a tree search algorithm for finding the optimal solution of the problem. Computational results of the proposed algorithm are given on a number of randomly generated test problems.

Traditionally, practical clique algorithms have been compared based on their performance on various random graphs. We propose a new testing methodology which permits testing to be completed in a fraction of the time required by previous methods. In addition, the range of testing can be extended to include problems that could not be attempted in the past because of being too time consuming. We accomplish this by applying the approach that makes dynamic programming a very e ective algorithmic technique: we use tabulated estimates for the time required to solve subproblems rather than timing each exactly. Our computational experiments validate this approach. The next step is to use this workbench to develop fast new algorithms for the maximum clique problem. A mixed algorithm is an algorithm which applies di erent strategies for the subproblems that arise. Using the dynamic programming approach again for timing, we mechanize the process of developing table driven algorithms which apply the best possible strategy to each subproblem. We demonstrate the utility of this approach by creating new algorithms for random graphs with varying densities. Our resulting clique algorithms are extremely simple, yet signi cantly faster than the best approach found in the literature to date.

Since gluons in QCD are interacting fundamental constituents just as quarks are, we expect that in addition to mesons made from a quark and an antiquark, there should also be glueballs and hybrids (bound states of quarks, antiquarks and gluons). In general, these states would mix strongly with the conventional qq mesons. However, they can also have exotic quantum numbers inaccessible to qq mesons. Confirmation of such states would give information on the role of "dynamical" color in low energy QCD. In the quenched approximation we present a lattice calculation of the masses of mesons with exotic quantum numbers. These hybrid mesons can mix with four quark (q qqq) states. The quenched approximation partially suppresses this mixing. Nonetheless, our hybrid interpolating fields also couple to four quark states. Using a four quark source operator, we demonstrate this mixing for the 1 -+ meson. Using the conventional Wilson quark action, we calculate both at reasonably light quark masses, intending to extrapolate to small quark mass, and near the charmed quark mass, where we calculate the masses of some ccg hybrid mesons. The hybrid meson masses are large -over 4 GeV for charmonium and more than twice the vector meson mass at our smallest quark mass, which is near the strange quark mass.

We present an exact algorithm, based on techniques from the field of Model Checking, for finding control policies for Boolean networks (BN) with control nodes. Given a BN, a set of starting states, I, a set of goal states, F , and a target time, t, our algorithm automatically finds a sequence of control signals that deterministically drives the BN from I to F at, or before time t, or else guarantees that no such policy exists. Despite recent hardness-results for finding control policies for BNs, we show that, in practice, our algorithm runs in seconds to minutes on over 13,400 BNs of varying sizes and topologies, including a BN model of embryogenesis in D. melanogaster with 15,360 Boolean variables. We then extend our method to automatically identify a set of Boolean transfer functions that reproduce the qualitative behavior of gene regulatory networks. Specifically, we automatically (re)learn a BN model of D. melanogaster embryogenesis in 5.3 seconds, from a space containing 6.9 × 10 10 possible models.

We present algorithms for finding control strategies in Boolean Networks (BN). Our approach uses symbolic techniques from the field of model checking. We show that despite recent hardness-results for finding control policies, a model checking-based approach is often capable of scaling to extremely large and complex models. We demonstrate the effectiveness of our approach by applying it to a BN model of embryogenesis in D. melanogaster with 15,360 Boolean variables.

We present a framework for solving some types of 0 -1 multi-stage scheduling/planning problems under uncertainty in the objective function coefficients and the right-hand-side. A scenario analysis scheme with full recourse is used. The solution offered for each scenario group at each stage takes into account all scenarios but without subordinating to any of them. The constraints are modelled by a splitting variables representation via scenarios. So, a 0 -1 model for each scenario is considered plus the non-anticipativity constraints that equate the 0 -1 variables from the same group of scenarios in each stage. The mathematical representation of the model is very amenable for the proposed framework to deal with the 0 -1 character of the variables. A Branch-and-Fix Coordination approach is introduced for coordinating the selection of the branching nodes and branching variables in the scenario subproblems to be jointly optimized. Some computational experience is reported for different types of problems.

