Round-up Property Research Articles

Constructing an instance of the cutting stock problem of minimum size which does not possess the integer round-up property

Constructing an instance of the cutting stock problem of minimum size which does not possess the integer round-up property

Constructing an Instance of the Cutting Stock Problem of Minimum Size Which Does Not Possess the Integer Round-Up Property

Constructing an Instance of the Cutting Stock Problem of Minimum Size Which Does Not Possess the Integer Round-Up Property

Complexity and approximations for submodular minimization problems on two variables per inequality constraints

We demonstrate here that submodular minimization (SM) problems subject to constraints containing up to two variables per inequality, SM2, are 2-approximable in polynomial time and a better approximation factor cannot be achieved in polynomial time unless NP=P. The 2-approximation holds when either the constraints have the round-up property (the rounding up of a feasible fractional solution is feasible) or, if the constraints do not have this property, for monotone submodular functions. The submodular minimization or supermodular maximization on constraints where the coefficients of the two variables in each constraint are of opposite signs (monotone constraints) is solvable in polynomial time. The 2-approximability and the polynomial time solvability for monotone constraints hold also for multi-sets that contain elements with integer multiplicity greater than 1, except that the running time is then pseudo-polynomial in that it depends on the range of the variables. This complexity cannot be improved unless NP=P.Our results indicate that SM2 problems are not much harder than the respective linear integer problems on two variables per constraint: For monotone constraints both problems are polynomial time solvable, and for non-monotone NP-hard problems, both problems have 2-approximation algorithms. For SM2 problems the factor 2 approximation is provably best possible, whereas for the respective linear integer problems it has not been established that the factor 2 is best possible, but this has been conjectured. On the other hand, for SM2 problems where the two variables constraints’ coefficients form a totally unimodular constraint matrix, the linear integer optimization problem is solved in polynomial time, whereas the submodular optimization is proved here to be NP-hard.The submodular minimization NP-hard problems for which our general purpose 2-approximation algorithm applies include submodular-vertex cover, submodular-2SAT, submodular-min satisfiability, submodular-edge deletion for clique, submodular-node deletion for biclique and others.

Integer round-up property for the chromatic number of some h-perfect graphs

We prove that every h-perfect line graph and every t-perfect claw-free graph G has the integer round-up property for the chromatic number: for every non-negative integral weight function c on the vertices of G, the weighted chromatic number of (G, c) can be obtained by rounding up its fractional relaxation. As a corollary, we obtain that the weighted chromatic number can be computed in polynomial-time for these graphs. Finally, we show a new example of a graph operation which preserves the integer round-up property for the chromatic number, and use it to provide a first example of a t-perfect 3-colorable graph which does not have this property.

Minimal proper non-IRUP instances of the one-dimensional cutting stock problem

We consider the well-known one dimensional cutting stock problem (1CSP). Based on the pattern structure of the classical ILP formulation of Gilmore and Gomory, we can decompose the infinite set of 1CSP instances, with a fixed number n of demanded pieces, into a finite number of equivalence classes. We show up a strong relation to weighted simple games. Studying the integer round-up property (IRUP) we use the proper LP relaxation of the Gilmore and Gomory model that allows us to consider the 1CSP as the bin packing problem (BPP). We computationally show that all 1CSP instances with n≤9 have the proper IRUP, while we give examples of proper non-IRUP instances with n=10 and proper gap 1. Proper gaps larger than 1 occur for n≥11. The largest known proper gap is raised from 1.003 to 1.0625. The used algorithmic approaches are based on exhaustive enumeration and integer linear programming. Additionally we give some theoretical bounds showing that all 1CSP instances with some specific parameters have the proper IRUP.

Friendly bin packing instances without Integer Round-up Property

It is well known that the gap between the optimal values of bin packing and fractional bin packing, if the latter is rounded up to the closest integer, is almost always null. Known counterexamples to this for integer input values involve fairly large numbers. Specifically, the first one was derived in 1986 and involved a bin capacity of the order of a billion. Later in 1998 a counterexample with a bin capacity of the order of a million was found. In this paper we show a large number of counterexamples with bin capacity of the order of a hundred, showing that the gap may be positive even for numbers which arise in customary applications. The associated instances are constructed starting from the Petersen graph and using the fact that it is fractionally, but not integrally, 3-edge colorable.

