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This paper describes methods to merge two cover inequalities and also simultaneously merge multiple cover inequalities in a multiple knapsack instance. Theoretical results provide conditions under which merged cover inequalities are valid. Polynomial time algorithms are created to find merged cover inequalities. A computational study demonstrates that merged inequalities improve the solution times for benchmark multiple knapsack instances by about 9% on average over CPLEX with default settings.

An integer program (IP) is a common type of optimization problem, defined as maximize

One frequently studied IP is the 0 - 1 knapsack problem (KP), defined as maximize

multiple knapsack constraints and is defined as maximize

Solutions to KP and MK problems support a wide variety of real-world applications, including examples in Ahuja and Cunha [

A half space is

spaces. A set

Let P be the set of feasible points of an integer program, where

multiple knapsack problems, respectively where

A well-known technique to improve solution times for IP problems is the generation of valid inequalities. An

inequality

valid inequality separates the linear relaxation solution from the convex hull of the IP, then it is called a cutting plane. The linear relaxation is the IP with the integrality restriction eliminated. The theoretically best cutting planes define facets of

For a MK problem, a cover cut may be generated in one or more of the m constraints. A set

cover for row

et al. [

Many such covers may exist and pseudo-costing strategies provide a prioritized variable ordering. Pseudo- costing strategies for integer programming problems were studied by Benichou, et al. in [

In some instances, cover inequalities may be strengthened through lifting. Gomory introduced the technique in [

Theoretical foundations for inequality merging were first introduced by Hickman and Easton in [

This paper extends the idea of inequality merging by focusing on cover inequalities in MK problems. Information from two or more cover inequalities in an MK instance may be merged into a single cutting plane. In some instances, simultaneous merging of cover inequalities may occur across multiple rows at the same time.

The next section describes the process of cover inequality merging for MK instances and provides theoretical results and examples. The third section offers the results of a computational study that highlights the computa- tional benefits of employing merged cover inequalities in test MK problems. The final section offers some directions for future research.

It is straightforward to find cover inequalities in MK instances and merging requires two covers, called host and donor. Let

the cover inequalities

Merging the host and donor cover inequalities occurs on binary variable

If the merged inequality is valid, then this inequality includes more nonzero coefficients than either

Theorem 1. Let

instance for each

is valid for

Proof. Let

Theorem 1 describes which indices can be used to create a donor cover. These candidate indices can be easily found based upon a

inequality as shown in the following theorem.

Theorem 2. Given a multiple knapsack instance, a host cover

Proof. Assume

First, assume

property that

Second, assume^{donor} is a cover in row s,

Consequently,

These two cases are exhaustive. Therefore every

To identify valid merged cover inequalities, the user must identify a host cover,

The input to the Reducing

Reducing

If the Reducing

condition that

mined overlapped variable

ficients may identify acceptable additional variables for use in

In some instances, the Reducing

Reducing

Observe that the Reducing

High values of

The Reducing

the algorithm runs in

This section presents a method to strengthen the previous results by merging on multiple donor covers at the same time. Conditions are provided to create valid inequalities from merging over three or more cover inequa- lities simultaneously. Another algorithm is presented to search for the strongest merging coefficients among multiple potential donor rows in the MK instance.

Simultaneous merging over multiple donor covers begins with a

maximum cardinality cover from any row.

The check of validity must assure that there does not exist a feasible point which violates this new inequality.

Prior to this result, define

Theorem 3. Let

associated set

the following conditions holds

1)

2)

Proof. Let

Assume 1) is true. If

Assume 2) is true. Let

An immediate result of Theorem 3 is an algorithm to merge over multiple donor covers simultaneously. This algorithm explores all rows to determine the smallest eligible covers of each merging variable in

Donor Coeffcient Strengthening Algorithm

DCSA identifies the smallest donor covers possible for each index in

instance using the indices sorted in each row by the a values. Observe that DCSA does not guarantee a valid inequality, but it does identify the strongest possible merged inequality. If the reported merged inequality satisfies a condition of Theorem 3, then it is a valid inequality.

