A Branch-And-Price Approach with Milp Formulation to Modularity Density Maximization on Graphs
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Optimal Placement by Branch-And-Price
Optimal Placement by Branch-and-Price Pradeep Ramachandaran1 Ameya R. Agnihotri2 Satoshi Ono2;3;4 Purushothaman Damodaran1 Krishnaswami Srihari1 Patrick H. Madden2;4 SUNY Binghamton SSIE1 and CSD2 FAIS3 University of Kitakyushu4 Abstract— Circuit placement has a large impact on all aspects groups of up to 36 elements. The B&P approach is based of performance; speed, power consumption, reliability, and cost on column generation techniques and B&B. B&P has been are all affected by the physical locations of interconnected applied to solve large instances of well known NP-Complete transistors. The placement problem is NP-Complete for even simple metrics. problems such as the Vehicle Routing Problem [7]. In this paper, we apply techniques developed by the Operations We have tested our approach on benchmarks with known Research (OR) community to the placement problem. Using optimal configurations, and also on problems extracted from an Integer Programming (IP) formulation and by applying a the “final” placements of a number of recent tools (Feng Shui “branch-and-price” approach, we are able to optimally solve 2.0, Dragon 3.01, and mPL 3.0). We find that suboptimality is placement problems that are an order of magnitude larger than those addressed by traditional methods. Our results show that rampant: for optimization windows of nine elements, roughly suboptimality is rampant on the small scale, and that there is half of the test cases are suboptimal. As we scale towards merit in increasing the size of optimization windows used in detail windows with thirtysix elements, we find that roughly 85% of placement. -
Linear Programming Notes X: Integer Programming
Linear Programming Notes X: Integer Programming 1 Introduction By now you are familiar with the standard linear programming problem. The assumption that choice variables are infinitely divisible (can be any real number) is unrealistic in many settings. When we asked how many chairs and tables should the profit-maximizing carpenter make, it did not make sense to come up with an answer like “three and one half chairs.” Maybe the carpenter is talented enough to make half a chair (using half the resources needed to make the entire chair), but probably she wouldn’t be able to sell half a chair for half the price of a whole chair. So, sometimes it makes sense to add to a problem the additional constraint that some (or all) of the variables must take on integer values. This leads to the basic formulation. Given c = (c1, . , cn), b = (b1, . , bm), A a matrix with m rows and n columns (and entry aij in row i and column j), and I a subset of {1, . , n}, find x = (x1, . , xn) max c · x subject to Ax ≤ b, x ≥ 0, xj is an integer whenever j ∈ I. (1) What is new? The set I and the constraint that xj is an integer when j ∈ I. Everything else is like a standard linear programming problem. I is the set of components of x that must take on integer values. If I is empty, then the integer programming problem is a linear programming problem. If I is not empty but does not include all of {1, . , n}, then sometimes the problem is called a mixed integer programming problem. -
Heuristic Search Viewed As Path Finding in a Graph
ARTIFICIAL INTELLIGENCE 193 Heuristic Search Viewed as Path Finding in a Graph Ira Pohl IBM Thomas J. Watson Research Center, Yorktown Heights, New York Recommended by E. J. Sandewall ABSTRACT This paper presents a particular model of heuristic search as a path-finding problem in a directed graph. A class of graph-searching procedures is described which uses a heuristic function to guide search. Heuristic functions are estimates of the number of edges that remain to be traversed in reaching a goal node. A number of theoretical results for this model, and the intuition for these results, are presented. They relate the e])~ciency of search to the accuracy of the heuristic function. The results also explore efficiency as a consequence of the reliance or weight placed on the heuristics used. I. Introduction Heuristic search has been one of the important ideas to grow out of artificial intelligence research. It is an ill-defined concept, and has been used as an umbrella for many computational techniques which are hard to classify or analyze. This is beneficial in that it leaves the imagination unfettered to try any technique that works on a complex problem. However, leaving the con. cept vague has meant that the same ideas are rediscovered, often cloaked in other terminology rather than abstracting their essence and understanding the procedure more deeply. Often, analytical results lead to more emcient procedures. Such has been the case in sorting [I] and matrix multiplication [2], and the same is hoped for this development of heuristic search. This paper attempts to present an overview of recent developments in formally characterizing heuristic search. -
