Constrained Integer Partitions

  • Christian Borgs
  • Jennifer T. Chayes
  • Stephan Mertens
  • Boris Pittel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)


We consider the problem of partitioning n integers into two subsets of given cardinalities such that the discrepancy, the absolute value of the difference of their sums, is minimized. The integers are i.i.d. random variables chosen uniformly from the set {1,...,M}. We study how the typical behavior of the optimal partition depends on n,M and the bias s, the difference between the cardinalities of the two subsets in the partition. In particular, we rigorously establish this typical behavior as a function of the two parameters \(\kappa :=n^{-1}{\rm log}_{2}M\) and b:=|s|/n by proving the existence of three distinct “phases” in the κb-plane, characterized by the value of the discrepancy and the number of optimal solutions: a “perfect phase” with exponentially many optimal solutions with discrepancy 0 or 1; a “hard phase” with minimal discrepancy of order Me  − Θ( n); and a “sorted phase” with an unique optimal partition of order Mn, obtained by putting the (s+n)/2 smallest integers in one subset.


Basis Solution Linear Programming Problem Hard Phase Partition Problem Saddle Point Equation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Christian Borgs
    • 1
  • Jennifer T. Chayes
    • 1
  • Stephan Mertens
    • 2
  • Boris Pittel
    • 3
  1. 1.Microsoft ResearchRedmond
  2. 2.Institut für Theoretische PhysikOtto-von-Guericke UniversitätMagdeburgGermany
  3. 3.Department of MathematicsOhio State UniversityColumbus

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