This brief addresses the problem of implementing very large constant multiplications by a single variable under the shift-adds architecture using a minimum number of adders/subtractors. Due to the intrinsic complexity of the problem, we introduce an approximate algorithm, called TÕLL, which partitions the very large constants into smaller ones. To reduce the number of operations, TÕLL incorporates graph-based and common subexpression elimination methods proposed for the shift-adds design of constant multiplications. It can also consider the delay of a multiplierless design defined in terms of the maximum number of operations in series, i.e., the number of adder-steps, while reducing the number of operations. High-level experimental results show that the adder-steps of a shift-adds design can be reduced significantly with a little overhead in the number of operations. Gate-level experimental results indicate that while the shift-adds design can lead to a 36.6% reduction in gate-level area with respect to a design using a multiplier, the delay-aware optimization can yield a 48.3% reduction in minimum achievable delay of the shift-adds design when compared to the area-aware optimization.

This article addresses the multiplication of one data sample with multiple constants using addition/subtraction and shift operations, i.e., the multiple constant multiplications (MCM) operation. In the last two decades, many efficient algorithms have been proposed to implement the MCM operation using the fewest number of addition and subtraction operations. However, due to the NP-hardness of the problem, almost all the existing algorithms have been heuristics. The main contribution of this article is the proposal of an exact depth-first search algorithm that, using lower and upper bound values of the search space for the MCM problem instance, finds the minimum solution consuming less computational resources than the previously proposed exact breadth-first search algorithm. We start by describing the exact breadth-first search algorithm that can be applied on real mid-size instances. We also present our recently proposed approximate algorithm that finds solutions close to the minimum and is able to compute better bounds for the MCM problem. The experimental results clearly indicate that the exact depth-first search algorithm can be efficiently applied to large size hard instances that the exact breadth-first search algorithm cannot handle and the heuristics can only find suboptimal solutions.

This paper addresses the multiplication of one data sample with multiple constants using addition/subtraction and shift operations, i.e., the multiple constant multiplications (MCM) problem. The MCM problem finds itself and its variants in many applications, such as digital finite impulse response (FIR) filters, linear signal transforms, and computer arithmetic. Although many efficient algorithms have been proposed to implement the MCM using the fewest number of operations, due to the NP-hardness of the problem, they have been heuristics, i.e., they cannot guarantee the minimum solution. In this work, we propose an exact algorithm based on the breadth-first search that finds the minimum number of operations solution of midsize MCM instances in a reasonable time. The proposed exact algorithm has been tested on a set of instances including FIR filter and randomly generated instances, and compared with the previously proposed efficient heuristics. It is observed from the experimental results that, even though the previously proposed heuristics obtain similar results with the minimum number of operations solutions, there are instances for which the exact algorithm finds better solutions than the prominent heuristics.

The last two decades have seen many efficient algorithms and architectures for the design of low-complexity bit-parallel Multiple Constant Multiplications (MCM) operation, that dominates the complexity of Digital Signal Processing (DSP) systems. On the other hand, digit-serial architectures offer alternative low-complexity designs, since digit-serial operators occupy less area and are independent of the data wordlength. This paper introduces the problem of designing a digit-serial MCM operation with minimal area at gate-level and presents the exact formalization of the area optimization problem as a 0-1 Integer Linear Programming (ILP) problem. Experimental results show the efficiency of the proposed algorithm and digitserial MCM designs in terms of area at gate-level.

Existing optimization algorithms for the multiplierless realization of multiple constant multiplications (MCM) typically target the minimization of the number of addition and subtraction operations. Since power dissipation is directly related to the amount of hardware, some power reduction is indirectly achieved by these algorithms. However, in many cases, glitching plays an equally important role in defining the power consumption. This is specially true for arithmetic circuits, and in particular to MCM due to high logic depth and large number of re-convergent paths. This paper introduces exact algorithms that search the optimal area of an MCM design at gate-level where each constant multiplication is implemented in its minimum depth. Experimental results show that the proposed algorithms lead to MCM designs consuming significantly less power with respect to those obtained by the MCM algorithms.