Branch-And-Bound Experiments in Convex Nonlinear Integer Programming
Noname manuscript No. (will be inserted by the editor) More Branch-and-Bound Experiments in Convex Nonlinear Integer Programming Pierre Bonami · Jon Lee · Sven Leyffer · Andreas W¨achter September 29, 2011 Abstract Branch-and-Bound (B&B) is perhaps the most fundamental algorithm for the global solution of convex Mixed-Integer Nonlinear Programming (MINLP) prob- lems. It is well-known that carrying out branching in a non-simplistic manner can greatly enhance the practicality of B&B in the context of Mixed-Integer Linear Pro- gramming (MILP). No detailed study of branching has heretofore been carried out for MINLP, In this paper, we study and identify useful sophisticated branching methods for MINLP. 1 Introduction Branch-and-Bound (B&B) was proposed by Land and Doig [26] as a solution method for MILP (Mixed-Integer Linear Programming) problems, though the term was actually coined by Little et al. [32], shortly thereafter. Early work was summarized in [27]. Dakin [14] modified the branching to how we commonly know it now and proposed its extension to convex MINLPs (Mixed-Integer Nonlinear Programming problems); that is, MINLP problems for which the continuous relaxation is a convex program. Though a very useful backbone for ever-more-sophisticated algorithms (e.g., Branch- and-Cut, Branch-and-Price, etc.), the basic B&B algorithm is very elementary. How- Pierre Bonami LIF, Universit´ede Marseille, 163 Av de Luminy, 13288 Marseille, France E-mail: [email protected] Jon Lee Department of Industrial and Operations Engineering, University -
Integer Linear Programming
Introduction Linear Programming Integer Programming Integer Linear Programming Subhas C. Nandy ([email protected]) Advanced Computing and Microelectronics Unit Indian Statistical Institute Kolkata 700108, India. Introduction Linear Programming Integer Programming Organization 1 Introduction 2 Linear Programming 3 Integer Programming Introduction Linear Programming Integer Programming Linear Programming A technique for optimizing a linear objective function, subject to a set of linear equality and linear inequality constraints. Mathematically, maximize c1x1 + c2x2 + ::: + cnxn Subject to: a11x1 + a12x2 + ::: + a1nxn b1 ≤ a21x1 + a22x2 + ::: + a2nxn b2 : ≤ : am1x1 + am2x2 + ::: + amnxn bm ≤ xi 0 for all i = 1; 2;:::; n. ≥ Introduction Linear Programming Integer Programming Linear Programming In matrix notation, maximize C T X Subject to: AX B X ≤0 where C is a n≥ 1 vector |- cost vector, A is a m× n matrix |- coefficient matrix, B is a m × 1 vector |- requirement vector, and X is an n× 1 vector of unknowns. × He developed it during World War II as a way to plan expenditures and returns so as to reduce costs to the army and increase losses incurred by the enemy. The method was kept secret until 1947 when George B. Dantzig published the simplex method and John von Neumann developed the theory of duality as a linear optimization solution. Dantzig's original example was to find the best assignment of 70 people to 70 jobs subject to constraints. The computing power required to test all the permutations to select the best assignment is vast. However, the theory behind linear programming drastically reduces the number of feasible solutions that must be checked for optimality. -
Cooperative and Adaptive Algorithms Lecture 6 Allaa (Ella) Hilal, Spring 2017 May, 2017 1 Minute Quiz (Ungraded)
Cooperative and Adaptive Algorithms Lecture 6 Allaa (Ella) Hilal, Spring 2017 May, 2017 1 Minute Quiz (Ungraded) • Select if these statement are true (T) or false (F): Statement T/F Reason Uniform-cost search is a special case of Breadth- first search Breadth-first search, depth- first search and uniform- cost search are special cases of best- first search. A* is a special case of uniform-cost search. ECE457A, Dr. Allaa Hilal, Spring 2017 2 1 Minute Quiz (Ungraded) • Select if these statement are true (T) or false (F): Statement T/F Reason Uniform-cost search is a special case of Breadth- first F • Breadth- first search is a special case of Uniform- search cost search when all step costs are equal. Breadth-first search, depth- first search and uniform- T • Breadth-first search is best-first search with f(n) = cost search are special cases of best- first search. depth(n); • depth-first search is best-first search with f(n) = - depth(n); • uniform-cost search is best-first search with • f(n) = g(n). A* is a special case of uniform-cost search. F • Uniform-cost search is A* search with h(n) = 0. ECE457A, Dr. Allaa Hilal, Spring 2017 3 Informed Search Strategies Hill Climbing Search ECE457A, Dr. Allaa Hilal, Spring 2017 4 Hill Climbing Search • Tries to improve the efficiency of depth-first. • Informed depth-first algorithm. • An iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solution by incrementally changing a single element of the solution. • It sorts the successors of a node (according to their heuristic values) before adding them to the list to be expanded. -
