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：受软件定义网络（SDN）最新进展中利用有限数量可配置光开关的启发，我们定义了一个在线问题——匹配中的缓存问题。该问题具有自然的组合结构，因此在理论和实践中可能具有更多应用。在匹配中的缓存问题中，我们的缓存由服务器间连接构成的$k$个匹配组成（形成二分图）。为缓存一个连接，我们将其插入$k$个匹配之一，可能从该匹配中最多驱逐两个其他连接。此问题类似于已知的连接缓存问题——在该问题中我们同样缓存连接，但唯一限制是这些连接构成一个度数不超过$k$的图。我们的结果揭示了两个问题之间令人惊讶的定性差异：匹配缓存中任何在线算法的竞争比必须依赖于图的大小。具体而言，我们给出了匹配缓存的确定性$O(nk)$竞争算法和随机化$O(n \log k)$竞争算法，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$\Omega(\max(\frac{n}{k},k))$，任何随机化算法的竞争比为$\Omega(\log \frac{n}{k^2 \log k} \cdot \log k)$。特别地，随机化算法的下界与$k$无关始终为$\Omega(\log n)$，且当$k=n^{1/3}$时可达$\Omega(\log^2 n)$。我们还证明，若允许算法使用至少$2k-1$个匹配（相较于最优解使用的$k$个匹配），则能达到与连接缓存问题相同的与$n$无关的竞争比。有趣的是，仅额外增加一个匹配即可使算法获得显著改进的界。