## 1 Introduction and overview

The number partitioning problem is defined as follows: Given a list of non-negative, integer numbers, find a partition such that the partition difference

(1) |

is minimized. In the constrained partition problem, the cardinality difference between and its complement,

(2) |

must obey certain constraints. The most common case is the balanced partitioning problem with the constraint .

Partitioning is of both theoretical and practical importance. It is one of Garey and Johnson’s six basic NP-complete problems that lie at the heart of the theory of NP-completeness [3]. Among the many practical applications one finds multiprocessor scheduling and the minimization of VLSI circuit size and delay [2, 13].

Due to the NP-hardness of the partitioning problem [7], it seems unlikely that there is an efficient exact solution. Numerical investigations have shown, however, that large instances of partitioning can be solved exactly within reasonable time [8, 4, 9]. This surprising fact is based on the existence of perfect partitions, partitions with

. The moment an algorithm finds a perfect partition, it can stop. For identically, independently distributed (i.i.d.) random numbers

, the number of perfect perfect partitions increases with , but in a peculiar way. For smaller than a critical value, there are no perfect partitions (with probability one). For

, the number of perfect partitions increases exponentially with . The critical value depends on the number of bits needed to encode the . For the unconstrained partitioning problem(3) |

where denotes the average over the distribution of the [12]. The corresponding equation for the balanced partitioning problem reads [11]

(4) |

For most practical applications the have a finite precision and Eq. 3 resp. Eq. 4 can be applied. Theoretical investigations consider real-valued i.i.d. numbers , i.e. numbers with infinite precision. In this case, there are no perfect partitions, and for a large class of real valued input distributions, the optimum partition has a median difference of for the unconstrained resp. for the balanced case [5]. Using methods from statistical physics, the average optimum difference has been calculated recently [12, 11]. It reads

(5) |

for the unconstrained and

(6) |

for the balanced partioning problem. These equations also describe the case of finite precision in the regime .

For both variants of the partitioning problem, the best heuristic algorithms are based on the Karmarkar-Karp differencing scheme and yield partitions with expected

when run with i.i.d. real valued input values [6, 14]. They run in polynomial time, but offer no way of improving their solutions given more running time. Korf [9] proposed an algorithm that yields the Karmarkar-Karp solution within polynomial time and finds better solutions the longer it is allowed to run, until it finally finds and proves the optimum solution. Algorithms with this property are referred to as anytime algorithms [1]. Korf’s anytime algorithm is very efficient, especially for problems with moderate values of . For numbers with twelve significant digits or less (), it can optimally solve partitioning problems of arbitrary size in practice, since it quickly finds a perfect partition for . For larger values of , several orders of magnitude improvement in solution quality compared to the Karmarkar-Karp heuristic can be obtained in short time.For practical applications of this NP-hard problem, this is almost more than one might expect. Korf’s algorithm is not very useful to find the optimum constrained partition, however. In this paper, we describe a modification of Korf’s algorithm, which is as efficient as the original, but solves the constrained partition problem. The next section comprises a description of Korf’s algorithm and the modifications for the balanced problem. In the third section we discuss some experimental results. The paper ends with a summary and some conclusions.

## 2 Algorithms

### 2.1 Differencing heuristics

The key ingredient to the most powerful partition heuristics is the differencing operation [6]: select two elements and and replace them by the element . Replacing and by is equivalent to making the decision that they will go into opposite subsets. Applying differencing operations times produces in effect a partition of the list . The value of its partition difference is equal to the single element left in the list.

Various partitions can be obtained by choosing different methods for selecting the pairs of elements to operate on. In the paired differencing method (PDM), the elements are ordered. The first operations are performed on the largest two elements, the third and the fourth largest, etc.. After these operations, the left-over elements are ordered and the procedure is iterated until there is only one element left.

Another example is the largest differencing method (LDM). Again the elements are ordered. The largest two elements are picked for differencing. The resulting set is ordered and the algorithm is iterated until there is only one element left.