Backtracking / Branch-And-Bound
Backtracking / Branch-and-Bound Optimisation problems are problems that have several valid solutions; the challenge is to find an optimal solution. How optimal is defined, depends on the particular problem. Examples of optimisation problems are: Traveling Salesman Problem (TSP). We are given a set of n cities, with the distances between all cities. A traveling salesman, who is currently staying in one of the cities, wants to visit all other cities and then return to his starting point, and he is wondering how to do this. Any tour of all cities would be a valid solution to his problem, but our traveling salesman does not want to waste time: he wants to find a tour that visits all cities and has the smallest possible length of all such tours. So in this case, optimal means: having the smallest possible length. 1-Dimensional Clustering. We are given a sorted list x1; : : : ; xn of n numbers, and an integer k between 1 and n. The problem is to divide the numbers into k subsets of consecutive numbers (clusters) in the best possible way. A valid solution is now a division into k clusters, and an optimal solution is one that has the nicest clusters. We will define this problem more precisely later. Set Partition. We are given a set V of n objects, each having a certain cost, and we want to divide these objects among two people in the fairest possible way. In other words, we are looking for a subdivision of V into two subsets V1 and V2 such that X X cost(v) − cost(v) v2V1 v2V2 is as small as possible. -
IEOR 269, Spring 2010 Integer Programming and Combinatorial Optimization
IEOR 269, Spring 2010 Integer Programming and Combinatorial Optimization Professor Dorit S. Hochbaum Contents 1 Introduction 1 2 Formulation of some ILP 2 2.1 0-1 knapsack problem . 2 2.2 Assignment problem . 2 3 Non-linear Objective functions 4 3.1 Production problem with set-up costs . 4 3.2 Piecewise linear cost function . 5 3.3 Piecewise linear convex cost function . 6 3.4 Disjunctive constraints . 7 4 Some famous combinatorial problems 7 4.1 Max clique problem . 7 4.2 SAT (satisfiability) . 7 4.3 Vertex cover problem . 7 5 General optimization 8 6 Neighborhood 8 6.1 Exact neighborhood . 8 7 Complexity of algorithms 9 7.1 Finding the maximum element . 9 7.2 0-1 knapsack . 9 7.3 Linear systems . 10 7.4 Linear Programming . 11 8 Some interesting IP formulations 12 8.1 The fixed cost plant location problem . 12 8.2 Minimum/maximum spanning tree (MST) . 12 9 The Minimum Spanning Tree (MST) Problem 13 i IEOR269 notes, Prof. Hochbaum, 2010 ii 10 General Matching Problem 14 10.1 Maximum Matching Problem in Bipartite Graphs . 14 10.2 Maximum Matching Problem in Non-Bipartite Graphs . 15 10.3 Constraint Matrix Analysis for Matching Problems . 16 11 Traveling Salesperson Problem (TSP) 17 11.1 IP Formulation for TSP . 17 12 Discussion of LP-Formulation for MST 18 13 Branch-and-Bound 20 13.1 The Branch-and-Bound technique . 20 13.2 Other Branch-and-Bound techniques . 22 14 Basic graph definitions 23 15 Complexity analysis 24 15.1 Measuring quality of an algorithm . -
A Branch-And-Bound Algorithm for Zero-One Mixed Integer
A BRANCH-AND-BOUND ALGORITHM FOR ZERO- ONE MIXED INTEGER PROGRAMMING PROBLEMS Ronald E. Davis Stanford University, Stanford, California David A. Kendrick University of Texas, Austin, Texas and Martin Weitzman Yale University, New Haven, Connecticut (Received August 7, 1969) This paper presents the results of experimentation on the development of an efficient branch-and-bound algorithm for the solution of zero-one linear mixed integer programming problems. An implicit enumeration is em- ployed using bounds that are obtained from the fractional variables in the associated linear programming problem. The principal mathematical result used in obtaining these bounds is the piecewise linear convexity of the criterion function with respect to changes of a single variable in the interval [0, 11. A comparison with the computational experience obtained with several other algorithms on a number of problems is included. MANY IMPORTANT practical problems of optimization in manage- ment, economics, and engineering can be posed as so-called 'zero- one mixed integer problems,' i.e., as linear programming problems in which a subset of the variables is constrained to take on only the values zero or one. When indivisibilities, economies of scale, or combinatoric constraints are present, formulation in the mixed-integer mode seems natural. Such problems arise frequently in the contexts of industrial scheduling, investment planning, and regional location, but they are by no means limited to these areas. Unfortunately, at the present time the performance of most compre- hensive algorithms on this class of problems has been disappointing. This study was undertaken in hopes of devising a more satisfactory approach. In this effort we have drawn on the computational experience of, and the concepts employed in, the LAND AND DoIGE161 Healy,[13] and DRIEBEEKt' I algorithms. -
