（多）对数空间中的逼近
Approximation in (Poly-) Logarithmic Space
摘要：
在空间约束下，我们对RAM模型中的经典图和集合问题提出了新的近似算法。作为我们的主要结果之一，我们设计了一个计算d-命中集的算法，该算法在时间n^{O（d^2 + d/ε}）}上运行，使用O（（d^2 + d/ε）log n）个空间位，并且对任意正的ε \leq 1和任意自然数d达到O（（d/ε）n^ε）.特别地，这产生了一个因子为O（log n）的近似算法，它在时间n^{O（log n）}上运行，并且使用O（log^2 n）位的空间（对于常数d）。作为推论，我们得到了类似的顶点覆盖和几个图删除问题的界。对于有界多重性问题的例子，我们可以做得更好。我们设计了一个最大度为Δ的图的顶点覆盖的因子2近似算法，以及一个计算最大独立集的算法，该算法在时间上为n^{O（Δ）}，在空间上为O（Δlog n）位.对于更一般的d-击中集问题，我们设计了一个因子d近似算法，它在时间n^{O（d δ^2）}上运行，在集合族上使用O（d δ ^2logn）位空间，其中每个元素最多出现在δ集合中。对于平均度为d的图的独立集，我们给出了一个在多项式时间内运行的、占用O（logn）比特空间的因子-（2d）近似算法.我们还设计了一个求解d-退化图上的支配集的因子O（d^2）近似算法，该算法在时间n^{O（logn）}内运行，所需空间为O（log^2n）比特.对于d-正则图，我们展示了如何将一个已知的随机因子-O（log d）近似算法去随机化，使其在时间上运行为n^{O（1）}，并使用O（log n）位空间。我们的结果使用了核化理论，分布式算法和随机算法的思想相结合。
英文摘要：
We develop new approximation algorithms for classical graph and set problems in the RAM model under space constraints. As one of our main results, we devise an algorithm for d-Hitting Set that runs in time n^{O(d^2 + d/ε})}, uses O((d^2 + d/ε) log n) bits of space, and achieves an approximation ratio of O((d/ε) n^ε) for any positive ε\leq 1 and any natural number d. In particular, this yields a factor-O(log n) approximation algorithm which runs in time n^{O(log n)} and uses O(log^2 n) bits of space (for constant d). As a corollary, we obtain similar bounds for Vertex Cover and several graph deletion problems. For bounded-multiplicity problem instances, one can do better. We devise a factor-2 approximation algorithm for Vertex Cover on graphs with maximum degree Δ, and an algorithm for computing maximal independent sets which both run in time n^{O(Δ)} and use O(Δlog n) bits of space. For the more general d-Hitting Set problem, we devise a factor-d approximation algorithm which runs in time n^{O(d δ^2)} and uses O(d δ^2 log n) bits of space on set families where each element appears in at most δsets. For Independent Set restricted to graphs with average degree d, we give a factor-(2d) approximation algorithm which runs in polynomial time and uses O(log n) bits of space. We also devise a factor-O(d^2) approximation algorithm for Dominating Set on d-degenerate graphs which runs in time n^{O(log n)} and uses O(log^2 n) bits of space. For d-regular graphs, we show how a known randomized factor-O(log d) approximation algorithm can be derandomized to run in time n^{O(1)} and use O(log n) bits of space. Our results use a combination of ideas from the theory of kernelization, distributed algorithms and randomized algorithms.
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