{"ID":2844800,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.04982","arxiv_id":"2511.04982","title":"Tight Bounds for Sampling q-Colorings via Coupling from the Past","abstract":"The Coupling from the Past (CFTP) paradigm is a canonical method for perfect sampling. For uniform sampling of proper $q$-colorings in graphs with maximum degree $Δ$, the bounding chains of Huber (STOC 1998) provide a systematic framework for efficiently implementing CFTP algorithms within the classical regime $q \\ge (1 + o(1))Δ^2$. This was subsequently improved to $q \u003e 3Δ$ by Bhandari and Chakraborty (STOC 2020) and to $q \\ge (8/3 + o(1))Δ$ by Jain, Sah, and Sawhney (STOC 2021). In this work, we establish the asymptotically tight threshold for bounding-chain-based CFTP algorithms for graph colorings. We prove a lower bound showing that all such algorithms satisfying the standard contraction property require $q \\ge 2.5Δ$, and we present an efficient CFTP algorithm that achieves this asymptotically optimal threshold $q \\ge (2.5 + o(1))Δ$ via an optimal design of bounding chains.","short_abstract":"The Coupling from the Past (CFTP) paradigm is a canonical method for perfect sampling. For uniform sampling of proper $q$-colorings in graphs with maximum degree $Δ$, the bounding chains of Huber (STOC 1998) provide a systematic framework for efficiently implementing CFTP algorithms within the classical regime $q \\ge (...","url_abs":"https://arxiv.org/abs/2511.04982","url_pdf":"https://arxiv.org/pdf/2511.04982v2","authors":"[\"Tianxing Ding\",\"Hongyang Liu\",\"Yitong Yin\",\"Can Zhou\"]","published":"2025-11-07T04:59:15Z","proceeding":"cs.DS","tasks":"[\"cs.DS\"]","methods":"[]","has_code":false}