DCSA’s computational effort required for the initialization is

The following example demonstrates the theoretical concepts discussed earlier. Consider multiple knapsack constraints of the form

and

Designate the first constraint as the host constraint,

No subset of

smaller

or equal to 5. In this case {5} is eliminated from the host cover, and the host cover adds an index with a coefficient between 5 and 9. Indices 10, 11, 12, and 13 are all suitable and index 11 is added to

The algorithm has now determined a host and donor cover that can be merged. Merging the host with the donor on

The following arguments demonstrate Theorems 1 and 2 in practice. Verifying the validity of (1) requires that

Observe that numerous other minimal donor covers exist when

Each of these merged inequalities remove linear relaxation points and are thus cutting planes. For instance,

the point (1,1,1,1,0,0,0,0,0,

simple to find points that are satisfied by two of the three merged inequalities, but eliminated by the other inequality. Thus, each merged inequality is eliminating distinct regions of the linear relaxation space.

Returning to the original host cover, it is also possible to generate new families of merged inequalities if merging on

The idea of

The authors believe that such constraints may be more useful computationally since they are incorporating

covers from multiple constraints to obtain validity. For instance, the linear relaxation point (1,1,1,

To demonstrate Theorem 3, an additional row is added to this example. Now consider the following multiple knapsack instance

and

Again, consider

For index 5, the smallest covers are

Thus the simultaneous merged inequality is

Observe that this new inequality dominates all of the previous inequalities. Furthermore, to achieve this inequality all rows are necessary. For instance, the smallest cover in row 3 containing index 6 has 6 indices and thus row two is necessary. Similarly, the smallest cover in row 2 containing index 7 has 6 indices and thus row 3 is necessary.

Index | Smallest Cover | Row | |
---|---|---|---|

5 | 3 | ||

6 | 2 | ||

7 | 3 | ||

8 | 3 | ||

9 | 3 | ||

10 | 2 | ||

12 | 2 | ||

13 | 3 |

To argue validity of (6), consider Theorem 3. Since

Determining the values for

Since

The final benefit of this example demonstrates that merging cover inequalities are not an immediate extension of known methods. There are similarities between inequality merging and some categories of lifting. Any type of sequential lifting has integer coefficients [

The general inequality merging presented by Hickman and Easton in [

A single call to DCSA creates (6) and requires

This computational study compares solution times for multiple knapsack problems both with and without the use of merged inequalities. The instances chosen for this study are the MK instances from the OR-Library [

Each file contains 30 instances divided into groups of 10 based upon a tightness ratio, which is equal to

ces, 0.5 for the second ten instances, and 0.75 for the final ten instances. For this computational study, the first ten instances are only considered. When the tightness ratio is 0.5 or higher,

The study considers a variety of implementation strategies including the number of merged inequalities added, the possibility of overlapping rows when multiple cuts are added, the option to use the Donor Coefficient Strengthening Algorithm when constructing merged inequalities, and different pseudocosting techniques. The psuedocosting techniques provide an order for selecting indices for cover inequalities. Three options are con- sidered: sorting on the reduced costs, sorting on the a coefficient values, and sorting on equal weights for both reduced costs and a coefficient values. More details of these methods and computational results are described in [

The experimentation compares computational effort to solve the MK instances with and without the inclusion of merged cover inequalities. CPLEX 12.5 [

The computational study considered the variations of each implementation strategy by testing both small and large instances. Solving all 10 smaller instances required from 10 to 15 minutes. Solving all 10 larger instances typically needed 1 to 2 days. Instead of reporting the time in seconds, the data below compares computational ticks in CPLEX, as this is more accurate. It should be noted that the time in seconds was highly correlated to ticks. The overall improvement in time was plus or minus two percent of the percent improvement in ticks.

Ticks provide a more accurate comparison between the experimental runs because the computational time in seconds is subject to variability on different computers. Fischetti, et al. argue the benefit of using ticks in [

Problems from the smaller MK instances (file mknapcb2) offered an excellent opportunity for extensive experimentation with each of the implementation strategies.

