The Modified Integer Round-Up Property of the One-Dimensional Cutting Stock Problem
|
|
- Jasmin Harrington
- 6 years ago
- Views:
Transcription
1 EJOR 84 (1995) The Modified Integer Round-Up Property of the One-Dimensional Cutting Stock Problem Guntram Scheithauer and Johannes Terno Institute of Numerical Mathematics, Dresden University of Technology Mommsenstr. 13, D Dresden, Germany Abstract A linear integer minimization problem (IP) has the modified integer round-up property (MIRUP) if the optimal value of any instance of IP is not greater than the optimal value of the corresponding LP relaxation problem rounded up plus one. The aim of this paper is to investigate numerically whether the MIRUP holds for the one-dimensional cutting stock problem. The computational experiments carried out with a lot of randomly generated instances of the one-dimensional cutting stock problem show, that for all instances an integer solution fulfills the MIRUP. Moreover, in most cases the optimal value equals the round-up optimal value of the corresponding LP relaxation. Similarly, the approach proposed to verify the MIRUP is usable as a new heuristic procedure for solving one-dimensional cutting stock problems. This heuristic always leads to very good solutions being optimal in the most cases. Keywords: cutting stock problem, modified integer round-up property 1
2 1 Introduction In this paper the classical one-dimensional cutting stock problem is investigated with respect to the relation between the optimal value and the corresponding linear programming (LP) relaxation bound. The numerical tests given below show that this difference is smaller than two. Therefore the proposed approach leads to a heuristic procedure for solving one-dimensional cutting stock problems. The starting point of our investigations can be summarized as follows. A frequently used method for solving one-dimensional cutting stock problems involves applying the column generation approach proposed by Gilmore and Gomory [6] and an appropriate rounding of the solution of the continuous relaxation problem [7], [14]. Gilmore and Gomory discuss briefly in [6] the possibility of round-down and using ad hoc -methods to fulfill the unfilled order quantities. In this paper we will take this much more into consideration since many numerical computations show an only small difference between the optimal value of the one-dimensional cutting stock problem and that of its corresponding linear programming (LP) relaxation [2], [15]. On the other hand from the theoretical point of view, Baum and Trotter [1] define an integer programming problem as having the integer round-up property (IRUP) if its optimal value is given by the smallest integer greater than or equal to the optimal value of its LP relaxation. They establish this property for certain classes of matrices arising in the context of the polymatroid theory. In [8] Marcotte investigates the cutting stock problem and integer rounding. She proves that certain classes of cutting stock problems have the IRUP. Furthermore, Marcotte gives in [9] an instance which does not fulfil the IRUP. But the gap between the optimal value and the corresponding LP lower bound always equals 1. Fieldhouse [5] presents an example of the one-dimensional cutting stock problem with a gap equal to Moreover, an example is given in [11] possessing a gap equal to This is the largest difference to have been found so far. In this paper we define the modified integer round-up property (MIRUP) of a (linear) integer minimization problem and investigate this property numerically for the classical one-dimensional cutting stock problem. Additionally we refer to [11] where the MIRUP is proven for such one-dimensional cutting stock problems where all cutting patterns fulfill a special density property. The paper is organized as follows. In Section 2 we formulate the one-dimensional cutting stock problem, give some definitions and state the main conjecture. Section 3 contains some theoretical background and the new approach to verify the MIRUP is proposed in Section 4. Results of numerical experiments are summarized in Section 5. Finally a conclusion is given which also includes consequences for practical applications. 2 The one-dimensional cutting stock problem The following cutting stock problem is investigated: 2
3 One-dimensional material objects (e.g. paper reels, iron slabs, wooden rods) of a given length L are divided into smaller pieces of desired lengths l 1,..., l m in order to fulfill the order demands b 1,..., b m. The goal is to minimize the total amount of stock material or, equivalently, to minimize the total waste. According to the typology of cutting and packing introduced by Dyckhoff [3] this is a problem of type 1/V/I/R. In the following standard model [6] an integer m-vector a j = (a 1j,..., a mj ) T with mi=1 l i a ij L represents a cutting pattern. The integer variable x j gives the number of times the cutting pattern a j is cut. n z = x j j=1 min s.t. n a ij x j b i, i = 1,..., m, (1) j=1 x j 0, integer, j = 1,..., n, where n denotes the number of cutting patterns. Model (1) is a special case of the model of the standard linear integer optimization (minimization) problem: z = c T x min s.t. Ax b, (2) x 0, integer. Here, an instance is given by the following Input-data: the number of variables n, the number of restrictions m (without non-negativity conditions), the matrix A of coefficients, the objective function c and the right hand side b. Hence to any instance E = (m, l, L, b) of the one-dimensional cutting stock problem an instance (m, n, A, c, b) of type (2) can be assigned. Without loss of generality, all Input-data are assumed to be integer, and to ensure the problem can be solved, we suppose max i=1,...,m l i L. A set P := P (I) = {(m i, n i, A i, c i, b i ) : i I, m i, n i Z +, A i Z m i n i, c i Z n i, b i Z m i } where I is a non-empty set of indices, defines a subproblem (or problem) of type (2). For a given instance E = (m, n, A, c, b) P of a problem P of type (2) let z (E) denote the optimal value of the integer problem and z c (E) the optimal value of the corresponding LP relaxation z = c T x min Then we give the following definitions. s.t. Ax b, x 0. (3) Definition 1 A linear integer minimization problem P has the integer property (IP) if z (E) = z c (E) for all E P. 3
