Grahams Schedules and the Number Partition Problem

We show the equivalence of the Number Partition Problem and the two processor scheduling problem. We establish a priori bounds on the completion times for the scheduling problem which are tighter than Graham’s but almost on par with a posteriori bounds of Coffman and Sethi. We conclude the paper with a characterization of the asymptotic behavior of the scheduling problem which relates to the spread of the processing times and the number of jobs.

💡 Research Summary

The paper “Graham’s Schedules and the Number Partition Problem” establishes a formal equivalence between the Number Partition Problem (NPP) and the two‑processor scheduling problem (2PS). Both problems aim to split a multiset of positive integers into two subsets: NPP minimizes the absolute difference of the subset sums, while 2PS minimizes the makespan of two identical machines processing the same jobs. By noting that the total sum of all jobs is invariant, the author shows that an optimal NPP solution yields an optimal 2PS schedule and vice‑versa; the proof is a short algebraic argument that the differences must be zero when both solutions are optimal.

Having reduced NPP to 2PS, the paper focuses on Graham’s classic Longest Processing Time (LPT) rule, which assigns the longest remaining job to the currently less‑loaded processor. Graham’s original a‑priori bound guarantees a worst‑case ratio C_G / C_O ≤ 7/6 for any instance, while Coffman and Sethi later gave a tighter a‑posteriori bound C_G / C_O ≤ 1 + 1/k – 1/(2k), where k is the number of jobs on the processor that finishes last.

The author proposes two new families of bounds. The first (Theorem 3) is an a‑priori bound that depends on the index L of the last job to finish in the LPT schedule. Defining M = ⌈L/2⌉ and a polynomial P = 24 M³ / (7 + 12 M + 24 M²), the bound reads
 C_G / C_O ≤ (P + 1)/P – 1/(2P).
The proof proceeds by assuming the bound is violated, deriving a condition t_L > C_O / P, and then using Graham’s 7/6 bound to obtain a contradiction that forces a specific choice of P. Although the reasoning is indirect, it yields a bound that is marginally tighter than 7/6 for many instances.

The second contribution introduces the “Possible Last Job” (PLJ). A job t_i is called dominant if its processing time is at least the sum of all smaller jobs, i.e., t_i ≥ Σ_{j=i+1}^n t_j. PLJ is defined as the largest index i that satisfies this condition. By construction, the true last job index L in the LPT schedule always satisfies L ≥ PLJ. Computing PLJ requires O(n²) time, but once PLJ is known, the same functional form as Theorem 3 can be applied with P′ = ⌈PLJ/2⌉, yielding an a‑priori bound (Theorem 4) that can be evaluated without constructing the actual schedule.

Experimental results are presented for randomly generated instances with 15, 20, and 25 jobs (processing times uniformly drawn from