Parameterized algorithmics for linear arrangement problems

In this paper we study a version of the Multidimensional Arrangement Problem (MAP) that embeds a graph into a multidimensional array minimizing the aggregated (Manhattan) distance of the embedded edges. This problem includes the minimum Linear Arrangement Problem (minLA) as a special case, among others. We propose JAM, a tabu-based two-stage simulated annealing heuristic for this problem. Our algorithm relies on existing techniques for the minimum linear arrangement (minLA) problem, which are non-trivially adapted to work in multiple dimensions. Due to the scarcity of specific benchmarks for MAP, we have tested the performance of our algorithm with benchmarks for the minLA and Quadratic Assignment Problems (with more than 80 graphs). For each graph in these benchmarks, we provide results for 1, 2 and 3-dimensional instances of MAP, enlarging, hence, the benchmarking resources for the research community. The results obtained show the practicality of JAM, often matching the best known result and even improving some of them.

Parallel algorithms for the minimum cut and the minimum length tree layout problems

We study the bipartite crossing number problem. When the minimum degree and the maximum degree of the graph are close to each other, we derive two polynomial time approximation algorithms for solving this problem, with approximation factors, O(log2 n), and O(log n log log n), from the optimal, respectively, where n is the number of vertices. This problem had been known to be NP-hard, but no approximation algorithm which could generate a provably good solution was known. An important aspect of our work has been relating this problem to the linear arrangement problem. Indeed using this relationship we also present an O(n 1.6) time algorithm for computing the bipartite crossing number of a tree.We also settle down the problem of computing a largest weighted biplanar subgraph of an acyclic graph by providing a linear time algorithm to it. This problem was known to be NP-hard when graph is planar and very sparse, and all weights are 1.KeywordsBipartite GraphLinear Time AlgorithmMedian PropertyLinear ArrangementPolynomial Time Approximation AlgorithmThese 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.

We study the optimal linear arrangement (OLA) problem on interval graphs. Several linear layout problems that are NP-hard on general graphs are solvable in polynomial time on interval graphs. We prove that, quite surprisingly, optimal linear arrangement of interval graphs is NP-hard. The same result holds for permutation graphs. We present a lower bound and a simple and fast 2-approximation algorithm based on any interval model of the input graph.

Mathematical model construction of complicate physical phenomenon often leads to the setting and solving problems of parameters optimal control in differential equations in partial derivatives. Chosen equation with boundary and initial conditions is usually mathematical model basis of the object, which is under analysis. Optimal control of right-hand side function in non-linear problem for inhomogeneous biharmonic has been investigated. With the help of various gradient methods the problems of parameters control in such equations are solved successfully. Herewith linear problem is solved with the potential method on every step. In this work biharmonic potentials with logarithmic singularity are under consideration. That is why parameters optimization in these problems are conducted together with elimination of their incorrectness. It should be noted that system optimal control problems, described with biharmonic equations in irregular shape region, are studied poorly. Therefore the object of investigation is one of boundary conditions for inhomogeneous biharmonic equation. The circumstance which complicated the problem was irregularity of function domain. Clearly that this problem can be solved with computational mathematics methods. Without precise solution of linear problem, it is impossible to use gradient method, build converging iterative process, and obtain precise solution of optimization problem. Algorithm for linear boundary value problem solution with boundary integral equations overcomes this problem successfully. Physical examples of numerical implementation have been presented, analysis of obtained solutions have been conducted. Their accuracy, algorithm simplicity and time spent evidence about this approach promising for practical results obtaining in plate theory and mathematical physics problems successful numerical solving

Optimization of the interaction distance between qubits to map a quantum circuit into one-dimensional quantum architectures is addressed. The problem is formulated as the Minimum Linear Arrangement (MinLA) problem. To achieve this, an interaction graph is constructed for a given circuit, and multiple instances of the MinLA problem for selected subcircuits of the initial circuit are formulated and solved. In addition, a lookahead technique is applied to improve the cost of the proposed solution which examines different subcircuit candidates. Experiments on quantum circuits for quantum Fourier transform and reversible benchmarks show the effectiveness of the approach.

