AI 中文总结
针对需求匹配问题，通过背包交集线性规划松弛和冲突消解方案，首次给出优于3的随机近似算法，达到约2.914的近似比，二分图情形可达2+ε。
AI 中文摘要
需求匹配问题（demand matching problem）同时推广了背包问题（knapsack problem）和 $b$-匹配问题（$b$-matching problem）。在该问题中，图的每条边具有一个需求（demand）和一个权重（weight），每个顶点具有一个容量（capacity）。目标是寻找一个最大权重的边子集，使得在每个顶点处，被选中的关联边的总需求不超过该顶点的容量。Parekh [IPCO 2011] 证明了，如果每条边单独可行，则需求匹配的自然线性规划松弛（natural LP relaxation）的整数间隙（integrality gap）至多为 $3$，从而得到一个 $3$-近似算法。该界对自然线性规划松弛是紧的，与 Shepherd 和 Vetta [Math. Oper. Res. 2007] 的下界相匹配。我们提出了一个随机化的 $(3/2 + \sqrt{2} + \varepsilon) \approx (2.914 + \varepsilon)$-近似算法，适用于每个 $\varepsilon > 0$，这是第一个严格优于 $3$ 的近似比。对于二分图，我们为每个 $\varepsilon > 0$ 获得了一个随机化的 $(2 + \varepsilon)$-近似算法。两个算法的运行时间在 $1/\varepsilon$ 和输入长度上都是多项式的。我们的算法使用了一个加强的线性规划松弛，该松弛基于与顶点相关联的整数背包多面体（integral knapsack polytopes）的交集，并结合了多选泛化（multiple-choice generalization）。作为一个关键组成部分，我们证明了对于每个 $q \in [0, 1]$，整数背包多面体存在一个 $(q, 1/(1+q))$-平衡的冲突消解方案（contention resolution scheme），这可能具有独立的意义。平衡保证 $1/(1+q)$ 在所有背包实例的最坏情况下是紧的。
英文摘要
The demand matching problem generalizes both the knapsack problem and the $b$-matching problem. In this problem, each edge of a graph has a demand and a weight, and each vertex has a capacity. The goal is to find a maximum weight subset of edges such that, at each vertex, the total demand of the incident selected edges does not exceed the vertex capacity. Parekh [IPCO 2011] proved that, if each edge is individually feasible, the natural LP relaxation for demand matching has integrality gap at most $3$, yielding a $3$-approximation algorithm. This bound is tight for the natural LP relaxation, matching the lower bound of Shepherd and Vetta [Math. Oper. Res. 2007]. We present a randomized $(3/2 + \sqrt{2} + \varepsilon) \approx (2.914 + \varepsilon)$-approximation algorithm for the demand matching problem for every $\varepsilon > 0$, giving the first approximation ratio strictly better than $3$. For bipartite graphs, we obtain a randomized $(2 + \varepsilon)$-approximation algorithm for every $\varepsilon > 0$. Both algorithms run in time polynomial in $1/\varepsilon$ and the input length. Our algorithms use a strengthened LP relaxation based on intersecting the integral knapsack polytopes associated with the vertices, together with a multiple-choice generalization. As a key ingredient, we prove the existence of a $(q, 1/(1+q))$-balanced contention resolution scheme for the integral knapsack polytope for every $q \in [0, 1]$, which may be of independent interest. The balance guarantee $1/(1+q)$ is tight in the worst case over all knapsack instances.
