arXiv 1809.093452018-09cs.CCcs.DMcs.DS

串图同态问题变体的次指数算法

Subexponential algorithms for variants of homomorphism problem in string graphs

Karolina Okrasa, Paweł Rzążewski

摘要：

我们考虑了从具有$n$顶点的曲线（弦图）的交图到固定图$H$的加权同态的复杂性。我们提供了一个完整的二分法的问题：如果$H$没有两个顶点共享两个共同的邻居，那么这个问题可以解决的时间$2^{O(n^{2/3} \log n)}$，否则没有算法工作的时间$2^{o(n)}$，即使在相交图的部分，除非ETH失败。这推广了有关几何求交图计算问题复杂性的几个已知结果。然后我们考虑了图同态问题的两个变体，称为局部单射同态和局部双射同态，其中我们要求同态在每个顶点的邻域上是单射或双射的。我们表明，对于每个目标图$H$，这两个问题总是可以解决的时间$2^{O(\sqrt{n} \log n)}$在字符串图。对于以类似方式定义的局部满同态，情况似乎更复杂。我们证明了最大度为2的简单连通图$H$的二分法。如果$H$同构于$P_3$或$C_4$，则从具有$n$顶点的字符串图到$H$的局部满射同态的存在性可以在时间$2^{O(n^{2/3} \log^{3/2} n)}$上判定，否则问题不能在时间$2^{o(n)}$上求解，除非ETH失败。作为副产品，我们得到了几个结果有关的复杂性变种的同态问题的$P_t$ -free图。特别地，我们得到了加权同态的二分法，类似于弦图的二分法。

英文摘要：

We consider the complexity of finding weighted homomorphisms from intersection graphs of curves (string graphs) with $n$ vertices to a fixed graph $H$. We provide a complete dichotomy for the problem: if $H$ has no two vertices sharing two common neighbors, then the problem can be solved in time $2^{O(n^{2/3} \log n)}$, otherwise there is no algorithm working in time $2^{o(n)}$, even in intersection graphs of segments, unless the ETH fails. This generalizes several known results concerning the complexity of computatational problems in geometric intersection graphs. Then we consider two variants of graph homomorphism problem, called locally injective homomorphism and locally bijective homomorphism, where we require the homomorphism to be injective or bijective on the neighborhood of each vertex. We show that for each target graph $H$, both problems can always be solved in time $2^{O(\sqrt{n} \log n)}$ in string graphs. For the locally surjecive homomorphism, defined in an analogous way, the situation seems more complicated. We show the dichotomy theorem for simple connected graphs $H$ with maximum degree 2. If $H$ is isomorphic to $P_3$ or $C_4$, then the existence of a locally surjective homomorphism from a string graph with $n$ vertices to $H$ can be decided in time $2^{O(n^{2/3} \log^{3/2} n)}$, otherwise the problem cannot be solved in time $2^{o(n)}$, unless the ETH fails. As a byproduct, we obtain several results concerning the complexity of variants of homomorphism problem in $P_t$-free graphs. In particular, we obtain the dichotomy theorem for weighted homomorphism, analogous to the one for string graphs.

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摘要：

英文摘要：

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