In this paper, we propose an exact algorithm for the problem of area optimization under a delay constraint in the synthesis of multiplierless FIR filters. To the best of our knowledge, the method presented in this paper is the only exact algorithm designed for this problem. We present the results of the algorithm on real-sized filter instances and compare with an improved version of a recently proposed exact algorithm designed for the minimization of area. We show that in many cases delay can be minimized without any area penalty. Additionally, we describe two approximate algorithms that can be applied to instances which cannot be solved, or take too long, with the exact algorithm. We show that these algorithms find similar solutions to the exact algorithm in less CPU time.

In the last two decades, many efficient algorithms and architectures have been introduced for the design of lowcomplexity bit-parallel multiple constant multiplications (MCM) operation which dominates the complexity of many digital signal processing systems. On the other hand, little attention has been given to the digit-serial MCM design that offers alternative lowcomplexity MCM operations albeit at the cost of an increased delay. In this paper, we address the problem of optimizing the gate-level area in digit-serial MCM designs and introduce highlevel synthesis algorithms, design architectures, and a computeraided design tool. Experimental results show the efficiency of the proposed optimization algorithms and of the digit-serial MCM architectures in the design of digit-serial MCM operations and finite impulse response filters.

This article addresses the problem of finding the fewest numbers of addition and subtraction operations in the multiplication of a constant matrix with an input vector---a fundamental operation in many linear digital signal processing transforms. We first introduce an exact common subexpression elimination (CSE) algorithm that formalizes the minimization of the number of operations as a 0-1 integer linear programming problem. Since there are still instances that the proposed exact algorithm cannot handle due to the NP-completeness of the problem, we also introduce a CSE heuristic algorithm that iteratively finds the most common 2-term subexpressions with the minimum conflicts among the expressions. Furthermore, since the main drawback of CSE algorithms is their dependency on a particular number representation, we propose a hybrid algorithm that initially finds promising realizations of linear transforms using a numerical difference method, and then applies the proposed CSE algorithm...

We consider a parallel-machine scheduling problem with a learning effect and the makespan objective. The impact of the learning effect on job processing times is modelled by the general DeJong's learning curve. For this NP-hard problem we propose two exact algorithms: a sequential branch-and-bound algorithm and a parallel branch-and-bound algorithm. We also present the results of experimental evaluation of these algorithms on a computational cluster. Finally, we use the exact algorithms to estimate the performance of two greedy heuristic scheduling algorithms for the problem.

Zhang, K., R. Statman and D. Shasha, On the editing distance between unordered labeled trees, Information Processing Letters 42 (1992) 133-139. This paper considers the problem of computing the editing distance between unordered, labeled trees. We give efficient polynomial-time algorithms for the case when one tree is a string or has a bounded number of leaves. By contrast, we show that the problem is NP-complete even for binary trees having a label alphabet of size two.

Range and k-nearest neighbor searching are core problems in pattern recognition. Given a database S of objects in a metric space M and a query object q in M , in a range searching problem the target is to find the objects of S within some threshold distance to q, whereas in a k-nearest neighbor searching problem, the k elements of S closest to q must be produced. These problems can obviously be solved with a linear number of distance calculations, by comparing the query object against every object in the database. However, the goal is to solve such problems much faster. We combine and extend ideas from the M-Tree, the Multi-Vantage Point structure, and the FQ-Tree to create a new structure in the "bisector tree" class, called the Antipole Tree. Bisection is based on the proximity to an "Antipole" pair of elements generated by a suitable linear randomized tournament. The final winners a, b of such a tournament are far enough apart to approximate the diameter of the splitting set. If dist(a, b) is larger than the chosen cluster diameter threshold, then the cluster is split.