For , i.e. in the regime where there are no perfect partitions, and for random i.i.d. input numbers, the expected partition differences are for PDM [10] and for LDM [14].

LDM, being superior to PDM, is not applicable to the constrained partioning problem. PDM on the other hand yields only perfectly balanced partitions. Yakir proposed a combination of both algorithms, which finds perfectly balanced partitions, but with an expected partition difference of [14]. In his balanced LDM (BLDM), the first iteration of PDM is applied to reduce the original -element list to elements. By doing so, it is assured that the final partition is balanced, regardless of which differencing operations are used thereafter. If one continues with LDM, a final difference of can be expected.

The time complexity of LDM, PDM and BLDM is , the space-complexity is .

### 2.2 Korf’s complete anytime algorithm

LDM and BLDM are the best known heuristics for the partioning problem, but they find approximate solutions only. Korf [9] showed, how the LDM can be extended to a complete anytime algorithm, i.e. an algorithm that finds better and better solutions the longer it is allowed to run, until it finally finds and proves the optimum solution: At each iteration, the LDM heuristic commits to placing the two largest numbers in different subsets, by replacing them with their difference. The only other option is to place them in the same subset, replacing them by their sum. This results in a binary tree, where each node replaces the two largest remaining numbers, : the left branch replaces them by their difference, while the right branch replaces them by their sum:

(7) |

Iterating both operations times generates a tree with terminal nodes. The terminal nodes are single element lists, whose elements are the valid partition differences . Korf’s complete Karmarkar-Karp (CKK) algorithm searches this tree depth-first and from left to right. CKK first returns the LDM solution, then continues to find better solutions as time allows. See Fig. 1 for the example of a tree generated by CKK.

There are two ways to prune the tree: At any node, where the difference between the largest element in the list and the sum of all other elements is larger than the current minimum partition difference, the node’s offspring can be ignored. If one reaches a terminal node with a perfect partition, , the entire search can be terminated. The dashed nodes in Fig. 1 are pruned by these rules.

In the regime , the number of nodes generated by CKK to find the optimum partition grows exponentially with . The first solution found, the LDM-solution, is significantly improved with much less nodes generated, however. In the regime , the running time decreases with increasing , due to the increasing number of perfect partitions. For , the running time is dominated by the time to construct the LDM-solution, which in this regime is almost always perfect.

### 2.3 A complete anytime algorithm for constrained partioning

The application of differencing and its opposite operation leads to lists, in which single elements represent several elements of the original list. In order to apply CKK to the constrained partitioning problem, one needs to keep track of the resulting cardinality difference. This can be achieved by introducing an effective cardinality for every list element . In the original list, all . The differencing operation and its opposite become

(8) |

Fig. 1 shows how the evolve if the branching rule 8 is applied to the list . The terminal nodes contain the partition difference and the cardinality difference. A simple approach to the constrained partition problem is to apply CKK with the branching rule 8 and to consider only solutions with matching . This can be very inefficient, as can be seen for the constraint . This extreme case is trivial, but CKK needs to search the complete tree to find the solution!

As a first improvement we note, that an additional pruning rule can be applied. Let and at a given node. The cardinality difference which can be found within the offspring of this node, is bounded by

(9) |

Comparing these bounds to the cardinality constraint, one can prune parts of the tree. Consider again the case as an example: The trivial solution is now found right away.

CKK finds the first valid partition (the LDM solution) after generating nodes. For the constrained partition problem, this can not be guaranteed – except in the case of balanced partitions, where we can use the BLDM strategy. Applying the first PDM operations to the original list leaves us with a -element list with all (resp. with a single if

is odd). CKK applied to this list produces only perfectly balanced partitions, the BLDM solution in first place. To keep the completeness of the algorithm, we have to consider the alternative to each of the PDM operations, i.e. to put a pair of subsequent numbers in the same subset.

