Authors:
Ruben Horn
1
;
Daan van den Berg
2
;
3
and
Pieter Adriaans
4
Affiliations:
1
Helmut-Schmidt-University, Hamburg, Germany, U.K.
;
2
Department of Computer Science, University of Amsterdam, Netherlands
;
3
Department of Computer Science, VU Amsterdam, Netherlands
;
4
Institute for Logic, Language, and Computation, University of Amsterdam, Netherlands
Keyword(s):
Subset-Sum Problem, Number Partitioning Problem, Fractal Analysis, Hilbert’s Hotel, NP-Hard.
Abstract:
It is known that the hardness of the (two-way) number partitioning problem (NPP) variant of the subset-sum problem (SSP) depends on the number and distribution of bits in the set of numbers, but beyond this, it is relatively unexplained for the SSP itself. Thus, we look at the solution space of various problem instances of the SSP using fractal analysis. Two methods to determine the dimension are used. Plotting the fractal dimension over the range and distributions of informational bits, we find that it is correlated with this linear model and also moderately correlated to the hardness of the NPP. This suggests that fractal analysis might be a useful tool in understanding the complexity of combinatorial problems and, we believe, may help further understand the hardness in NP. Finally, we introduce a thought experiment derived from the famous Hilbert’s hotel, which we call Hilbert’s hotel with elevators, to intuitively illustrate how the complexity of the solutions space and the compu
tational hardness may relate across combinatorial problems.
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