4 Definition 2 (Baum/Trotter) A linear integer minimization problem P has the integer round-up property (IRUP) if z (E) = z c (E) for all E P. Definition 3 A linear integer minimization problem P has the modified integer roundup property (MIRUP) if z (E) z c (E) + 1 for all E P. Remark: With respect to the integer property we have the well-known result that a problem P has the integer property if and only if for any instance of P the coefficient matrix is totally unimodular (see in [10]). In many practical computations it has been observed that for instances E of onedimensional cutting stock problems P the difference between z (E) and z c (E) is small [2], [15]. The investigations of Marcotte [9], Fieldhouse [5] and ourselves show that the standard one-dimensional cutting stock problem does not possess the IRUP. But there is no counterexample known to the MIRUP. For that reason we formulate the following Conjecture: The standard one-dimensional cutting stock problem has the modified integer round-up property. Remark: In [12] it is shown that the one-dimensional cutting stock problem has the MIRUP when m 5. There the MIRUP is also proven for a wide range of instances of the onedimensional cutting stock problem independently from the number m of small lengths. 3 Theoretical background Using the LP relaxation to obtain feasible integer solutions, one has to overcome the difficulties which arise when rounding the solution of the LP relaxation. Simply rounding up leads always to feasible integer solutions but these are in general worse and can be improved easily by more intelligent rounding procedures. Such methods are proposed in [7] and [14]. In order to investigate the MIRUP we consider a concept which consists in defining and investigating a so-called residual problem which is defined in accordance with a solution of the LP relaxation. The approach is based on the following lemmas. Let E = (m, l, L, b) be an instance of the one-dimensional cutting stock problem and let a j, j = 1,..., n, be the corresponding cutting patterns. Furthermore let A denote the coefficient matrix according to (2). Let x c denote an optimal solution of the corresponding LP relaxation. Rounding down of x c yields an integer vector x with x j = x c j, j = 1,..., n. If x x c then a residual problem can be defined with right hand side b := b Ax. The instance of the residual problem is E := (m, l, L, b). In comparison with the original problem E we have a new (reduced) demand vector b. Lemma 1 Let E be an instance of the one-dimensional cutting stock problem and E a corresponding residual instance. Then it holds that: if z(e) z c (E) + 1 then z (E) z c (E) + 1, i.e. if the residual problem has the MIRUP then the initial problem has this property too. 4
5 Proof: Let e = (1,..., 1) T R n. Then it holds: z c (E) = e T x c = e T x + e T {x c } z (E) e T x + z(e) e T x + z c (E) + 1 = e T x + z c (E) + 1 = z c (E) + 1. It is easy to verify that a similar statement is true if x c is replaced by any feasible solution x of the LP relaxation fulfilling e T x = z c (E). Therefore in some cases the LP bound does not need to be computed exactly. In order to verify whether a solution x with e T x = z c (E) is found, a dual-like problem has to be solved. For more details we refer to [13]. In order to decide whether an instance or a residual instance has the IRUP or the MIRUP we use the following two lemmas which are proven in [12]. Lemma 2 Let E be an instance of the one-dimensional cutting stock problem and E a corresponding residual instance. Then it holds that: if z c (E) 1.5 then E has the IRUP, if 1.5 < z c (E) 3 then E has the MIRUP. Lemma 3 Let E be an instance of the one-dimensional cutting stock problem and E a corresponding residual instance. Furthermore let x c denote an optimal solution of the LP relaxation problem of E. Then it holds that: if z c (E) > κ 1 then E has the IRUP, if z c (E) > κ 2 then E has the MIRUP, where κ := j sign(x c j). Remark: Using Lemma 1 we can verify that if the residual instance possesses the MIRUP, then the initial instance possesses the MIRUP too. On the other hand, such a conclusion cannot be stated in the case that if E does not have the IRUP then neither does E, since instances E of the one-dimensional cutting stock problem exist whereby z c (E) = z (E) < z + z(e) = z c (E) + 1. Such an instance is L = 396, l 1 = 132, l 2 = 99, l 3 = 44, l 4 = 36 with b 1 = 2, b 2 = 3, b 3 = 9 and b 4 = 6. The continuous solution is = with z c (E) = 391 since the applied patterns contain no waste, and z = 1. The residual 132 problem E with b = (2, 3, 0, 6) T has the optimal value z(e) = 3 but z (E) = 3. A further helpful tool for estimations and bound computations in a branch and bound algorithm is based on the next lemma which is simple to verify. 5
6 Lemma 4 Let E be an instance of the one-dimensional cutting stock problem. Then there exists an optimal solution x which fulfills the additional constraint j with a j b = x j = 0. (4) In comparison to the integer problems, the restriction (4) often leads to a stronger lower LP bound in comparison to the bound obtained from the LP relaxation of (1) especially in the case of residual instances. 4 The new approach In the following, we describe the concept for computational investigations with respect to the MIRUP for the one-dimensional cutting stock problems. Let E = (m, l, L, b) be an instance of the one-dimensional cutting stock problem. Then E is investigated with respect to the MIRUP as follows. If the LP relaxation does not lead to an integer solution we define a residual instance E as described above. Then because of Lemma 2 or 3 we are either able to conclude the MIRUP for E or further investigations are necessary. In the latter case we try to find an appropriate integer solution for E using two heuristics. If this also fails a stronger lower bound for E is computed in accordance to Lemma 4. The new bound may lead to a termination of the process in some cases or to a further (second) residual instance Ê. Investigating Ê analogously, may lead the termination. If all this fails Ê or/and E has to be solved using a branch&bound-algorithm until a solution is found which proves the MIRUP of E, or until an instance is found which does not possess the MIRUP. Moreover, during the investigations we distinguish between the knowledge of exact solution and the MIRUP. In the description of the procedure we consider only that case when the LP relaxation problems are solved exactly. The modifications with respect to the remark of Lemma 1 are omitted for the sake of simplicity. The two heuristics are described briefly below. At first we give the general algorithm for testing an instance with respect to the MIRUP. Algorithm MIRUP test for instance E 1. Computation of an optimal solution x c of the corresponding LP relaxation. If an integer solution is found - STOP. Otherwise build the residual instance E. Let z := e T x c. Then z c (E) = z c (E) z. If E has the MIRUP because of Lemma 2 or 3 - STOP. 2. Heuristic solution of E yields the value z h. If z h = z c (E) then an optimal solution to E is found - STOP. If z h = z c (E) + 1 then E has the MIRUP - STOP. 3. Computation of an optimal solution x r of the LP relaxation of E in consideration of the condition i {1,..., m} : a ij > b i x j = 0. If x r is integer and e T x r = z c (E) then an optimal solution of E is found - STOP. If x r is integer and e T x r = z c (E) + 1 then E has the MIRUP - STOP. 6