# Merged | Overlap | Pseudo-Costing Strategy | Total Ticks | Percent | ||
---|---|---|---|---|---|---|

Cuts | Rows | Red. Costs | Balanced | a Values | (10 probs.) | Improv. |

Baseline | Baseline | 0 | 0 | 0 | 81,497 | Baseline |

1 | N/A | 1 | 0 | 0 | 70,895 | 13.0% |

1 | N/A | 0 | 0 | 1 | 69,669 | 14.5% |

1 | N/A | 0 | 1 | 0 | 75,868 | 6.9% |

2 | Yes | 2 | 0 | 0 | 72,376 | 11.2% |

2 | Yes | 0 | 0 | 2 | 78,840 | 3.3% |

2 | Yes | 0 | 2 | 0 | 71,668 | 12.1% |

2 | Yes | 1 | 0 | 1 | 71,305 | 12.5% |

2 | Yes | 0 | 1 | 1 | 67,634 | 17.0% |

2 | Yes | 1 | 1 | 0 | 76,272 | 6.4% |

2 | No | 2 | 0 | 0 | 81,022 | 0.6% |

2 | No | 0 | 0 | 2 | 76,956 | 5.6% |

2 | No | 0 | 2 | 0 | 77,947 | 4.4% |

2 | No | 1 | 0 | 1 | 64,417 | 21.0% |

2 | No | 0 | 1 | 1 | 72,356 | 11.2% |

2 | No | 1 | 1 | 0 | 79,088 | 3.0% |

3 | Yes | 3 | 0 | 0 | 74,985 | 8.0% |

3 | Yes | 0 | 0 | 3 | 72,123 | 11.5% |

3 | Yes | 0 | 3 | 0 | 67,794 | 16.8% |

3 | Yes | 1 | 1 | 1 | 72,593 | 10.9% |

3 | No | 3 | 0 | 0 | 80,178 | 1.6% |

3 | No | 0 | 0 | 3 | 75,445 | 7.4% |

3 | No | 0 | 3 | 0 | 77,490 | 4.9% |

3 | No | 1 | 1 | 1 | 77,448 | 5.0% |

Merged Average | 74,099 | 9.1% |

# Merged | Overlap | Pseudo-Costing Strategy | Total Ticks | Percent | ||
---|---|---|---|---|---|---|

Cuts | Rows | Red. Costs | Balanced | a Values | (10 probs.) | Improv. |

Baseline | Baseline | 0 | 0 | 0 | 81,497 | Baseline |

4 | Yes | 4 | 0 | 0 | 80,230 | 1.6% |

4 | Yes | 0 | 0 | 4 | 79,756 | 2.1% |

4 | Yes | 0 | 4 | 0 | 80,494 | 1.2% |

4 | Yes | 1 | 2 | 1 | 73,751 | 9.5% |

4 | No | 4 | 0 | 0 | 80,606 | 1.1% |

4 | No | 0 | 0 | 4 | 81,981 | −0.6% |

4 | No | 0 | 4 | 0 | 72,744 | 10.7% |

4 | No | 1 | 2 | 1 | 74,820 | 8.2% |

5 | Yes | 5 | 0 | 0 | 82,279 | −1.0% |

5 | Yes | 0 | 0 | 5 | 72,882 | 10.6% |

5 | Yes | 0 | 5 | 0 | 82,423 | −1.1% |

5 | Yes | 1 | 3 | 1 | 76,820 | 5.7% |

5 | No | 5 | 0 | 0 | 77,944 | 4.4% |

5 | No | 0 | 0 | 5 | 78,817 | 3.3% |

5 | No | 0 | 5 | 0 | 83,201 | −2.1% |

5 | No | 1 | 3 | 1 | 75,256 | 7.7% |

Merged Average | 78,375 | 3.8% |

from these experiments on the smaller MK instances. Since there are 5 rows in the smaller test problems, each implementation strategy was tested with the inclusion of 1 - 5 merged inequalities.

Observe that inequality merging outperformed the baseline CPLEX computational ticks for all strategies in

Merged inequalities almost always improved the computational time, regardless of the overlapping strategy. It appears that deliberate overlapping of rows provides even stronger results if multiple cutting planes are added. This is consistent with the theory motivating Theorem 3. Overlapping allows the algorithm to search in rows that had previously been used to generate a host cover inequality for an earlier merged cut. If DCSA is employed, the algorithm may also search all candidate rows including those that had previously generated a host inequality. Thus, all future experimentation overlaps rows.

As the problems increased in size, the computational time quickly increased. The same implementation strategies tended to yield the strongest results with larger problems, as shown in this section. Solving all 10 MK instances required from 1 to 2 days to solve.

