Vertex packing problem application to the design of electronic testing fixtures

In this paper, we report on the use of combinatorial optimization techniques to design testing fixtures for Printed Circuit Boards (PCBs). For testing the functionality of a PCB, nail-like testing devices (probes) on the surface of a testing fixture are brought in contact with prespectified test points (pads) on the surface of the PCB. The two design decisions for the testing fixture are: (a) to select from an available set of pads the ones to test (this determines the location of the probes on the fixture) subject to the restriction that in prespecified subsets of the set of pads (these subsets are referred to as nets) and a priori determined number of pads have to be tested (this is referred to as the net restriction), and (b) to choose the probe size to be used for testing each pad (only two available sizes: a large (100 mil) and a small (50 mil)) subject to the considerations that larger size probes are more reliable for testing purposes and probes that are assigned to pads in close proximity should not come in physical contact with each other (it creates short circuits and erroneous test result). Thus, the problem the testing engineer faces is to assign the maximum number of 100 mil probes to an appropriately selected set of pads in a way that avoids the creation of short circuits and accounts for the net restriction. We develop an efficient algorithm to solve the problem using results from the vertex packing literature, which exploit the special structure of an appropriate geometric graph we can define in this application. The algorithm can handle the large size real problem within 2–3 minutes of real time on a microcomputer.     

A maximization problem and its application to canonical correlation

Let Σ be an n × n positive definite matrix with eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λ_n > 0 and let M = {x, y | x?Rⁿ, y?Rⁿ, x ≠ 0, y ≠ 0, x′y = 0}. Then for x, y in M, we have that $\text{x′Σy}\text{(x′Σxy′Σy)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$ and the inequality is sharp. If

$\text{∑=}\text{∑}{}_{11}\text{∑}{}_{12}\text{∑}{}_{21}\text{∑}{}_{22}$

is a partitioning of Σ, let θ₁ be the largest canonical correlation coefficient. The above result yields $\text{θ}{}_1\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$.     

Sigma-subdifferential and its application to minimization problem

Positivity - In this paper, we study $$\sigma $$ -subdifferentials of $$\sigma $$ -convex functions. Two equivalent conditions for $$\sigma $$ -convexity are given. The formula for the $$\sigma $$...     

Approximation algorithms for general packing problems and their application to the multicast congestion problem

In this paper we present approximation algorithms based on a Lagrangian decomposition via a logarithmic potential reduction to solve a general packing or min–max resource sharing problem with M non-negative convex constraints on a convex set B. We generalize a method by Grigoriadis et al. to the case with weak approximate block solvers (i.e., with only constant, logarithmic or even worse approximation ratios). Given an accuracy , we show that our algorithm needs calls to the block solver, a bound independent of the data and the approximation ratio of the block solver. For small approximation ratios the algorithm needs calls to the block solver. As an application we study the problem of minimizing the maximum edge congestion in a multicast communication network. Interestingly the block problem here is the classical Steiner tree problem that can be solved only approximately. We show how to use approximation algorithms for the Steiner tree problem to solve the multicast congestion problem approximately. This work was done in part when the second author was studying at the University of Kiel. This paper combines our extended abstracts of the 2nd IFIP International Conference on Theoretical Computer Science, TCS 2002, Montréal, Canada and the 3rd Workshop on Approximation and Randomization Algorithms in Communication Networks, ARACNE 2002, Roma, Italy. This research was supported in part by the DFG - Graduiertenkolleg, Effiziente Algorithmen und Mehrskalenmethoden; by the EU Thematic Network APPOL I + II, Approximation and Online Algorithms, IST-1999-14084 and IST-2001-32007; by the EU Research Training Network ARACNE, Approximation and Randomized Algorithms in Communication Networks, HPRN-CT-1999-00112; by the EU Project CRESCCO, Critical Resource Sharing for Cooperation in Complex Systems, IST-2001-33135. The second author was also supported by an MITACS grant of Canada; and by the NSERC Discovery Grant DG 5-48923.     

