H-着色问题空间中的逼近距离
Approximability Distance in the Space of H-Colourability Problems
摘要：
图同态是一个顶点映射，它将源图中的边带到目标图中的边。研究了加权最大H-可积子图问题（MAX H-COL）的可逼近性。这个问题的例子是边权图G，目标是找到G的一个具有最大总边权的子图，条件是该子图与H有同态;注意，对于H=K_k，这个问题等价于MAX k-CUT。为此，我们引入了一个度量结构的图形空间，使我们能够扩展以前已知的逼近结果更大的类图。具体地说，近似算法的最大切割由Goemans和Williamson和最大k-CUT由Frieze和Jerrum可以用来产生非平凡的近似结果的最大H-COL。对于各种图，我们显示了近最优性的结果下的唯一游戏猜想。我们还使用我们的方法比较Frieze & Jerrum的算法与Hastad的近似算法的性能一般MAX 2-CSP。这种比较在大多数情况下有利于Frieze & Jerrum。
英文摘要：
A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. We study the approximability properties of the Weighted Maximum H-Colourable Subgraph problem (MAX H-COL). The instances of this problem are edge-weighted graphs G and the objective is to find a subgraph of G that has maximal total edge weight, under the condition that the subgraph has a homomorphism to H; note that for H=K_k this problem is equivalent to MAX k-CUT. To this end, we introduce a metric structure on the space of graphs which allows us to extend previously known approximability results to larger classes of graphs. Specifically, the approximation algorithms for MAX CUT by Goemans and Williamson and MAX k-CUT by Frieze and Jerrum can be used to yield non-trivial approximation results for MAX H-COL. For a variety of graphs, we show near-optimality results under the Unique Games Conjecture. We also use our method for comparing the performance of Frieze & Jerrum's algorithm with Hastad's approximation algorithm for general MAX 2-CSP. This comparison is, in most cases, favourable to Frieze & Jerrum.
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