Exact and heuristic solutions for combinatorial optimization problems

This is a summary of the Ph.D. thesis defended by the author on July 2012 at the University of Modena and Reggio Emilia. The thesis was supervised by Mauro Dell’Amico and co-supervised by Manuel Iori. The manuscript is written in English and is available from the author upon request at jose.diaz@unimore.it. The thesis addresses relevant optimization problems: the Bin Packing Problem, the Quadratic Assignment Problem, the large-scale Energy Management Problem, and the Node, Edge and Arc Routing Problem. To solve these problems exact, heuristic and local search techniques are proposed. The first problem studied is the Bin Packing Problem with Precedence Constraints (BPP-P): given a set of identical capacitated bins, a set of weighted items and a set of precedences among those items, we are interested in determining the minimum number of bins that can accommodate all items and can be ordered in such a way that all precedences are satisfied. The problem has a very intriguing combinatorial structure and models many assembly and scheduling issues. According to our knowledge the BPP-P has received little attention in the literature, and in the thesis we address it for the first time with exact solution methods. In particular, we develop reduction criteria, a large set of lower bounds, a variable neighborhood search upper bounding technique and a branch-and-bound algorithm (see Dell’Amico M., Diaz Diaz J.C. and Iori M. (2012): The Bin Packing Problem with Precedence Constraints, Operations Research, doi:10.1287/opre.1120.1109). We show the effectiveness of the proposed algorithms by means of extensive computational tests on benchmark instances and comparison with standard integer linear programming techniques. The second problem studied is to find the Friendly Bin Packing Instances without Integer Round-up Property. It is well known that the gap between the optimal values of bin packing and fractional bin packing, if the latter is rounded up to the closest integer, is almost always null. Known counterexamples to this for integer input values involve fairly large numbers. Specifically, the first one was derived in 1986 and involved a bin capacity of the order of a billion. Later in 1998 a counterexample with a bin capacity of the order of a million was found. In the thesis we show a large number of counterexamples with bin capacity of the order of a hundred, showing that the gap may be positive even for numbers which arise in customary applications. The associated instances are constructed starting from the Petersen graph and taking advantage of the fact that it is fractionally, but not integrally, 3-edge colorable. The third problem studied is the Single-finger Keyboard Layout Problem. The problem of designing new keyboards layouts able to improve the typing speed of an

A simple [formula omitted] algorithm for cutting stock under the modified integer round-up property assumption

We present a simple algorithm for the cutting stock problem with objects of a constant number d of different sizes. Our algorithm produces solutions of value at most OPT + 1 in time d O ( d ) ( log 7 n + s 3.5 ) , where OPT is the value of an optimum solution, n is the number of objects, and s is the total number of bits needed to encode the object sizes. This algorithm works under the assumption that the modified integer round-up property of Scheithauer and Terno for the cutting stock problem holds.

Sufficient Conditions for the Integer Round-Up Property to Be Violated for the Linear Cutting Stock Problem

The integer round-up property of the linear cutting stock problem is investigated. An algorithm for verifying the sufficient conditions under which the problem has no integer round-up property is designed.

Families of non-IRUP instances of the one-dimensional cutting stock problem

In case of the one-dimensional cutting stock problem (CSP) one can observe for any instance a very small gap between the integer optimal value and the continuous relaxation bound. These observations have initiated a series of investigations. An instance possesses the integer round-up property (IRUP) if its gap is smaller than 1. In the last 15 years, some few instances of the CSP were published possessing a gap greater than 1. In this paper, various families of non-IRUP instances are presented and methods to construct such instances are given, showing in this way, there exist much more non-equivalent non-IRUP instances as computational experiments with randomly generated instances suggest. Especially, an instance with gap equal to 10 9 is obtained. Furthermore, an equivalence relation for instances of the CSP is considered to become independent from the real size parameters.