Clearly a focus on reduced costs had the best impact for this particular grouping of larger MK instances, but that may not be the case in general. Previous analysis from

Pseudo-Costing Strategy | ||||
---|---|---|---|---|

All Reduced Costs | All Balanced | All a Values | Mixture of Strategies | |

Average Ticks | 77,835 | 76,625 | 76,274 | 73,569 |

% Improvement | 4.5% | 6.0% | 6.4% | 9.8% |

# Merged | Pseudo-Costing Strategy | Total Ticks | Percent | ||
---|---|---|---|---|---|

Cuts Added | Red. Costs | Balanced | a Values | (10 problems) | Improvement |

Baseline | 0 | 0 | 0 | 30,994,459 | Baseline |

1 | 1 | 0 | 0 | 29,949,459 | 3.4% |

1 | 0 | 0 | 1 | 30,268,076 | 2.3% |

1 | 0 | 1 | 0 | 29,614,573 | 4.5% |

2 | 2 | 0 | 0 | 20,166,265 | 34.9% |

2 | 0 | 0 | 2 | 29,347,409 | 5.3% |

2 | 0 | 2 | 0 | 30,881,549 | 0.4% |

2 | 1 | 0 | 1 | 28,518,016 | 8.0% |

2 | 0 | 1 | 1 | 29,975,494 | 3.3% |

2 | 1 | 1 | 0 | 29,718,811 | 4.1% |

3 | 3 | 0 | 0 | 20,412,908 | 34.1% |

3 | 0 | 0 | 3 | 29,362,710 | 5.3% |

3 | 0 | 3 | 0 | 30,908,925 | 0.3% |

3 | 1 | 1 | 1 | 29,903,185 | 3.5% |

Merged Average | 28,260,350 | 8.8% |

Problem | Baseline | Merging | Percent | Implementation Strategy | ||
---|---|---|---|---|---|---|

# | Ticks | Ticks | Improv. | Cuts | Pseudo-cost | Donor Rows |

1 | 1,955,055 | 128,467 | 93.4% | 3 cuts | Reduced Costs | All |

2 | 203,122 | 160,209 | 21.1% | 1 cuts | Balanced | Specified |

3 | 316,729 | 265,573 | 16.2% | 3 cuts | a Values | Specified |

4 | 1,964,804 | 1,710,877 | 12.9% | 2 cuts | Red. Cost & a Val. | Specified |

5 | 6,735,442 | 6,300,815 | 6.4% | 2 cuts | Red. Cost & a Val. | Specified |

6 | 331,058 | 288,987 | 12.7% | 1 cut | a Values | Specified |

7 | 224,004 | 208,500 | 6.9% | 1 cut | a Values | All |

8 | 17,630,931 | 5,993,211 | 66.0% | 5 cuts | Reduced Costs | Specified |

9 | 651,113 | 563,288 | 13.5% | 2 cuts | Red. Cost & a Val. | All |

10 | 982,201 | 895,267 | 8.8% | 3 cuts | a Values | Specified |

Average | 25.8% |

These larger problems are excellent representatives of difficult, real-world problems. Thus, the observed reductions in computational requirements validated the theoretical advancements in this research as effective methods to help decrease computational effort for modern MK problems.

This paper provides the theoretical foundations needed to build merged cover inequalities in MK instances. The theorems generate conditions for validity, using the

The computational study validates inequality merging as an effective technique that reduces computational time for multiple knapsack problems. Preferred implementation strategies should generate 1, 2, or 3 cuts and overlap the rows. These strategies provide the strongest results, yielding an average reduction of computational effort by about 9%. The computational study provides strong evidence that inequality merging yields productive cutting planes for MK problems, and it is likely that similar computational improvements will be achieved for other IPs.

Three ideas present themselves as excellent candidates for future research extensions. In this paper, inequality merging occurs on a single variable. The theory may be extended to merge on multiple variables. Since this paper focuses on cover inequalities and MK instances, another theoretical extension may merge other classes of cutting planes in general IPs.

All of the computational analysis in this research was performed on the first 10 problems of each file provided by Chu and Beasley [

RandalHickman,ToddEaston, (2015) On Merging Cover Inequalities for Multiple Knapsack Problems. Open Journal of Optimization,04,141-155. doi: 10.4236/ojop.2015.44014