Module 5: Backtracking
Module-5 : Backtracking Contents 1. Backtracking: 3. 0/1Knapsack problem 1.1. General method 3.1. LC Branch and Bound solution 1.2. N-Queens problem 3.2. FIFO Branch and Bound solution 1.3. Sum of subsets problem 4. NP-Complete and NP-Hard problems 1.4. Graph coloring 4.1. Basic concepts 1.5. Hamiltonian cycles 4.2. Non-deterministic algorithms 2. Branch and Bound: 4.3. P, NP, NP-Complete, and NP-Hard 2.1. Assignment Problem, classes 2.2. Travelling Sales Person problem Module 5: Backtracking 1. Backtracking Some problems can be solved, by exhaustive search. The exhaustive-search technique suggests generating all candidate solutions and then identifying the one (or the ones) with a desired property. Backtracking is a more intelligent variation of this approach. The principal idea is to construct solutions one component at a time and evaluate such partially constructed candidates as follows. If a partially constructed solution can be developed further without violating the problem’s constraints, it is done by taking the first remaining legitimate option for the next component. If there is no legitimate option for the next component, no alternatives for any remaining component need to be considered. In this case, the algorithm backtracks to replace the last component of the partially constructed solution with its next option. It is convenient to implement this kind of processing by constructing a tree of choices being made, called the state-space tree. Its root represents an initial state before the search for a solution begins. The nodes of the first level in the tree represent the choices made for the first component of a solution; the nodes of the second level represent the choices for the second component, and soon. -
Matroid Partitioning Algorithm Described in the Paper Here with Ray’S Interest in Doing Ev- Erything Possible by Using Network flow Methods
Chapter 7 Matroid Partition Jack Edmonds Introduction by Jack Edmonds This article, “Matroid Partition”, which first appeared in the book edited by George Dantzig and Pete Veinott, is important to me for many reasons: First for per- sonal memories of my mentors, Alan J. Goldman, George Dantzig, and Al Tucker. Second, for memories of close friends, as well as mentors, Al Lehman, Ray Fulker- son, and Alan Hoffman. Third, for memories of Pete Veinott, who, many years after he invited and published the present paper, became a closest friend. And, finally, for memories of how my mixed-blessing obsession with good characterizations and good algorithms developed. Alan Goldman was my boss at the National Bureau of Standards in Washington, D.C., now the National Institutes of Science and Technology, in the suburbs. He meticulously vetted all of my math including this paper, and I would not have been a math researcher at all if he had not encouraged it when I was a university drop-out trying to support a baby and stay-at-home teenage wife. His mentor at Princeton, Al Tucker, through him of course, invited me with my child and wife to be one of the three junior participants in a 1963 Summer of Combinatorics at the Rand Corporation in California, across the road from Muscle Beach. The Bureau chiefs would not approve this so I quit my job at the Bureau so that I could attend. At the end of the summer Alan hired me back with a big raise. Dantzig was and still is the only historically towering person I have known. -
Mathematical Programming Lecture 19 OR 630 Fall 2005 November 1, 2005 Notes by Nico Diener
Mathematical Programming Lecture 19 OR 630 Fall 2005 November 1, 2005 Notes by Nico Diener Column and Constraint Generation Idea The column and constraint generation algorithms exploit the fact that the revised simplex algorithm only needs very limited access to data of the LP problem to solve some large-scale LPs. Illustration The one-dimensional cutting-stock problem (Chapter 6 of Bertsimas and Tsit- siklis, Chapters 13 and 26 of Chvatal). Problem A paper manufacturer produces paper in large rolls of width W and has a demand for narrow rolls of widths say w1, ..., wn, where 0 < wi < W . The demand is for bi rolls of width wi. Figure 1: We want to use as few large rolls as possible to satisfy the demand. We consider patterns, i.e., ways of cutting a large roll into smaller rolls: each pattern j corresponds to a vector m aj ∈ R with aij equal to the number of rolls of width wi in pattern j. 1 0 0 Example: a = 1 . 0 . . 1 If we could list all patterns say 1, 2, ..., N (N can be very large), then the problem would become: PN min j=1 xj PN j=1 ajxj = b, (1) xj ≥ 0 ∀j. Actually: We want xj, the number of rolls cut in pattern j, integer but we are just going to consider the LP relaxation. Problem N is large and A (the matrix made of columns aj) is known only implicitly: m PN Any vector a ∈ R satisfying i=1 wiai ≤ W , ai ≥ 0 and integer, defines a column of A.