None of the available minimizers for 2-level hazard-free l o gic minimization can synthesize very large circuits. This limitation has forced researchers to resort to manual and automated circuit partitioning techniques. This paper introduces two new 2-level logic minimizers: Espresso-HF, a heuristic method which is loosely based on Espresso-II, and Impymin, an exact method based on implicit data structures. Both minimizers can solve all currently available examples, which range up to 32 inputs and 33 outputs. These include examples that have never been solved before. For examples that can be solved by other minimizers our methods are several orders of magnitude faster. As by-products of these algorithms, we also present two additional results. First, we introduce a fast new algorithm to check if a hazard-free covering problem can feasibly be solved. Second, we introduce a novel formulation of the 2-level hazard-free logic minimization problem by capturing hazard-freedom constraints within a synchronous function by adding new variables.

This article develops a robust, exact algorithm for the maximal covering problem (MCP) using dual-based solution methods and greedy heuristics in branch and bound. Based on tests using randomly generated problems with problem parameters similar to those in the existing literature, the hybrid approach developed in this work appears to be effective over a wide range of MCP model parameters. The method is further validated on problems constructed from three real-world data sets. The extensive computational study compares the new method with other existing exact methods using problems that are as big, or larger than, those used in previous work on MCP. The results show that the proposed method is effective in most instances of MCP. In particular, it is shown that bounding schemes using Lagrangian relaxation are effective on MCP as a method of obtaining both exact and heuristic solutions.

In this paper we present a survey of results concerning algorithms, complexity, and applications of the maximum clique problem. We discuss enumerative and exact algorithms, heuristics, and a variety of other proposed methods. An up to date bibliography on the maximum clique and related problems is also provided.

This article reports an experimental study on a given structural property of connectedness of optimal solutions for two variants of the bicriteria knapsack problem. A local search algorithm that explores this property is then proposed and its performance is compared against exact algorithms in terms of running time and number of optimal solutions found. The experimental results indicate that this simple local search algorithm is able to find a representative set of optimal solutions in most of the cases, and in much less time than exact approaches.

Systematic conservation planning applications based solely on the presence/absence of a large number of species are not sufficient to guarantee their persistence in highly fragmented landscapes. Recent developments have thus incorporated much desired spatial design considerations, and reserve-network connectivity has received increased attention. Nonetheless, connectivity is often determined without regard to species-specific responses to habitat fragmentation. But species differ in their dispersal ability and habitat requirements, making proximate priority areas necessary for some species, while undesirable for others. We present a novel approach that incorporates species-specific connectivity needs in reserve-network design. Importantly, our method differs from previous approaches in that connectivity is not part of the objective function, but part of the constraints, thus avoiding typical undesirable trade-offs that may result in high connectivity for some species but null connectivity for others. We use graphs to describe the dispersal pattern of each species and our goal is to iden-

We give experimental and theoretical results on the problem of computing the treewidth of a graph by exact exponential time algorithms using exponential space or using only polynomial space. We first report on an implementation of a dynamic programming algorithm for computing the treewidth of a graph with running time O * (2 n ). This algorithm is based on the old dynamic programming method introduced by Held and Karp for the Traveling Salesman problem. We use some optimizations that do not affect the worst case running time but improve on the running time on actual instances and can be seen to be practical for small instances. However, our experiments show that the space used by the algorithm is an important factor to what input sizes the algorithm is effective. For this purpose, we settle the problem of computing treewidth under the restriction that the space used is only polynomial. In this direction we give a simple O * (4 n ) algorithm that requires polynomial space. We also show that with a more complicated algorithm, using balanced separators, Treewidth can be computed in O * (2.9512 n ) time and polynomial space.