7 4. If ẑ := e T x r b > 0 then build a further residual instance Ê with right hand side := b A x r, otherwise go to step 5. Heuristic solution of Ê yields ẑh. If ẑ + ẑ h = z c (E) then an optimal solution to E is found - STOP. If ẑ + ẑ h = z c (E) + 1 then E has the MIRUP - STOP. Exact solution of Ê by using a branch and bound algorithm yields ẑ. If ẑ + ẑ = z c (E) then an optimal solution to E is found - STOP. If ẑ + ẑ = z c (E) + 1 then E has the MIRUP - STOP. 5. Exact solution of E by using a branch and bound algorithm yields z. If z = z c (E) then an optimal solution to E is found - STOP. If z = z c (E) + 1 then E has the MIRUP - STOP. 6. If this point is reached then E is an instance of the one-dimensional cutting stock problem which does not have the MIRUP. Remark: The last step takes into consideration the fact that the MIRUP is not proven for the standard one-dimensional cutting stock problem so far. In order to decide in such a situation whether E has the MIRUP or not, further investigations are required. For the description of the heuristics used in Step 2 and 4 let E = (m, l, L, b) denote an instance of the one-dimensional cutting stock problem. Without loss of generality we may assume L l 1 >... > l m > 0. These heuristics are used to get a feasible cutting pattern a Z m taking into consideration the right hand side b. In the first heuristic the cutting pattern is constructed by using the greedy method. There the pieces are packed in the given sequence if there is sufficient length but no more than required. Heuristic 1 (L, b, a). L := L, b := b; for i := 1 to m do a i := min{ b i, L l i }; b i := b i a i ; L := L a i l i ; In the second heuristic the cutting pattern is constructed by using a modified greedy method. Let β := z c (E) e T x c (E). That means, β is the sum of all fractional parts of the optimal solution x c of the corresponding linear relaxation problem. Using the weight 1/β a more equalized cutting pattern is constructed in comparison to Heuristic 1. Heuristic 2 (L, b, a); L := L; b := b, β := max{1, β}; for i := 1 to m do a i := min{ b i, L }; b β l i := b i a i L := L a i l i ; i if L l T a l m then heuristic1 ( L, b, a), a := a + a. 7
8 In order to get a feasible solution for the instance E the first or second heuristic are repeatedly applied until b = 0. As the computational experiments show it is not necessary to use the branch&boundalgorithm and therefore we will not go into the details of a description. Instead we refer to [13]. 5 Computational results In order to investigate the one-dimensional cutting stock problem with respect to the MIRUP, we solved series of randomly generated instances. Thereby the Input-data are chosen from a uniform distribution on some ranges given below. For a given material length L and a chosen m [m, m] the lengths l i are in [ L/(m 2), L/2] and the order quantities b i are in [2m, 10m]. The LP relaxation problems are solved using the simplex method with column generation where the new pattern is obtained by the greedy algorithm, and if this fails the corresponding knapsack problem is solved exactly using a dynamic programming forward state algorithm. The generation process is terminated (columns termination) if the optimality condition is fulfilled (column opt) or, secondly, if a given maximum number of solved knapsack problems is exceeded (column iter) or, thirdly, if the decrease of the objective function value is smaller then 0.1 within the last m/2 iteration steps (column eps). In the cases iter and eps it is ensured that the current objective function value z fulfills the condition z c (E) = z in accordance to the remark to Lemma 1. The columns value characterize the best found integer solution. exact gives the number of instances which are exactly solved because IRUP is fulfilled. MIRUP gives the number of instances where a solution proving the MIRUP is found but optimality is not ensured. The column E characterizes the first residual problem. lb counts the number of terminations because of Lemma 3 and h1 gives the number of instances where Heuristic 1 leads to a termination due to a solution which proves the MIRUP. Since Heuristic 1 is used before Heuristic 2 and since in any case after applying Heuristic 1 a solution is found proving the MIRUP, Heuristic 2 is not needed. But in the reverse order, Heuristic 1 has to be used in some cases after Heuristic 2. The last column describes the average time per instance in seconds on a PC 486 DX, 50 MHz. The following tables summarize the results for L = 1000 (Table 1), L = 2000 (Table 2) and L = 3000 (Table 3). For each range [m, m] 20 instances were generated. The average times for the different L-values reflect the dependence of the knapsack procedures on the absolute size of the Input-data. Since the solution of the instances with more than m = 110 pieces requires a large amount of computer time and otherwise, since the instances with m 110 make clear the dependence of the computer time and the stock length L, we omit the solution of instances with L = 2000, L = 3000 and m >
9 Table 1: L = 1000 m termination value E average m m opt iter eps exact MIRUP lb h1 time Table 2: L = 2000 m termination value E average m m opt iter eps exact MIRUP lb h1 time Table 3: L = 3000 m termination value E average m m opt iter eps exact MIRUP lb h1 time
