# Efficient approximation algorithms for the subset-sums equality problem

## Abstract

We investigate the problem of finding two nonempty disjoint subsets of a set of *n* positive integers, with the objective that the sums of the numbers in the two subsets be as close as possible. In two versions of this problem, the quality of a solution is measured by the ratio and the difference of the two partial sums, respectively.

Answering a problem of Woeginger and Yu (1992) in the affirmative, we give a fully polynomial-time approximation scheme for the case where the value to be optimized is the ratio between the sums of the numbers in the two sets. On the other hand, we show that in the case where the value of a solution is the positive difference between the two partial sums, the problem is not 2^{n} ^{k} -approximable in polynomial time unless *P=NP*, for any constant *k*. In the positive direction, we give a polynomial-time algorithm that finds two subsets for which the difference of the two sums does not exceed *K/n* ^{Ω(log n)} where *K* is the greatest number in the instance.

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## References

- 1.C. Bazgan, M. Santha and Zs. Tuza,
*On the approximation of finding a(nother*)*Hamiltonian cycle in cubic Hamiltonian graphs*, 15th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Vol. 1373 (1998), 276–286.MathSciNetCrossRefGoogle Scholar - 2.O. H. Ibarra and C. E. Kim,
*Fast approximation algorithms for the Knapsack and Sum of Subset problems*, J. ACM, 22:4 (1975), 463–468.MATHMathSciNetCrossRefGoogle Scholar - 3.R. M. Karp,
*Reducibility among combinatorial problems*, in: Complexity of Computer Computations (R. E. Miller and J. W. Thatcher, eds.) (1972), 85–103.Google Scholar - 4.H. Kellerer, R. Mansini, U. Pferschy and M. G. Speranza,
*An efficient fully polynomial approximation scheme for the Subset-Sum problem*, manuscript, 1997.Google Scholar - 5.N. Megiddo and C. Papadimitriou,
*On total functions, existence theorems and computational complexity*, Theoretical Computer Science, 81 (1991), 317–324.MATHMathSciNetCrossRefGoogle Scholar - 6.C. Papadimitriou,
*On the complexity of the parity argument and other inefficient proofs of existence*, Journal of Computer and System Sciences 48 (1994), 498–532.MATHMathSciNetCrossRefGoogle Scholar - 7.G. J. Woeginger and Z. Yu,
*On the equal-subset-sum problem*, Information Processing Letters 42 (1992), 299–302.MATHMathSciNetCrossRefGoogle Scholar