We study linear and non-linear small divisors problems in analytic and non-analytic regularity. We observe that the Bruno arithmetic condition, which is usually attached to non-linear analytic problems, can also be characterized as the optimal condition to solve the linear problem in some fixed non-quasi-analytic class. Based on this observation, it is natural to conjecture that the optimal arithmetic condition for the linear problem is also optimal for non-linear small divisors problems in any reasonable non-quasi-analytic classes. Our main result proves this conjecture in a representative non-linear problem, which is the linearization of vector fields on the torus, in the most natural non-quasi-analytic class, which is the Gevrey class. The proof follows Moser’s argument of approximation by analytic functions, and uses works of Popov, Rüssmann and Pöschel in an essential way.

In this paper, we propose Simplified Space based Neural Architecture Search (SSNAS), an efficient approach for automatic architecture search. Inspired by the popular convolutional neural networks which have strong capability for feature extraction with small convolutional kernels, we design a simplified search space of convolutional operations with small kernel and construct a large model based on it. Furthermore, we use an Long Short-Term Memory (LSTM) to sample child model from the large model in a way of selective activation. The probability distribution of selective activation is obtained by training the LSTM with reinforcement learning for maximizing the expected reward of selected child model on the validation set. Moreover, the trained weights are saved in a large model which concludes all child models. Each weight of new sampled child model will be restored when being selected by another one again instead of training from scratch so that SSNAS can greatly reduce the expensive computation than traditional approaches for automatic model design. Extensive experiments on CIFAR-10, ImageNet and VOC show that the proposed approach excels in discovering transferable architectures with high performance.

When there are several different types of heat-storage ceramic bricks (checkers) that can be arranged in a regenerative combustion system, one must find an optimal arrangement (with the highest long-term Waste Heat Recovery Ratio) of these checkers, possibly of different types, in this regenerative combustion system. However, the number of possible arrangements of checkers in a heat regenerator could be huge. For example, when 5 different types of checkers are available for each of 14 positions in a heat regenerator, the total number of possible arrangements of checkers is 6,103,515,625. It is impractical to completely evaluate the efficiency of each of the 6,103,515,625 arrangements of checkers by 3D CFD simulations on Ansys Fluent. Here, we propose an optimization algorithm by tree search to tackle this optimization problem. This tree search method is motivated by the recent applications of Artificial Intelligence, based on combination of Deep Learning with Monte-Carlo Tree Search, to the incredibly complicated board game Go. Empirical evidence shows that this simple tree search algorithm leads to fast convergence of an optimization search and successfully suggests the optimal arrangement of checkers. This simple tree search method/algorithm may effectively enhance the thermal efficiency of a regenerative combustion system.

We propose a genetic algorithm (GA) for solving the maximization version of the Optimal Linear Arrangement problem and we also demonstrate how solutions found by it can be used for constructing smart initial layouts for force-directed graph drawing. Effectively, we show that our GA can be used as a first step in force-directed graph drawing for achieving more aesthetically pleasing graph layouts at the end. We present experimental results which show that the initial layouts based on the solutions of our GA reduce the number of edge crossings in force-directed graph layouts.

In this paper we first present a sequential linear algorithm for a linear arrangement problem on trees, MINSUMCUT, and then an O(log n)-time parallel algorithm for MINSUMCUT on trees, which uses n2/(logn) processors.

Decorous combinatorial lower bounds for row layout problems

We present a new integer linear formulation using \(O(n^2)\) variables, called adjacency variables, to solve the Minimum Linear Arrangement problem (MinLA). We give a couple of valid equalities and inequalities for this formulation, some of them deriving from on a new general partitioning approach that is not limited to our formulation. We numerically tested the lower bound provided by the linear relaxation using instances of the matrix market library. Our results are compare with the best known lower bounds, in terms of quality, as well computing times.

More than 30 years ago, Shiloach published an algorithm to solve the minimum linear arrangement problem for undirected trees. Here we fix a small error in the original version of the algorithm and discuss its effect on subsequent literature. We also improve some aspects of the notation.

Scatter search (SS) and path relinking (PR) are evolutionary methods that have been successfully applied to a wide range of hard optimization problems. The fundamental concepts and principles of the methods were first proposed in the 1970s and 1980s, and were based on formulations, dating back to the 1960s, for combining decision rules and problem constraints. The methods use strategies for search diversification and intensification that have proved effective in a variety of optimization problems and that have sometimes been embedded in other evolutionary methods to yield improved performance. This paper examines the scatter search and path relinking methodologies from both conceptual and practical points of view, and identifies certain connections between their strategies and those adopted more recently by particle swarm optimization. The authors describe key elements of the SS & PR approaches and apply them to a hard combinatorial optimization problem: the minimum linear arrangement problem, which has been used in applications of structural engineering, VLSI and software testing.