10 As the tables show, all randomly generated instances of the one-dimensional cutting stock problem fulfill the MIRUP. Moreover, a corresponding integer solution was found by the greedy heuristic. Applying the branch and bound algorithm was not necessary. In many cases (see column exact) an optimal solution was obtained with this approach. For 80.4 percent of the instances the optimality is proved for the solution found. Hence, solving one-dimensional cutting stock problems via column generation and computing of a greedy integer solution for the residual problem leads to an integer solution whose value does not diverse more than one unit in comparison to the optimal solution. Additionally to the investigations with respect to the MIRUP, we have paid attention to the exact difference between the optimal value and the LP bound. For that reason we define the MAXGAP problem as follows. Let P be a problem of type (2). Definition 4 The maximum gap problem (MAXGAP) of a problem P consists in determining the maximum difference between the optimal value z (E) and the LP lower bound z c (E) with respect to all instances E of P : (P ) := max E P {z (E) z c (E)}. In the case of the standard one-dimensional cutting stock problem we have (s. [11]) (1D CSP) = This gap arises, e.g., in the residual problem E given in Section 3. However, our conjecture is (1D CSP) = In order to investigate the one-dimensional cutting stock problem numerically with respect to MAXGAP a similar concept as in Section 4 can be used. But all the numerical experiments did not lead to an instance with a larger gap. Therefore our conjecture seems to be still more realistic. We refer to [13] for a branch&bound algorithm for solving onedimensional cutting stock problems exactly. 6 Conclusional Remarks The extensive computational tests support the conjecture that the one-dimensional cutting stock problem fulfills the MIRUP. If this conjecture can be proven there arises some new insights in Integer Programming, especially for problems with submodular objective functions. Based on the optimistic numerical results, investigations are motivated with respect to theoretical statements and for practical applications in the case of one-dimensional cutting stock problems as well as for higher dimensional cutting stock problems. Moreover, the proposed approach can be used directly as a heuristic procedure for solving instances of the one-dimensional cutting stock problem. And the objective function value of the obtained solution differs from the optimal value by 1 at the most. Although the algorithm is not polynomial the obtained results show that instances of medium size can be solved within an acceptable running time. 10
11 Acknowledgement The authors wish to thank Uta Sommerwei and Jan Riehme for implementing the algorithm and doing the extensive computational tests. Furthermore we thank the editors and the anonymous referees for their helpful comments. References [1] Baum, S., and Trotter, L.E., Jr., Integer rounding for polymatroid and branching optimization problems, SIAM J. Alg. Disc. Meth. 2 (1981) 4, [2] Diegel, A., Integer LP solution for large trim problem, Working Paper, University of Natal, South Africa, [3] Dyckhoff, H., A typology of cutting and packing problems, it EJOR 44 (1990) [4] Dyckhoff, H., and Finke, U., Cutting and packing in production and distribution, Physica Verlag, Heidelberg, [5] Fieldhouse, M., The duality gap in trim problems, SICUP-Bulletin No. 5, [6] Gilmore, P.C., and Gomory, R.E., A linear programming approach to the cutting stock problem, Operations Res. 9 (1961) [7] Johnston, R.E., Rounding algorithms for cutting stock problems, Asia-Pacific J. of OR 3 (1986) [8] Marcotte, O., The cutting stock problem and integer rounding, Mathematical Programming 33 (1985) [9] Marcotte, O., An instance of the cutting stock problem for which the rounding property does not hold, Oper. Res. Lett. 4 (1986) 5, [10] Nemhauser, G.L., and Woolsey, L.A., Integer and Combinatorial Optimization, John Wiley & Sons, New York [11] Scheithauer, G., and Terno, J., About the gap between the optimal values of the integer and continuous relaxation one-dimensional cutting stock problem, Operations Research Proceedings 1991, Springer-Verlag, Berlin, Heidelberg, [12] Scheithauer, G., and Terno, J., Theoretical investigations on the MIRUP (modified integer round-up property) for the one-dimensional cutting stock problem, Preprint MATH-NM , TU Dresden, 1993 (submitted for publication). [13] Scheithauer, G., and Terno, J., A branch&bound algorithm for solving onedimensional cutting stock problems exactly, Working Paper, TU Dresden, 1994 (submitted for publication). 11
12 [14] Terno, J., Lindemann, R., und Scheithauer, G., Zuschnittprobleme und ihre praktische Lösung, Verlag Harry Deutsch, Thun und Frankfurt/Main, und Fachbuchverlag Leipzig [15] Wscher, G., and Gau, T., Two approaches to the cutting stock problem, IFORS 93 Conference, Lisboa
A BRANCH&BOUND ALGORITHM FOR SOLVING ONE-DIMENSIONAL CUTTING STOCK PROBLEMS EXACTLY
APPLICATIONES MATHEMATICAE 23,2 (1995), pp. 151 167 G. SCHEITHAUER and J. TERNO (Dresden) A BRANCH&BOUND ALGORITHM FOR SOLVING ONE-DIMENSIONAL CUTTING STOCK PROBLEMS EXACTLY Abstract. Many numerical computations
More informationAbout the gap between the optimal values of the integer and continuous relaxation one-dimensional cutting stock problem
Operations Research Proceedings 1991, Springer-Verlag Berlin Heidelberg (1992) 439 444 About the gap between the optimal values of the integer and continuous relaxation one-dimensional cutting stock problem
More informationA Node-Flow Model for 1D Stock Cutting: Robust Branch-Cut-and-Price
A Node-Flow Model for 1D Stock Cutting: Robust Branch-Cut-and-Price Gleb Belov University of Dresden Adam N. Letchford Lancaster University Eduardo Uchoa Universidade Federal Fluminense August 4, 2011
More informationLecture 8: Column Generation
Lecture 8: Column Generation (3 units) Outline Cutting stock problem Classical IP formulation Set covering formulation Column generation A dual perspective 1 / 24 Cutting stock problem 2 / 24 Problem description
More informationInteger Solutions to Cutting Stock Problems
Integer Solutions to Cutting Stock Problems L. Fernández, L. A. Fernández, C. Pola Dpto. Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005 Santander, Spain, laura.fernandezfern@alumnos.unican.es,
More information1 Column Generation and the Cutting Stock Problem
1 Column Generation and the Cutting Stock Problem In the linear programming approach to the traveling salesman problem we used the cutting plane approach. The cutting plane approach is appropriate when
More informationLecture 8: Column Generation
Lecture 8: Column Generation (3 units) Outline Cutting stock problem Classical IP formulation Set covering formulation Column generation A dual perspective Vehicle routing problem 1 / 33 Cutting stock
More informationA Branch-and-Cut-and-Price Algorithm for One-Dimensional Stock Cutting and Two-Dimensional Two-Stage Cutting
A Branch-and-Cut-and-Price Algorithm for One-Dimensional Stock Cutting and Two-Dimensional Two-Stage Cutting G. Belov,1 G. Scheithauer University of Dresden, Institute of Numerical Mathematics, Mommsenstr.