The superstar packing problem

Hell and Kirkpatrick proved that in an undirected graph, a maximum size packing by a set of non-singleton stars can be found in polynomial time if this star-set is of the form {S ₁, S ₂, ..., S _k } for some k∈ℤ₊ (S _i is the star with i leaves), and it is NP-hard otherwise. This may raise the question whether it is possible to enlarge a set of stars not of the form {S ₁, S ₂, ..., S _k } by other non-star graphs to get a polynomially solvable graph packing problem. This paper shows such families of depth 2 trees. We show two approaches to this problem, a polynomial alternating forest algorithm, which implies a Berge-Tutte type min-max theorem, and a reduction to the degree constrained subgraph problem of Lovász. Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No. 504438.     

Fuzzy bin packing problem

This paper deals with the fuzzy bin packing problem that is a packing problem of non-rigid rectangles into an open rectangular bin. This problem is different from the conventional bin packing problem, which considers only rigid rectangles. The goal of the fuzzy bin packing problem is to minimize both the height of a packing and the extra cost due to the reduction of each piece. The total cost of the problem is represented as the sum of the height cost and the extra cost due to reductions of the pieces, which is called reduction cost. Because the conventional bin packing problem itself is an NP-hard problem, the presented optimization method assumes that an initial packing for non-reduced pieces has already been found. A closed form solution is presented for fuzzy bin packing problems, in which fuzzy numbers are triangular and the reduction cost is given by a quadratic function.     

Stochastic set packing problem

In this paper a stochastic version of the set packing problem (SPP), is studied via scenario analysis. We consider a one-stage recourse approach to deal with the uncertainty in the coefficients. It consists of maximizing in the stochastic SPP a composite function of the expected value minus the weighted risk of obtaining a scenario whose objective function value is worse than a given threshold. The splitting variable representation is decomposed by dualizing the nonanticipativity constraints that link the deterministic SPP with a 0-1 knapsack problem for each scenario under consideration. As a result a (structured) larger pure 0-1 model is created. We present several procedures for obtaining good feasible solutions, as well as a preprocessing approach for fixing variables. The Lagrange multipliers updating is performed by using the Volume Algorithm. Computational experience is reported for a broad variety of instances, which shows that the new approach usually outperforms a state-of-the-art optimization engine, producing a comparable optimality gap with smaller (several orders of magnitude) computing time.     

A heuristic algorithm for the strip packing problem

The two-dimensional strip packing problem is to pack a given set of rectangles into a strip with a given width and infinite height so as to minimize the required height of the packing. From the computational point of view, the strip packing problem is an NP-hard problem. With the B*-tree representation, this paper first presents a heuristic packing strategy which evaluates the positions used by the rectangles. Then an effective local search method is introduced to improve the results and a heuristic algorithm (HA) is further developed to find a desirable solution. Computational results on randomly generated instances and popular test instances show that the proposed method is efficient for the strip packing problem.     

A note on a selfish bin packing problem

In this paper, we consider a selfish bin packing problem, where each item is a selfish player and wants to minimize its cost. In our new model, if there are k items packed in the same bin, then each item pays a cost 1/k, where k ≥ 1. First we find a Nash Equilibrium (NE) in time O(n log n) within a social cost at most 1.69103OPT + 3, where OPT is the social cost of an optimal packing; where n is the number of items or players; then we give tight bounds for the worst NE on the social cost; finally we show that any feasible packing can be converged to a Nash Equilibrium in O(n ²) steps without increasing the social cost.     

A heuristic for the three-dimensional strip packing problem

The contribution presents a heuristic for the three-dimensional strip packing problem (3D-SPP) with rectangular pieces (boxes). The considered 3D-SPP can be formulated as follows: for a given set of boxes and a given longitudinal open container, determine an arrangement of all boxes within the container so that the required container length is minimized.     

A one-dimensional bin packing problem with shelf divisions

Given bins of size B, non-negative values d and Δ, and a list L of items, each item e∈L with size s_e and class c_e, we define a shelf as a subset of items packed inside a bin with total item sizes at most Δ such that all items in this shelf have the same class. Two subsequent shelves must be separated by a shelf division of size d. The size of a shelf is the total size of its items plus the size of the shelf division. The class constrained shelf bin packing problem (CCSBP) is to pack the items of L into the minimum number of bins, such that the items are divided into shelves and the total size of the shelves in a bin is at most B. We present hybrid algorithms based on the First Fit (Decreasing) and Best Fit (Decreasing) algorithms, and an APTAS for the problem CCSBP when the number of different classes is bounded by a constant C.     

