稀疏图上弱鸽巢原理与完美匹配公式的指数级 Resolution 下界
Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs
AI总结：
本文通过扩展 pseudo-width 方法至稀疏图，证明了高度不平衡稀疏 expander 图上 PHP 公式与完美匹配公式的 resolution 证明长度具有指数级下界。
摘要：
我们证明了在高度不平衡的稀疏 expander 图上，鸽巢原理（PHP）公式和完美匹配公式的 resolution 证明长度具有指数级下界，从而回应了在 [Ben-Sasson and Wigderson '01] 的平衡常数度 expander 与 [Raz '04] 及 [Razborov '03, '04] 的高度不平衡稠密图之间建立强下界的挑战。我们通过重新审视 Razborov 针对稠密图上 PHP 公式的 pseudo-width 方法，并将其扩展至稀疏图来得出结果。这进一步展示了 pseudo-width 方法的威力，我们相信它可能有助于攻克 resolution 及其他证明系统中的其他长期开放问题。
英文摘要：
We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov's pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