More informationTechnische Universität Dresden Herausgeber: Der Rektor
Als Manuskript gedruckt Technische Universität Dresden Herausgeber: Der Rektor Models with Variable Strip Widths for Two-Dimensional Two-Stage Cutting G. Belov, G. Scheithauer MATH-NM-17-2003 October 8,
More informationMVE165/MMG631 Linear and integer optimization with applications Lecture 8 Discrete optimization: theory and algorithms
MVE165/MMG631 Linear and integer optimization with applications Lecture 8 Discrete optimization: theory and algorithms Ann-Brith Strömberg 2017 04 07 Lecture 8 Linear and integer optimization with applications
More informationColumn Generation. MTech Seminar Report. Soumitra Pal Roll No: under the guidance of
Column Generation MTech Seminar Report by Soumitra Pal Roll No: 05305015 under the guidance of Prof. A. G. Ranade Computer Science and Engineering IIT-Bombay a Department of Computer Science and Engineering
More informationImproving Branch-And-Price Algorithms For Solving One Dimensional Cutting Stock Problem
Improving Branch-And-Price Algorithms For Solving One Dimensional Cutting Stock Problem M. Tech. Dissertation Submitted in partial fulfillment of the requirements for the degree of Master of Technology
More information3.10 Column generation method
3.10 Column generation method Many relevant decision-making problems can be formulated as ILP problems with a very large (exponential) number of variables. Examples: cutting stock, crew scheduling, vehicle
More information3.10 Column generation method
3.10 Column generation method Many relevant decision-making (discrete optimization) problems can be formulated as ILP problems with a very large (exponential) number of variables. Examples: cutting stock,
More informationmaxz = 3x 1 +4x 2 2x 1 +x 2 6 2x 1 +3x 2 9 x 1,x 2
ex-5.-5. Foundations of Operations Research Prof. E. Amaldi 5. Branch-and-Bound Given the integer linear program maxz = x +x x +x 6 x +x 9 x,x integer solve it via the Branch-and-Bound method (solving
More informationIntroduction to integer programming II
Introduction to integer programming II Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects of Optimization
More informationThe Number of Setups (Different Patterns) in One-Dimensional Stock Cutting
G. Belov and G. Scheithauer. Setup Minimization in 1D Stock Cutting 1 The Number of Setups (Different Patterns) in One-Dimensional Stock Cutting G. Belov and G. Scheithauer Dresden University Institute
More informationColumn Generation. i = 1,, 255;
Column Generation The idea of the column generation can be motivated by the trim-loss problem: We receive an order to cut 50 pieces of.5-meter (pipe) segments, 250 pieces of 2-meter segments, and 200 pieces
More informationInterior-Point versus Simplex methods for Integer Programming Branch-and-Bound
Interior-Point versus Simplex methods for Integer Programming Branch-and-Bound Samir Elhedhli elhedhli@uwaterloo.ca Department of Management Sciences, University of Waterloo, Canada Page of 4 McMaster
More informationColumn Generation I. Teo Chung-Piaw (NUS)
Column Generation I Teo Chung-Piaw (NUS) 21 st February 2002 1 Outline Cutting Stock Problem Slide 1 Classical Integer Programming Formulation Set Covering Formulation Column Generation Approach Connection
More informationmin3x 1 + 4x 2 + 5x 3 2x 1 + 2x 2 + x 3 6 x 1 + 2x 2 + 3x 3 5 x 1, x 2, x 3 0.
ex-.-. Foundations of Operations Research Prof. E. Amaldi. Dual simplex algorithm Given the linear program minx + x + x x + x + x 6 x + x + x x, x, x. solve it via the dual simplex algorithm. Describe
More informationA packing integer program arising in two-layer network design
A packing integer program arising in two-layer network design Christian Raack Arie M.C.A Koster Zuse Institute Berlin Takustr. 7, D-14195 Berlin Centre for Discrete Mathematics and its Applications (DIMAP)
More informationTechnische Universität Dresden Herausgeber: Der Rektor
Als Manuskript gedruckt Technische Universität Dresden Herausgeber: Der Rektor Setup and Open Stacks Minimization in One-Dimensional Stock Cutting G. Belov, G. Scheithauer MATH-NM-16-2003 June 24, 2004
More informationSection Notes 9. IP: Cutting Planes. Applied Math 121. Week of April 12, 2010
Section Notes 9 IP: Cutting Planes Applied Math 121 Week of April 12, 2010 Goals for the week understand what a strong formulations is. be familiar with the cutting planes algorithm and the types of cuts
More informationColumn Generation. ORLAB - Operations Research Laboratory. Stefano Gualandi. June 14, Politecnico di Milano, Italy
ORLAB - Operations Research Laboratory Politecnico di Milano, Italy June 14, 2011 Cutting Stock Problem (from wikipedia) Imagine that you work in a paper mill and you have a number of rolls of paper of
More informationIntroduction to Integer Linear Programming
Lecture 7/12/2006 p. 1/30 Introduction to Integer Linear Programming Leo Liberti, Ruslan Sadykov LIX, École Polytechnique liberti@lix.polytechnique.fr sadykov@lix.polytechnique.fr Lecture 7/12/2006 p.
More informationSection Notes 9. Midterm 2 Review. Applied Math / Engineering Sciences 121. Week of December 3, 2018
Section Notes 9 Midterm 2 Review Applied Math / Engineering Sciences 121 Week of December 3, 2018 The following list of topics is an overview of the material that was covered in the lectures and sections
More informationIntroduction to Bin Packing Problems
Introduction to Bin Packing Problems Fabio Furini March 13, 2015 Outline Origins and applications Applications: Definition: Bin Packing Problem (BPP) Solution techniques for the BPP Heuristic Algorithms
More informationLarge-scale optimization and decomposition methods: outline. Column Generation and Cutting Plane methods: a unified view
Large-scale optimization and decomposition methods: outline I Solution approaches for large-scaled problems: I Delayed column generation I Cutting plane methods (delayed constraint generation) 7 I Problems
More informationInteger Programming ISE 418. Lecture 8. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 8 Dr. Ted Ralphs ISE 418 Lecture 8 1 Reading for This Lecture Wolsey Chapter 2 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.4.1, II.4.2, II.5.4 Duality for Mixed-Integer
More informationBOUNDS FOR THE NAKAMURA NUMBER
BOUNDS FOR THE NAKAMURA NUMBER JOSEP FREIXAS AND SASCHA KURZ ABSTRACT. The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of
More informationNetwork Flows. 6. Lagrangian Relaxation. Programming. Fall 2010 Instructor: Dr. Masoud Yaghini
In the name of God Network Flows 6. Lagrangian Relaxation 6.3 Lagrangian Relaxation and Integer Programming Fall 2010 Instructor: Dr. Masoud Yaghini Integer Programming Outline Branch-and-Bound Technique
More informationIntroduction to Integer Programming
Lecture 3/3/2006 p. /27 Introduction to Integer Programming Leo Liberti LIX, École Polytechnique liberti@lix.polytechnique.fr Lecture 3/3/2006 p. 2/27 Contents IP formulations and examples Total unimodularity
More informationInteger Programming. Wolfram Wiesemann. December 6, 2007
Integer Programming Wolfram Wiesemann December 6, 2007 Contents of this Lecture Revision: Mixed Integer Programming Problems Branch & Bound Algorithms: The Big Picture Solving MIP s: Complete Enumeration
More informationFeasibility Pump Heuristics for Column Generation Approaches
