{"slug": "iterated-logarithmic-additive-hardness-for-bin-packing", "title": "Iterated-Logarithmic Additive Hardness for Bin Packing", "summary": "A new proof establishes that, unless P=NP, no deterministic polynomial-time bin-packing algorithm can guarantee a packing using at most OPT + c·log*OPT bins for some absolute constant c>0, ruling out any additive error of o(log*OPT). The result addresses the third of ten open problems posed in Williamson and Shmoys' \"The Design of Approximation Algorithms\" and improves on OpenAI's earlier [Ope26] hardness showing that no deterministic polynomial-time algorithm achieves OPT + O(1) bins. The best known upper bound remains OPT + O(log OPT), from Hoberg and Rothvoss [HR17], a randomized algorithm with expected polynomial running time; the authors conjecture the optimal additive error is Θ(log OPT).", "body_md": "The question of obtaining a constant-additive approximation for bin packing was highlighted in the third of ten open problems in Williamson and Shmoys' The Design of Approximation Algorithms [SW11, Chapter~17, Problem~3]. We prove that there is an absolute constant $c>0$ such that, unless $\\mathrm P=\\mathrm{NP}$, no deterministic polynomial-time packing using at most $\\mathrm{OPT}+c\\log^*\\mathrm{OPT}$ bins, where $\\mathrm{OPT}$ is the minimum number of bins. In particular, no such algorithm has additive error $o(\\log^*\\mathrm{OPT})$. \nThe best known upper bound, due to Hoberg and Rothvoss [HR17], is $\\mathrm{OPT}+O(\\log\\mathrm{OPT})$, achieved by a randomized algorithm with expected polynomial running time. OpenAI [Ope26] previously showed that, unless $\\mathrm P=\\mathrm{NP}$, no deterministic polynomial-time algorithm guarantees a packing using at most $\\mathrm{OPT}+O(1)$ bins. \nWe conjecture that the optimal additive approximation error for bin packing is $\\Theta(\\log \\mathrm{OPT})$.", "url": "https://wpnews.pro/news/iterated-logarithmic-additive-hardness-for-bin-packing", "canonical_source": "https://eprint.iacr.org/2026/2644", "published_at": "2026-10-26 08:05:04+00:00", "updated_at": "2026-10-26 10:46:42.192329+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "machine-learning"], "entities": ["OpenAI", "Williamson", "Shmoys", "Hoberg", "Rothvoss"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/iterated-logarithmic-additive-hardness-for-bin-packing", "markdown": "https://wpnews.pro/news/iterated-logarithmic-additive-hardness-for-bin-packing.md", "text": "https://wpnews.pro/news/iterated-logarithmic-additive-hardness-for-bin-packing.txt", "jsonld": "https://wpnews.pro/news/iterated-logarithmic-additive-hardness-for-bin-packing.jsonld"}}