The field of mathematics plays a vital role in various fields. One of the most important areas in mathematics is Graph theory. It is a sophisticated mathematical framework for studying the properties and connections of graphs that allows data to be imaged and helps to simplify the complex structure of data with sets of vertices and edges. The significant advances have been made in graph theory and its applications in recent years, leading to novel techniques and real-world implementations. This summary provides an overview of these recent developments. The emergence of Graphical Neural Networks is a significant development that enables them to learn graph-structured data representations. The GNNs have been successfully applied in various domains, including node classification and link prediction. The community detection algorithms have also improved accuracy and efficiency. It will benefit social network analysis and other related fields. Graph embedding techniques have also progressed, facilitating learning low-dimensional vector representations for nodes or subgraphs. This advancement has enhanced tasks such as link prediction and node classification. Furthermore, the temporal graph analysis has gained attention for studying dynamic graphs, allowing for predicting future states and anomaly detection. The application of graph theory in social networks has yielded valuable insights into sentiment dynamics, influence maximization, and information diffusion. These findings aid to understand the social behavior's and designing effective interventions. Network alignment and graph matching techniques have been developed to integrate and analyze data from multiple sources. The finding applications in diverse fields, including biological networks and crossdomain data analysis. Graph theory has significantly contributed to transportation and urban planning, addressing challenges like traffic flow optimization, route planning, and public transportation design. Furthermore, the application of graph theory in bioinformatics and drug discovery has led to advancements in drug-target interaction and protein function prediction. Recent advances in graph theory have opened up new avenues for research and practical applications. The growing availability of large-scale Graph data and the need to extract insights from complex interconnected systems continue the evolution of this field. These developments have great potential for addressing real-world challenges in various domains.

Motivation: Multiple sequence alignment is an important tool in computational biology. In order to solve the task of computing multiple alignments in affordable time, the most commonly used multiple alignment methods have to use heuristics. Nevertheless, the computation of optimal multiple alignments is important in its own right, and it provides a means of evaluating heuristic approaches or serves as a subprocedure of heuristic alignment methods. Results: We present an algorithm that uses the divide-and-conquer alignment approach together with recent results on search space reduction to speed up the computation of multiple sequence alignments. The method is adaptive in that depending on the time one wants to spend on the alignment, a better, up to optimal alignment can be obtained. To speed up the computation in the optimal alignment step, we apply the \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyl...

A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1s on each row are consecutive. These matrices are used for DNA physical mapping and ancestral genome reconstruction in computational biology on the other hand they represents a class of convex bipartite graphs and are of interest of algorithm graph theory researchers. Tucker gave a forbidden submartices characterization of matrices that have C1P property in 1972. Booth and Lucker (1976) gave a first linear time recognition algorithm for matrices with C1P property and then in 2002, Habib, et al. gave a simpler linear time recognition algorithm. There has been substantial amount of works on efficiently finding minimum size forbidden submatrix. Our algorithm is at least n times faster than the existing algorithm where n is the number of columns of the input matrix.

A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1s on each row are consecutive. These matrices are used for DNA physical mapping and ancestral genome reconstruction in computational biology on the other hand they represents a class of convex bipartite graphs and are of interest of algorithm graph theory researchers. Tucker gave a forbidden submartices characterization of matrices that have C1P property in 1972. Booth and Lucker (1976) gave a first linear time recognition algorithm for matrices with C1P property and then in 2002, Habib, et al. gave a simpler linear time recognition algorithm. There has been substantial amount of works on efficiently finding minimum size forbidden submatrix. Our algorithm is at least n times faster than the existing algorithm where n is the number of columns of the input matrix.

In this paper, we consider the problem of maximizing total tardiness on a single machine, where the first job starts at time zero and idle times between the processing of jobs are not allowed. We present a modification of an exact pseudo-polynomial algorithm based on a graphical approach, which has a polynomial running time.

In this paper, we consider two scheduling problems on a single machine, where a specific objective function has to be maximized in contrast to usual minimization problems. We propose exact algorithms for the single machine problem of maximizing total tardiness 1|| max T j and for the problem of maximizing the number of tardy jobs 1|| max U j . In both cases, it is assumed that the processing of the first job starts at time zero and there is no idle time between the jobs. We show that problem 1|| max U j is polynomially solvable. For several special cases of problem 1|| max T j , we present exact polynomial algorithms. Moreover, we give an exact pseudo-polynomial algorithm for the general case of the latter problem and an alternative exact algorithm.