1 / 29 Feasibility Pump Heuristics for Column Generation Approaches Ruslan Sadykov 2 Pierre Pesneau 1,2 Francois Vanderbeck 1,2 1 University Bordeaux I 2 INRIA Bordeaux Sud-Ouest SEA 2012 Bordeaux, France,
More information3.4 Relaxations and bounds
3.4 Relaxations and bounds Consider a generic Discrete Optimization problem z = min{c(x) : x X} with an optimal solution x X. In general, the algorithms generate not only a decreasing sequence of upper
More informationIntroduction to optimization and operations research
Introduction to optimization and operations research David Pisinger, Fall 2002 1 Smoked ham (Chvatal 1.6, adapted from Greene et al. (1957)) A meat packing plant produces 480 hams, 400 pork bellies, and
More informationLinear and Integer Programming - ideas
Linear and Integer Programming - ideas Paweł Zieliński Institute of Mathematics and Computer Science, Wrocław University of Technology, Poland http://www.im.pwr.wroc.pl/ pziel/ Toulouse, France 2012 Literature
More informationMulticommodity Flows and Column Generation
Lecture Notes Multicommodity Flows and Column Generation Marc Pfetsch Zuse Institute Berlin pfetsch@zib.de last change: 2/8/2006 Technische Universität Berlin Fakultät II, Institut für Mathematik WS 2006/07
More informationLOWER BOUNDS FOR THE UNCAPACITATED FACILITY LOCATION PROBLEM WITH USER PREFERENCES. 1 Introduction
LOWER BOUNDS FOR THE UNCAPACITATED FACILITY LOCATION PROBLEM WITH USER PREFERENCES PIERRE HANSEN, YURI KOCHETOV 2, NENAD MLADENOVIĆ,3 GERAD and Department of Quantitative Methods in Management, HEC Montréal,
More informationLinear Programming. Scheduling problems
Linear Programming Scheduling problems Linear programming (LP) ( )., 1, for 0 min 1 1 1 1 1 11 1 1 n i x b x a x a b x a x a x c x c x z i m n mn m n n n n! = + + + + + + = Extreme points x ={x 1,,x n
More informationOperations Research Lecture 6: Integer Programming
Operations Research Lecture 6: Integer Programming Notes taken by Kaiquan Xu@Business School, Nanjing University May 12th 2016 1 Integer programming (IP) formulations The integer programming (IP) is the
More informationNotes on Dantzig-Wolfe decomposition and column generation
Notes on Dantzig-Wolfe decomposition and column generation Mette Gamst November 11, 2010 1 Introduction This note introduces an exact solution method for mathematical programming problems. The method is
More informationA Strongly Polynomial Simplex Method for Totally Unimodular LP
A Strongly Polynomial Simplex Method for Totally Unimodular LP Shinji Mizuno July 19, 2014 Abstract Kitahara and Mizuno get new bounds for the number of distinct solutions generated by the simplex method
More informationInteger Programming. The focus of this chapter is on solution techniques for integer programming models.
Integer Programming Introduction The general linear programming model depends on the assumption of divisibility. In other words, the decision variables are allowed to take non-negative integer as well
More informationFRACTIONAL PACKING OF T-JOINS. 1. Introduction
FRACTIONAL PACKING OF T-JOINS FRANCISCO BARAHONA Abstract Given a graph with nonnegative capacities on its edges, it is well known that the capacity of a minimum T -cut is equal to the value of a maximum
More informationA primal-simplex based Tardos algorithm
A primal-simplex based Tardos algorithm Shinji Mizuno a, Noriyoshi Sukegawa a, and Antoine Deza b a Graduate School of Decision Science and Technology, Tokyo Institute of Technology, 2-12-1-W9-58, Oo-Okayama,
More informationA p-median Model for Assortment and Trim Loss Minimization with an Application to the Glass Industry
A p-median Model for Assortment and Trim Loss Minimization with an Application to the Glass Industry Claudio Arbib, Fabrizio Marinelli Dipartimento di Informatica, Università degli Studi di L Aquila, via
More informationOn Counting Lattice Points and Chvátal-Gomory Cutting Planes
On Counting Lattice Points and Chvátal-Gomory Cutting Planes Andrea Lodi 1, Gilles Pesant 2, and Louis-Martin Rousseau 2 1 DEIS, Università di Bologna - andrea.lodi@unibo.it 2 CIRRELT, École Polytechnique
More informationLecture 23 Branch-and-Bound Algorithm. November 3, 2009
Branch-and-Bound Algorithm November 3, 2009 Outline Lecture 23 Modeling aspect: Either-Or requirement Special ILPs: Totally unimodular matrices Branch-and-Bound Algorithm Underlying idea Terminology Formal
More informationInteger Linear Programming Models for 2-staged Two-Dimensional Knapsack Problems. Andrea Lodi, Michele Monaci
Integer Linear Programming Models for 2-staged Two-Dimensional Knapsack Problems Andrea Lodi, Michele Monaci Dipartimento di Elettronica, Informatica e Sistemistica, University of Bologna Viale Risorgimento,
More informationInteger Programming ISE 418. Lecture 16. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 16 Dr. Ted Ralphs ISE 418 Lecture 16 1 Reading for This Lecture Wolsey, Chapters 10 and 11 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.3.7, II.5.4 CCZ Chapter 8
More informationSemi-Infinite Relaxations for a Dynamic Knapsack Problem
Semi-Infinite Relaxations for a Dynamic Knapsack Problem Alejandro Toriello joint with Daniel Blado, Weihong Hu Stewart School of Industrial and Systems Engineering Georgia Institute of Technology MIT
More informationCombinatorial Auction: A Survey (Part I)
Combinatorial Auction: A Survey (Part I) Sven de Vries Rakesh V. Vohra IJOC, 15(3): 284-309, 2003 Presented by James Lee on May 10, 2006 for course Comp 670O, Spring 2006, HKUST COMP670O Course Presentation
More informationOnline Scheduling of Parallel Jobs on Two Machines is 2-Competitive
Online Scheduling of Parallel Jobs on Two Machines is 2-Competitive J.L. Hurink and J.J. Paulus University of Twente, P.O. box 217, 7500AE Enschede, The Netherlands Abstract We consider online scheduling
More informationRelations between capacity utilization and minimal bin number
Als Manuskript gedruckt Technische Universität Dresden Herausgeber: Der Rektor A note on Relations between capacity utilization and minimal bin number Torsten Buchwald and Guntram Scheithauer MATH-NM-05-01
More informationSubmodular Functions and Their Applications
Submodular Functions and Their Applications Jan Vondrák IBM Almaden Research Center San Jose, CA SIAM Discrete Math conference, Minneapolis, MN June 204 Jan Vondrák (IBM Almaden) Submodular Functions and
More informationto work with) can be solved by solving their LP relaxations with the Simplex method I Cutting plane algorithms, e.g., Gomory s fractional cutting
Summary so far z =max{c T x : Ax apple b, x 2 Z n +} I Modeling with IP (and MIP, and BIP) problems I Formulation for a discrete set that is a feasible region of an IP I Alternative formulations for the
More informationLecture 11 October 7, 2013
CS 4: Advanced Algorithms Fall 03 Prof. Jelani Nelson Lecture October 7, 03 Scribe: David Ding Overview In the last lecture we talked about set cover: Sets S,..., S m {,..., n}. S has cost c S. Goal: Cover
More informationStochastic Submodular Cover with Limited Adaptivity
Stochastic Submodular Cover with Limited Adaptivity Arpit Agarwal Sepehr Assadi Sanjeev Khanna Abstract In the submodular cover problem, we are given a non-negative monotone submodular function f over
More informationCombinatorial Optimization
Combinatorial Optimization Lecture notes, WS 2010/11, TU Munich Prof. Dr. Raymond Hemmecke Version of February 9, 2011 Contents 1 The knapsack problem 1 1.1 Complete enumeration..................................