An outline of the complete BLDM algorithm can be seen in Fig. 2. Note that in an actual implementation several modifications should be applied to improve the performance. Instead of sorting the list at every node in the LDM phase, it is much more efficient to sort only when switching from PDM to LDM and insert the new element in the LDM-phase such that the order is preserved. The and of and should be calculated only once and then locally updated when the list is modified

## 3 Experimental results

We implemented the complete BLDM algorithm to test its performance as an exact solver, a polynomial heuristic and an anytime algorithm. For all computer experiments we use i.i.d. random numbers

, uniformly distributed from

to , i.e. -bit integers.To measure the performance of the algorithm as an exact solver, we count the number of nodes generated until the optimum solution has been found and proven. The result for -bit integers is shown in Fig. 3. Each data point is the average of random problem instances. The horizontal axes shows the number of integers partitioned, the vertical axes show the number of nodes generated (left) and the fraction of instances that have a perfect partition (right). Note that we counted all nodes of the tree, not just the terminal nodes. We observe three distinct regimes: for , the number of nodes grows exponentially with , for it decreases with increasing , reaching a minimum and starting to increase again slowly for very large values of .

Eq. 4 yields for our experimental setup, in good agreement with the numerical result, that the probability of having a perfect partition is one for and drops sharply to zero for smaller values of . In the regime , the algorithm has to search an exponential number of nodes in order to prove the optimality of a partition. For it finds a perfect partition and stops the search prematurely. The number of perfect partitions increases with increasing , making it easier to find one of them. This explains the decrease of searching costs. For , the very first partition found already is perfect. The construction of this BLDM solution requires nodes.

We have seen that for the BLDM heuristic yields perfect partitions. How does it behave in the other extreme, the “infinite precision limit”, ? Yakir [14] proved that in this limit BLDM yields an expected partition difference of . For a numerical check we applied BLDM to partition -bit integers to ensure that . The partition difference is then divided by to simulate infinite precision real numbers from the interval . Fig. 4 shows the resulting partition difference. Each data point is averaged over random instances. Due to the numerical fit in Fig. 4 it is tempting to conjecture

(10) |

If we want better solutions than the BLDM solution we let the complete BLDM run as long as time allows and take the best solution found. We applied this approach to partition random -bit integers. Perfect partitions do not exist (with probability one), and the true optimum is definitely out of reach. The results can be seen in Fig. 5. The horizontal axis is the number of nodes generated, and the vertical axes is the ratio of the initial BLDM solution to the best solution found in the given number of node generations, both on a logarithmic scale. The entire horizontal scale represents about 90 minutes of real time, measured on a Sun SPARC 20. The fact that the number of nodes per second is a factor of smaller than reported by Korf for the CKK [9] on -bit integers is probably due to the fact that we had to use a multi-precision package for the -bit arithmetic while Korf could stick to the fast arithmetic of built-in data types. Even with this slow node generation speed we observe a several order of magnitude improvement relative to the BLDM solution in a few minutes. A least square fit to the data of runs gives

(11) |

but the actual data vary considerably.

## 4 Summary and conclusions

The main contribution of this paper is to develop a complete anytime algorithm for the constrained number partioning problem. The complete Karmarkar-Karp algorithm CKK, proposed by Korf for the unconstrained partitioning problem, can be adapted to the constrained case simply by keeping book of the effective cardinalities and by extending the BLDM heuristic to a complete algorithm. The first solution the algorithm finds is the BLDM heuristic solution, and as it continues to run it finds better and better solutions, until it eventually finds and verifies an optimal solution.

The basic operation of the complete BLDM is very similar to Korf’s CKK. The additional processing of the effective cardinalities has only a minor impact on the runtime. The pruning based on estimating the cardinality difference leads to a gain in speed, on the other hand. Therefore we adopt Korf’s claim: For numbers with twelve significant digits or less, complete BLDM can optimally solve balanced partitioning problems of arbitrary size in practice.

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