In this paper, we consider a graphical realization of dynamic programming. The concept is discussed on the partition and knapsack problems. In contrast to dynamic programming, the new algorithm can also treat problems with non-integer data without necessary transformations of the corresponding problem. We compare the proposed method with existing algorithms for these problems on small-size instances of the partition problem with n ≤ 10 numbers. For almost all instances, the new algorithm considers on average substantially less ''stages'' than the dynamic programming algorithm.

In this paper, we consider two scheduling problems on a single machine, where a specific objective function has to be maximized in contrast to usual minimization problems. We propose exact algorithms for the single machine problem of maximizing total tardiness 1|| max T j and for the problem of maximizing the number of tardy jobs 1|| max U j . In both cases, it is assumed that the processing of the first job starts at time zero and there is no idle time between the jobs. We show that problem 1|| max U j is polynomially solvable. For several special cases of problem 1|| max T j , we present exact polynomial algorithms. Moreover, we give an exact pseudo-polynomial algorithm for the general case of the latter problem and an alternative exact algorithm.

This report addresses the problem of scheduling for real-time systems that include both hard and soft tasks. In order to capture the relative importance of soft tasks and how the quality of results is affected when missing a soft deadline, we use utility functions associated to soft tasks. Thus the aim is to find the execution order of tasks that makes the total utility maximum and guarantees hard deadlines. We consider intervals rather than fixed execution times for tasks. Since a purely off-line solution is too pessimistic and a purely on-line approach incurs an unacceptable overhead due to the high complexity of the problem, we propose a quasi-static approach where a number of schedules are prepared at design-time and the decision of which of them to follow is taken at run-time based on the actual execution times. We propose an exact algorithm as well as different heuristics for the problem addressed in this report.

We use simple argwnents from Galois theory to prove the impossibility of exact algorilhms for problems under various models of computation. In particular we show that lhere exist applied computational problems for which there are no closed form solutions over models such as Q(+,", .,/, v), Q (+, _, '10, /, tv), and Q(+, -, '1<,/, k...J, q(x», where Q is the field of rationals and q(x)eQ[x] are polynomials with non-solvable Galois groups.

Variable neighborhood search (VNS) is a recent metaheuristic for solving combinatorial and global optimization problems whose basic idea is systematic change of neighborhood within a local search. In this survey paper we present basic rules of VNS and some of its extensions. Moreover, applications are briefly summarized. They comprise heuristic solution of a variety of optimization problems, ways to accelerate exact algorithms and to analyze heuristic solution processes, as well as computer-assisted discovery of conjectures in graph theory. Résumé La Recherche à voisinage variable (RVV) est une métaheuristique récente pour la résolution de problèmes d'optimisation combinatoire et globale, dont l'idée de base est le changement systématique de voisinage au sein d'une recherche locale. Dans ce chaptre, nous présentons les règles de base de RVV et de certaines de ses extensions. De plus, des applications sont brièvement résumées. Elles comprennent la résolution approchée d'un ensemble de problèmes d'optimisation, des manières d'accélerer les algorithmes exacts et d'analyser le processus de résolution des heuristiques ainsi qu'un système automatique de découverte de conjectures en théorie des graphes. S, X, x and f are solution space, feasible set, feasible solution and real valued function, respectively. If S is a finite but large set a combinatorial optimization problem is defined.

Task allocation over distributed real time system, for parallel applications, is a vital segment, where policy for task allocation should be chosen in very appropriate manner. By efficient allocation of the tasks, the throughput and the overall processor utilization can be maximized. Task allocation is NP-hard or NP-complete problem. To improve the performance of the system, a new heuristic has been suggested and implemented in this paper. The number of modules which can be assigned on the processor is limited and the memory is also having certain limit. So these two constraints have been taken into consideration in the algorithm discussed in this paper. The dynamic load sharing policy has been used to improve the performance i.e. at the time of assignment, the required constraints must be check and fulfilled. For clustering task k-mean clustering is used and the proposed model is implemented in matlab.