More informationOn the Chvatál-Complexity of Binary Knapsack Problems. Gergely Kovács 1 Béla Vizvári College for Modern Business Studies, Hungary
On the Chvatál-Complexity of Binary Knapsack Problems Gergely Kovács 1 Béla Vizvári 2 1 College for Modern Business Studies, Hungary 2 Eastern Mediterranean University, TRNC 2009. 1 Chvátal Cut and Complexity
More informationLecture #21. c T x Ax b. maximize subject to
COMPSCI 330: Design and Analysis of Algorithms 11/11/2014 Lecture #21 Lecturer: Debmalya Panigrahi Scribe: Samuel Haney 1 Overview In this lecture, we discuss linear programming. We first show that the
More information5 Integer Linear Programming (ILP) E. Amaldi Foundations of Operations Research Politecnico di Milano 1
5 Integer Linear Programming (ILP) E. Amaldi Foundations of Operations Research Politecnico di Milano 1 Definition: An Integer Linear Programming problem is an optimization problem of the form (ILP) min
More informationA Bound for the Number of Different Basic Solutions Generated by the Simplex Method
ICOTA8, SHANGHAI, CHINA A Bound for the Number of Different Basic Solutions Generated by the Simplex Method Tomonari Kitahara and Shinji Mizuno Tokyo Institute of Technology December 12th, 2010 Contents
More informationConnectedness of Efficient Solutions in Multiple. Objective Combinatorial Optimization
Connectedness of Efficient Solutions in Multiple Objective Combinatorial Optimization Jochen Gorski Kathrin Klamroth Stefan Ruzika Communicated by H. Benson Abstract Connectedness of efficient solutions
More informationDual fitting approximation for Set Cover, and Primal Dual approximation for Set Cover
duality 1 Dual fitting approximation for Set Cover, and Primal Dual approximation for Set Cover Guy Kortsarz duality 2 The set cover problem with uniform costs Input: A universe U and a collection of subsets
More informationDiscrete (and Continuous) Optimization WI4 131
Discrete (and Continuous) Optimization WI4 131 Kees Roos Technische Universiteit Delft Faculteit Electrotechniek, Wiskunde en Informatica Afdeling Informatie, Systemen en Algoritmiek e-mail: C.Roos@ewi.tudelft.nl
More informationThe Strength of Multi-row Aggregation Cuts for Sign-pattern Integer Programs
The Strength of Multi-row Aggregation Cuts for Sign-pattern Integer Programs Santanu S. Dey 1, Andres Iroume 1, and Guanyi Wang 1 1 School of Industrial and Systems Engineering, Georgia Institute of Technology
More informationand to estimate the quality of feasible solutions I A new way to derive dual bounds:
Lagrangian Relaxations and Duality I Recall: I Relaxations provide dual bounds for the problem I So do feasible solutions of dual problems I Having tight dual bounds is important in algorithms (B&B), and
More informationThe use of predicates to state constraints in Constraint Satisfaction is explained. If, as an
of Constraint Satisfaction in Integer Programming H.P. Williams Hong Yan Faculty of Mathematical Studies, University of Southampton, Southampton, UK Department of Management, The Hong Kong Polytechnic
More informationDiscrete Optimization 2010 Lecture 7 Introduction to Integer Programming
Discrete Optimization 2010 Lecture 7 Introduction to Integer Programming Marc Uetz University of Twente m.uetz@utwente.nl Lecture 8: sheet 1 / 32 Marc Uetz Discrete Optimization Outline 1 Intro: The Matching
More informationSemi-Simultaneous Flows and Binary Constrained (Integer) Linear Programs
DEPARTMENT OF MATHEMATICAL SCIENCES Clemson University, South Carolina, USA Technical Report TR2006 07 EH Semi-Simultaneous Flows and Binary Constrained (Integer Linear Programs A. Engau and H. W. Hamacher
More informationOutline. Relaxation. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Lagrangian Relaxation. Lecture 12 Single Machine Models, Column Generation
Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. Lagrangian Relaxation Lecture 12 Single Machine Models, Column Generation 2. Dantzig-Wolfe Decomposition Dantzig-Wolfe Decomposition Delayed Column
More informationCarathéodory Bounds for Integer Cones
Carathéodory Bounds for Integer Cones Friedrich Eisenbrand, Gennady Shmonin Max-Planck-Institut für Informatik Stuhlsatzenhausweg 85 66123 Saarbrücken Germany [eisen,shmonin]@mpi-inf.mpg.de 22nd September
More informationMonoidal Cut Strengthening and Generalized Mixed-Integer Rounding for Disjunctions and Complementarity Constraints
Monoidal Cut Strengthening and Generalized Mixed-Integer Rounding for Disjunctions and Complementarity Constraints Tobias Fischer and Marc E. Pfetsch Department of Mathematics, TU Darmstadt, Germany {tfischer,pfetsch}@opt.tu-darmstadt.de
More informationa 1 a 2 a 3 a 4 v i c i c(a 1, a 3 ) = 3
AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 17 March 30th 1 Overview In the previous lecture, we saw examples of combinatorial problems: the Maximal Matching problem and the Minimum
More informationRelative Improvement by Alternative Solutions for Classes of Simple Shortest Path Problems with Uncertain Data
Relative Improvement by Alternative Solutions for Classes of Simple Shortest Path Problems with Uncertain Data Part II: Strings of Pearls G n,r with Biased Perturbations Jörg Sameith Graduiertenkolleg
More informationCOT 6936: Topics in Algorithms! Giri Narasimhan. ECS 254A / EC 2443; Phone: x3748
COT 6936: Topics in Algorithms! Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 giri@cs.fiu.edu https://moodle.cis.fiu.edu/v2.1/course/view.php?id=612 Gaussian Elimination! Solving a system of simultaneous
More informationInteger Linear Programming (ILP)
Integer Linear Programming (ILP) Zdeněk Hanzálek, Přemysl Šůcha hanzalek@fel.cvut.cz CTU in Prague March 8, 2017 Z. Hanzálek (CTU) Integer Linear Programming (ILP) March 8, 2017 1 / 43 Table of contents
More informationImproved Algorithms for Machine Allocation in Manufacturing Systems
Improved Algorithms for Machine Allocation in Manufacturing Systems Hans Frenk Martine Labbé Mario van Vliet Shuzhong Zhang October, 1992 Econometric Institute, Erasmus University Rotterdam, the Netherlands.
