具有集中性的拟阵冲突解决
Matroid Contention Resolution with Concentration
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中文总结 AI 辅助
研究为Adamczyk和Włodarczyk的随机顺序CRS（AW）输出推导下尾界，通过“强λ-有界性”分析，引入顺序选择过程并证明其下尾界，应用于含覆盖约束问题，如改进k-拟阵交色数近似及给出k个拟阵约束下单调子模最大化双准则算法。
中文摘要 AI 辅助
冲突解决方案（CRS）是用于对受组合约束的分数解进行舍入的基本且广泛应用的工具。已知对CRS的分析通常仅保证期望值的下界和上尾的集中性，而无下尾集中性。因此，CRS通常不适用于包含覆盖约束的问题。本文主要贡献是为特定冲突解决方案——Adamczyk和Włodarczyk的随机顺序CRS（称为AW）的输出推导下尾界。通过新性质“强λ-有界性”驱动分析，加强了AW的已知λ-有界性。引入捕获AW的随机过程——顺序选择过程，并证明了任何强λ-有界顺序选择过程的下尾界。将新尾界应用于两个涉及覆盖约束的问题，得到了k-拟阵交色数的O(k log k)近似（改进了之前的O(k^2)）以及k个拟阵约束下单调子模最大化的首个双准则近似算法。
英文摘要
Contention resolution schemes (CRS) are a fundamental and widely applied tool for rounding fractional solutions subject to combinatorial constraints. However, the known analyses of CRS generally only guarantee lower bounds on the expected value and concentration on the upper tail, but no concentration on the lower tail. Thus, CRS are generally not applicable to problems that contain covering constraints, since certifying a covering constraint holds requires a lower tail bound. Our main contribution is to derive lower tail bounds for the output of a particular contention resolution scheme, the random-order CRS of Adamczyk and Włodarczyk, which we call AW. We show that every linear function of the rounded solution attains a constant fraction of its expectation with a failure probability that is dimension-free, depending only on the expected value and on the number of matroids, but not on the size of the ground set. Our analysis is driven by a new property we call \emph{strong $λ$-boundedness}, which strengthens the known $λ$-boundedness of AW by providing two-sided control on how rounding propagates between elements. We then introduce a random process capturing AW, a \emph{sequential selection process}, that may be of independent interest. We prove lower tail bounds for any strongly $λ$-bounded sequential selection process. To demonstrate the applicability of our new tail bounds, we apply them to two problems involving covering constraints. The first result is an $O(k \log k)$-approximation for $k$-matroid intersection coloring (improving the prior $O(k^2)$) when the chromatic number of at least one matroid is $Ω(k^3 \log n)$, where $n$ is the number of elements. The second is the first bicriteria approximation algorithm for monotone submodular maximization under $k$ matroid constraints together with packing and covering constraints.
