AI 中文总结
该研究针对需求匹配问题及k-超图需求匹配问题，提出多种双准则近似算法，并给出对应下界，刻画了权重近似与加性容量违规的权衡关系。
AI 中文摘要
需求匹配问题是背包问题和b-匹配问题的泛化：图中每条边有一个需求和权重，每个顶点有一个容量，目标是找到一个最大权重的边子集，使得每个顶点处的总关联需求不超过其容量。我们研究(α, β)-双准则近似算法，这类算法返回的解权重至少为最优解的1/α，同时允许容量的加性违规不超过最大边需求的β倍。针对需求匹配问题，我们提出一种迭代松弛算法，该算法利用自然线性规划（LP）松弛的严格分数极点的结构特征，将剩余的舍入问题简化为奇环实例；结合双选更好的舍入策略，该算法分别为一般图和二分图提供了(7/6, 1)-和(1, 1)-双准则近似算法。我们进一步将该方法泛化为参数化算法族，其中包含(1, 4/3)-双准则近似。此外，针对更泛化的k-超图需求匹配问题，我们提出一种贪心组合型(k, 1)-双准则近似算法。我们还补充了与自然LP松弛匹配的下界：针对β=0及所有β≥1的情况，完全刻画了该范围内权重近似与加性容量违规之间的权衡关系。
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, each vertex has a capacity, and the goal is to find a maximum weight subset of edges whose total incident demand at every vertex does not exceed its capacity. We study $(α, β)$-bicriteria approximation algorithms, which return a solution of weight at least $1/α$ times the optimum while allowing an additive capacity violation of at most $β$ times the maximum edge demand. We give an iterative relaxation algorithm for the demand matching problem that exploits a structural characterization of strictly fractional extreme points of the natural LP relaxation, which reduces the residual rounding problem to odd-cycle instances. Combined with a better-of-two rounding strategy, this yields $(7/6, 1)$- and $(1, 1)$-bicriteria approximation algorithms for general and bipartite graphs, respectively. We further generalize this approach to obtain a parametric family of algorithms, including a $(1, 4/3)$-bicriteria approximation. Separately, for the more general $k$-hypergraph demand matching problem, we give a greedy, combinatorial $(k, 1)$-bicriteria approximation algorithm. We complement these algorithmic results with matching lower bounds relative to the natural LP relaxation for $β= 0$ and all $β\geq 1$, completely characterizing the trade-off between weight approximation and additive capacity violation in this range.