More informationInteger Programming Chapter 15
Integer Programming Chapter 15 University of Chicago Booth School of Business Kipp Martin November 9, 2016 1 / 101 Outline Key Concepts Problem Formulation Quality Solver Options Epsilon Optimality Preprocessing
More informationWeek 8. 1 LP is easy: the Ellipsoid Method
Week 8 1 LP is easy: the Ellipsoid Method In 1979 Khachyan proved that LP is solvable in polynomial time by a method of shrinking ellipsoids. The running time is polynomial in the number of variables n,
More informationOptimization of Submodular Functions Tutorial - lecture I
Optimization of Submodular Functions Tutorial - lecture I Jan Vondrák 1 1 IBM Almaden Research Center San Jose, CA Jan Vondrák (IBM Almaden) Submodular Optimization Tutorial 1 / 1 Lecture I: outline 1
More informationOptimisation and Operations Research
Optimisation and Operations Research Lecture 15: The Greedy Heuristic Matthew Roughan http://www.maths.adelaide.edu.au/matthew.roughan/ Lecture_notes/OORII/ School of
More informationKNAPSACK PROBLEMS WITH SETUPS
7 e Conférence Francophone de MOdélisation et SIMulation - MOSIM 08 - du 31 mars au 2 avril 2008 - Paris - France Modélisation, Optimisation et Simulation des Systèmes : Communication, Coopération et Coordination
More informationOn NP-Completeness for Linear Machines
JOURNAL OF COMPLEXITY 13, 259 271 (1997) ARTICLE NO. CM970444 On NP-Completeness for Linear Machines Christine Gaßner* Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität, F.-L.-Jahn-Strasse
More informationOptimization Methods in Management Science
Optimization Methods in Management Science MIT 15.05 Recitation 8 TAs: Giacomo Nannicini, Ebrahim Nasrabadi At the end of this recitation, students should be able to: 1. Derive Gomory cut from fractional
More informationInteger program reformulation for robust branch-and-cut-and-price
Integer program reformulation for robust branch-and-cut-and-price Marcus Poggi de Aragão Informática PUC-Rio Eduardo Uchoa Engenharia de Produção Universidade Federal Fluminense Outline of the talk Robust
More informationOverview of course. Introduction to Optimization, DIKU Monday 12 November David Pisinger
Introduction to Optimization, DIKU 007-08 Monday November David Pisinger Lecture What is OR, linear models, standard form, slack form, simplex repetition, graphical interpretation, extreme points, basic
More information3.10 Lagrangian relaxation
3.10 Lagrangian relaxation Consider a generic ILP problem min {c t x : Ax b, Dx d, x Z n } with integer coefficients. Suppose Dx d are the complicating constraints. Often the linear relaxation and the
More informationAdvanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Advanced Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture - 3 Simplex Method for Bounded Variables We discuss the simplex algorithm
More informationSpring 2018 IE 102. Operations Research and Mathematical Programming Part 2
Spring 2018 IE 102 Operations Research and Mathematical Programming Part 2 Graphical Solution of 2-variable LP Problems Consider an example max x 1 + 3 x 2 s.t. x 1 + x 2 6 (1) - x 1 + 2x 2 8 (2) x 1,
More informationStabilized Branch-and-cut-and-price for the Generalized Assignment Problem
Stabilized Branch-and-cut-and-price for the Generalized Assignment Problem Alexandre Pigatti, Marcus Poggi de Aragão Departamento de Informática, PUC do Rio de Janeiro {apigatti, poggi}@inf.puc-rio.br
More informationA Dynamic Programming Heuristic for the Quadratic Knapsack Problem
A Dynamic Programming Heuristic for the Quadratic Knapsack Problem Franklin Djeumou Fomeni Adam N. Letchford March 2012 Abstract It is well known that the standard (linear) knapsack problem can be solved
More informationInteger Linear Programming
Integer Linear Programming Solution : cutting planes and Branch and Bound Hugues Talbot Laboratoire CVN April 13, 2018 IP Resolution Gomory s cutting planes Solution branch-and-bound General method Resolution
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should
More informationInteger programming, Barvinok s counting algorithm and Gomory relaxations
Integer programming, Barvinok s counting algorithm and Gomory relaxations Jean B. Lasserre LAAS-CNRS, Toulouse, France Abstract We propose an algorithm based on Barvinok s counting algorithm for P max{c
More informationReconnect 04 Introduction to Integer Programming
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, Reconnect 04 Introduction to Integer Programming Cynthia Phillips, Sandia National Laboratories Integer programming
More informationKnapsack. Bag/knapsack of integer capacity B n items item i has size s i and profit/weight w i
Knapsack Bag/knapsack of integer capacity B n items item i has size s i and profit/weight w i Goal: find a subset of items of maximum profit such that the item subset fits in the bag Knapsack X: item set
More information