The master bay plan problem: a solution method based on its connection to the three-dimensional bin packing problem

This paper addresses the problem of determining stowage plansfor containers in a ship, referred to as the Master Bay PlanProblem (MBPP). MBPP is NP-complete. We present a heuristic method for solvingMBPP based on its relation with the three-dimensional bin packingproblem (3D-BPP), where items are containers and bins are differentportions of the ship. Our aim is to find stowage plans, takinginto account structural and operational constraints relatedto both the containers and the ship, that minimize the timerequired for loading all containers on board. A validation of the proposed approach with some test casesis given. The results of real instances of the problem involvingmore than 1400 containers show the effectiveness of the proposedapproach for large scale applications.     

Lagrangian relaxation and its application to the unit-commitment-economic-dispatch problem

This study is concerned with the optimal scheduling of an electricitypower system consisting of both hydro and thermal units. UsingLagrangian relaxation, the original (primal) problem may bewritten in a dual formulation; the problem then admits decompositioninto more tractable sub-problems. Furthermore, the primal solutioncan be approximated closely by the dual solution, by using theduality gap as a termination criterion. A heuristic has beenused to construct nearly optimal solutions to the primal problembased on the information provided by the dual problem. Thispaper highlights three main points: improved computational times,the economic significance of the Lagrange multipliers, and theimplicit parallelism of this algorithm.     

A lexicographic pricer for the fractional bin packing problem

We propose an exact lexicographic dynamic programming pricing algorithm for solving the Fractional Bin Packing Problem with column generation. The new algorithm is designed for generating maximal columns of minimum reduced cost which maximize, lexicographically, one of the measures of maximality we investigate. Extensive computational experiments reveal that a column generation algorithm based on this pricing technique can achieve a substantial reduction in the number of columns and the computing time, also when combined with a classical smoothing technique from the literature.     

Stochastic linear knapsack programming problem and its application to a portfolio selection problem

In this paper a probability maximization model of a stochastic linear knapsack problem is considered where the random variables consist of several groups with mutually correlated ones. We propose a solution algorithm to the equivalent nonlinear fractional programming problem with a simple ranking method. This approach will be effectively applied to one of the portfolio selection problems.     

A robust APTAS for the classical bin packing problem

Bin packing is a well studied problem which has many applications. In this paper we design a robust APTAS for the problem. The robust APTAS receives a single input item to be added to the packing at each step. It maintains an approximate solution throughout this process, by slightly adjusting the solution for each new item. At each step, the total size of items which may migrate between bins must be bounded by a constant factor times the size of the new item. We show that such a property cannot be maintained with respect to optimal solutions. A preliminary version of this paper appeared in Proceedings of the 33rd International Colloquium on Automata, Languages and Programming (ICALP2006), part I, pp. 214–225.     

A hybrid evolutionary algorithm for the two-dimensional packing problem

This paper presents a hybrid evolutionary algorithm for the two-dimensional non-guillotine packing problem. The problem consists of packing many rectangular pieces into a single rectangular sheet in order to maximize the total area of the pieces packed. Moreover, there is a constraint on the maximum number of times that a piece may be used in a packing pattern. The set of packing patterns is processed by an evolutionary algorithm. Three mutation operators and two types of quality functions are used in the algorithm. The best solution obtained by the evolutionary algorithm is used as the initial solution in a tree search improvement procedure. This approach is tested on a set of benchmark problems taken from the literature and compared with the results published by other authors.     

The variable-width strip packing problem

Here, we focus on a generalized version of the strip packing problem; namely we have several open-end strips with different widths, and we wish to pack rectangular items into these strips without overlapping such that we have to minimize either the makespan (i.e. the top of the topmost item), or the total area used. We investigate the online variant of the problem, where the items are arriving one-by-one, and we have to make irrevocable decisions on their packing. A similar framework was proposed by Ye and Mei (On-line scheduling of parallel jobs in heterogeneous multiple clusters. Frontiers in algorithmics and algorithmic aspects in information and management, Springer, Berlin, pp 139–148, 2012. https://doi.org/10.1007/978-3-642-29700-7_13) for scheduling models, and they studied the absolute competitive ratio of their algorithm. Our contribution is to define a new objective function and several algorithms by combining so-called shelf algorithms with techniques taken from the areas of the variable-sized bin packing problem and scheduling. We analyzed the asymptotic competitive ratio of our algorithms.