The round-up property of the fractional chromatic number for proper circular arc graphs

Let G = (V, E) be a graph and let k be a nonnegative integer. A vector c ∈ ℤ is called k-colorable iff there exists a coloring of G with k colors that assigns exactly c(v) colors to vertex v ∈ V. Denote by χ (G) and χf(G) the chromatic number and fractional chromatic number, respectively. We prove that χ(G) = ⌈χf(G)⌉ holds for every proper circular arc graph G. For this purpose, a more general round-up property is characterized by means of a polyhedral description of all k-colorable vectors. Both round-up properties are equivalent for proper circular arc graphs. The polyhedral description is established and, as a by-product, a known coloring algorithm is generalized to multicolorings. The round-up properties do not hold for the larger classes of circular arc graphs and circle graphs, unless P = NP. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 256–267, 2000

The round-up property of the fractional chromatic number for proper circular arc graphs

The round-up property of the fractional chromatic number for proper circular arc graphs

Tighter relaxations for the cutting stock problem

In the cutting stock problem (CSP) a given order for smaller pieces has to be cut from larger stock material in such a way that the number of stock material needed is minimal. Based on the classical integer linear programming model the common solution technique consists of solving the corresponding continuous relaxation problem followed by several heuristics which construct integer solutions. In many cases an optimal solution can be obtained quickly in this way. But for instances which do not possess the integer round-up property the optimality of the solution obtained cannot be verified by means of the LP bound. In order to overcome this non-satisfactory situation, two tighter relaxations of the CSP are proposed, and results of theoretical and numerical investigations are presented.

New cases of the cutting stock problem having MIRUP

The modified integer round-up property (MIRUP) for a linear integer minimization problem means that the optimal value of this problem is not greater than the optimal value of the corresponding LP relaxation rounded up plus one. In earlier papers the MIRUP was shown to hold for the so-called divisible case and some other subproblems of the one-dimensional cutting stock problem. In this paper we extend the results by using special transformations, proving the existence of non-elementary patterns in the solution, and using more detailed greedy techniques.

Theoretical investigations on the modified integer round-up property for the one-dimensional cutting stock problem

Many numerical computations show a small difference only between the optimal value of the one-dimensional cutting stock problem and that of its corresponding linear programming relaxation. In this paper we investigate the one-dimensional cutting stock problem with respect to the modified integer round-up property (MIRUP) and present some results on subproblems having the MIRUP.

The modified integer round-up property of the one-dimensional cutting stock problem

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.

The cutting stock problem and integer rounding

An integer programming problem is said to have the integer round-up property if its optimal value is given by the least integer greater than or equal to the optimal value of its linear programming relaxation. In this paper we prove that certain classes of cutting stock problems have the integer round-up property. The proof of these results relies upon the decomposition properties of certain knapsack polyhedra.

A polynomial algorithm for integer programming covering problems satisfying the integer round-up property

LetA be a non-negative matrix with integer entries and no zero column. The integer round-up property holds forA if for every integral vectorw the optimum objective value of the generalized covering problem min{1y: yA ź w, y ź 0 integer} is obtained by rounding up to the nearest integer the optimum objective value of the corresponding linear program. A polynomial time algorithm is presented that does the following: given any generalized covering problem with constraint matrixA and right hand side vectorw, the algorithm either finds an optimum solution vector for the covering problem or else it reveals that matrixA does not have the integer round-up property.

Finite checkability for integer rounding properties in combinatorial programming problems

LetA be a nonnegative integral matrix with no zero columns. Theinteger round-up property holds forA if for each nonnegative integral vectorw, the solution value to the integer programming problem min{1 źy: yA ź w, y ź 0, y integer} is obtained by rounding up to the nearest integer the solution value to the corresponding linear programming problem min{1 źy: yA ź w, y ź 0}. Theinteger round-down property is similarly defined for a nonnegative integral matrixB with no zero rows by considering max{1 źy: yB ≤ w, y ź 0, y integer} and its linear programming correspondent. It is shown that the integer round-up and round-down properties can be checked through a finite process. The method of proof motivates a new and elementary proof of Fulkerson's Pluperfect Graph Theorem.

The NP-Completeness of Edge-Coloring

We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.