[Paper Review] On the Worst-case Performance of the Sum-of-Squares Algorithm for Bin Packing

János Csirik, David S. Johnson+1 more|ArXiv.org|Sep 12, 2005

Optimization and Packing Problems2 references3 citations

TL;DR

This paper improves the asymptotic worst-case performance ratio of the Sum-of-Squares (SS) bin packing algorithm from 3 to 25/9 ≈ 2.777..., using a refined analysis based on pairing bin levels modulo the size of the last small item placed. The method leverages level-based averaging and structural constraints to show that, except for at most two bins, all others are filled to at least 9/25 of the bin capacity on average.

ABSTRACT

The Sum of Squares algorithm for bin packing was defined in [2] and studied in great detail in [1], where it was proved that its worst case performance ratio is at most 3. In this note, we improve the asymptotic worst case bound to 2.7777...

Motivation & Objective

• To tighten the asymptotic worst-case performance bound of the Sum-of-Squares (SS) algorithm for bin packing beyond the previously known factor of 3.
• To analyze the structure of SS packings by focusing on the last insertion of small items, particularly those of size ≤ αB and < δB, to derive tighter fill-level guarantees.
• To establish that, except for at most two bins, all other bins in an SS packing have an average fill level of at least δB, with δ = 9/25.
• To demonstrate that the proof technique based on level pairing and modular congruence classes can yield improved bounds, though further gains may require new insights.

Proposed method

• Introduces parameters α and δ such that 0 < α < δ ≤ 1/2, satisfying three inequalities (2.1)–(2.3) to control the fill-level analysis.
• Defines 'small' and 'large' levels relative to δB, and pairs levels j + is' and j + (c_j - i)s' modulo s' to compare bin counts and ensure level balance.
• Applies Lemma 1, which states that if a new bin is opened, then n_j(P) ≤ n_{j+s}(P) for j ≤ B−s, to ensure monotonicity in bin counts across levels.
• Partitions bins into congruence classes modulo s or s' and introduces 'honorary members' to extend level-pairing to bins with levels ≤ Δ.
• Analyzes average bin content in each class D_h by grouping bins into subclasses E_i and deriving lower bounds X_i on average fill levels.
• Uses inequalities involving d_h (number of levels in a class), h (base level), and s (item size) to prove that X_i ≥ δB for all i, ensuring sufficient average fill.

Experimental results

Research questions

• RQ1 Can the asymptotic worst-case performance ratio of the Sum-of-Squares algorithm be improved beyond the known bound of 3?
• RQ2 What structural properties of SS packings—particularly related to the last insertion of small items—can be exploited to derive tighter fill-level guarantees?
• RQ3 To what extent can level-pairing and modular arithmetic be used to bound the average bin fill level in SS packings?
• RQ4 What is the tightest possible bound achievable using the current proof framework based on last-insertion analysis of two or more items?
• RQ5 How do constraints on α and δ affect the achievable δ for which the average bin fill is at least δB?

Key findings

• The asymptotic worst-case performance ratio of the Sum-of-Squares algorithm is at most 25/9 ≈ 2.777..., improving upon the prior upper bound of 3.
• For any list L, the number of bins used by SS satisfies SS(L) < (25/9)·s(L)/B + 2 ≤ (25/9)·OPT(L) + 2.
• The bound is achieved by showing that, except for at most two bins, all other bins in the SS packing have an average fill level of at least δB with δ = 9/25.
• The analysis relies on pairing bin levels modulo the size of the last small item inserted and proving that such pairs maintain sufficient average fill.
• The proof technique is limited by the current framework: further improvements would require new structural insights beyond the current level-pairing and congruence class analysis.
• The bound 25/9 is tight under the current method, as no better δ can be obtained using the same constraints and assumptions.