In the restricted Santa Claus problem we are given resources $\mathcal{R}$ and players $\mathcal{P}$. Every resource $j\in \mathcal{R}$ has a value $v_j$ and every player $i\in \mathcal{P}$ desires a set of resources $\mathcal{R}(i)$. We are interested in distributing the resources to players that desire them. The quality of a solution is measured by the least happy player, i.e., the lowest sum of resource values. This value should be maximized. The local search algorithm by Asadpour et al. [1] and its connection to the configuration LP has proved itself to be a very influential technique for this and related problems.

Asadpour et al. used a local search to obtain a bound of 4 for the ratio of the fractional to the integral optimum of the configuration LP (the integrality gap). This bound is non-constructive, since the local search has not been shown to terminate in polynomial time. On the negative side, the worst instance known has an integrality gap of 2.

Although much progress was made in this area, neither bound has been improved since. We present a better analysis that shows the integrality gap is not worse than $3+5/6\approx 3.8333$.

在受限的圣诞老人问题中，我们得到了资源 $\mathcal{[R}$ 和球员 $\mathcal{P}$。每种资源$Ĵ\in \mathcal{[R}$ 有价值 $v_{Ĵ}$ 和每个玩家 $\mathrm{一世}\in \mathcal{P}$ 渴望一套资源 $\mathcal{[R}（\mathrm{一世}）$。我们有兴趣将资源分配给需要他们的玩家。解决方案的质量由最不高兴的参与者（即资源值的最低总和）衡量。该值应最大化。Asadpour等人的本地搜索算法。[1]及其与配置LP的连接已证明是解决此问题和相关问题的一种非常有影响力的技术。

Asadpour等。使用局部搜索为配置LP的分数最优值与积分最优值之比（积分差）获得4的边界。此边界是非构造性的，因为尚未显示局部搜索在多项式时间内终止。不利的一面是，已知的最坏情况的积分差距为2。

尽管在这方面取得了很大进展，但此后两者都没有得到改善。我们提出了一个更好的分析，表明完整性差距不小于$3+5/6\approx 3.8333$。