Analysis of the Karmarkar-Karp Differencing Algorithm

The Karmarkar-Karp differencing algorithm is the best known polynomial time heuristic for the number partitioning problem, fundamental in both theoretical computer science and statistical physics. We analyze the performance of the differencing algorithm on random instances by mapping it to a nonlinear rate equation. Our analysis reveals strong finite size effects that explain why the precise asymptotics of the differencing solution is hard to establish by simulations. The asymptotic series emerging from the rate equation satisfies all known bounds on the Karmarkar-Karp algorithm and projects a scaling $n^{-c\ln n}$, where $c=1/(2\ln2)=0.7213…$. Our calculations reveal subtle relations between the algorithm and Fibonacci-like sequences, and we establish an explicit identity to that effect.

💡 Research Summary

The paper presents a thorough average‑case analysis of the Karmarkar‑Karp differencing algorithm (also known as the Largest Differencing Method, LDM), which is the most effective polynomial‑time heuristic for the number partitioning problem (NPP). The authors begin by describing the algorithm as an iterative process that repeatedly selects the two largest numbers in a list, replaces them by their absolute difference, and continues until a single number remains; this final number equals the discrepancy of a particular partition.

To facilitate analytical treatment, the authors replace the usual i.i.d. uniform inputs with partial sums of i.i.d. exponential variables of mean one. Because the ratios of these partial sums have the same distribution as order statistics of uniform variables, the algorithm’s statistical behavior is unchanged, but the exponential representation has the crucial property that sums and differences of exponential variables remain exponential. This allows the authors to express each iteration of LDM in terms of a λ‑vector, where λ_i is the rate parameter of the i‑th exponential component.

Lemma 1 gives the exact conditional probabilities for the ordering of two exponentials and shows that, conditioned on a particular ordering, the involved variables decompose into independent exponentials with updated rates. Using this lemma, the authors derive explicit transition probabilities (Equations 10‑13) for the λ‑vector when the newly created difference is inserted at any possible rank k in the sorted list. Repeating these transitions recursively reduces the problem to a two‑element λ‑vector, whose second component λ_2 determines the final discrepancy distribution (an exponential with rate λ_2).

Averaging over the stochastic choices yields a nonlinear rate equation (Equation 21) that governs the evolution of the expected λ‑vector over discrete time steps t. Numerical integration of this rate equation reveals a wave‑like propagation pattern in the λ‑components, suggesting a simplifying Ansatz. Applying the Ansatz reduces the complex dynamics to a Fibonacci‑like recurrence, from which the authors obtain an explicit asymptotic form for the expected discrepancy:

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