Motivated by the desire to utilize a limited number of configurable optical switches by recent advances in Software Defined Networks (SDNs), we define an online problem which we call the Caching in Matchings problem. This problem has a natural combinatorial structure and therefore may find additional applications in theory and practice. In the Caching in Matchings problem our cache consists of $k$ matchings of connections between servers that form a bipartite graph. To cache a connection we insert it into one of the $k$ matchings possibly evicting at most two other connections from this matching. This problem resembles the problem known as Connection Caching, where we also cache connections but our only restriction is that they form a graph with bounded degree $k$. Our results show a somewhat surprising qualitative separation between the problems: The competitive ratio of any online algorithm for caching in matchings must depend on the size of the graph. Specifically, we give a deterministic $O(nk)$ competitive and randomized $O(n \log k)$ competitive algorithms for caching in matchings, where $n$ is the number of servers and $k$ is the number of matchings. We also show that the competitive ratio of any deterministic algorithm is $\Omega(\max(\frac{n}{k},k))$ and of any randomized algorithm is $\Omega(\log \frac{n}{k^2 \log k} \cdot \log k)$. In particular, the lower bound for randomized algorithms is $\Omega(\log n)$ regardless of $k$, and can be as high as $\Omega(\log^2 n)$ if $k=n^{1/3}$, for example. We also show that if we allow the algorithm to use at least $2k-1$ matchings compared to $k$ used by the optimum then we match the competitive ratios of connection catching which are independent of $n$. Interestingly, we also show that even a single extra matching for the algorithm allows to get substantially better bounds.

翻译：受限于软件定义网络中可配置光交换机数量有限这一实际需求的驱动，我们定义了一个名为“匹配中的缓存”（Caching in Matchings）的在线问题。该问题具有天然的组合结构，因此在理论和实践中可能具有更广泛的应用价值。在匹配缓存问题中，缓存由服务器之间形成二分图的k个连接匹配组成。要缓存一个连接，需将其插入k个匹配之一，同时可能从该匹配中驱逐至多两个其他连接。该问题与已知的“连接缓存”（Connection Caching）问题类似，后者中缓存连接的唯一约束是它们构成一个度为k的图。我们的研究结果显示这两个问题之间存在一个令人惊讶的定性差异：匹配缓存问题的任何在线算法的竞争比都必然依赖于图的大小。具体而言，我们为匹配缓存问题提出了确定性O(nk)竞争比算法和随机化O(n log k)竞争比算法（其中n为服务器数量，k为匹配数量）。同时证明任何确定性算法的竞争比为Ω(max(n/k, k))，任何随机化算法的竞争比为Ω(log(n/(k² log k))·log k)。特别地，随机化算法的下界与k无关且为Ω(log n)，例如当k=n^{1/3}时可达Ω(log² n)。我们还证明，若允许算法使用至少2k-1个匹配（而最优方案仅使用k个匹配），则其竞争比可达到与n无关的连接缓存问题水平。有趣的是，即使仅允许算法额外使用一个匹配，也能获得显著更优的界。