We consider subexponential algorithms finding weighted homomorphisms from intersection graphs of curves (string graphs) with n vertices to a fixed graph H. We provide a complete dichotomy: 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 subexponential algorithm, assuming the ETH. Then we consider locally constrained homomorphisms. We show that for each target graph H, the locally injective and locally bijective homomorphism problems can be solved in time $2^{O(\sqrt{n}\log n)}$ in string graphs. For locally surjective homomorphisms we show a dichotomy for H being a path or a cycle. If H is $P_3$ or $C_4$, then the problem can be solved in time $2^{O(n^{2/3}{\log }^{3/2}n)}$ in string graphs; otherwise, assuming the ETH, there is no subexponential algorithm. As corollaries, we obtain new results concerning the complexity of homomorphism problems in $P_t$-free graphs.

我们考虑用次指数算法从具有n个顶点的曲线（字符串图）的交点图到固定图H的位置找到加权同态。我们提供了一个完整的二分法：如果H没有两个顶点共享两个共同的邻居，那么这个问题可以及时解决。$2^{Ø（{ñ}^{2/3}\mathrm{日志}ñ）}$，否则假设ETH，则没有次指数算法。然后，我们考虑局部约束的同态。我们表明，对于每个目标图H，可以及时解决局部内射同质和局部双射同构问题$2^{Ø（\sqrt{ñ}\mathrm{日志}ñ）}$在字符串图中。对于局部异形同态，我们将H表示为路径或循环是二分法。如果H是$P_3$ 要么 $C_4$，那么问题就可以及时解决 $2^{Ø（{ñ}^{2/3}{\mathrm{日志}}^{3/2}ñ）}$在字符串图中；否则，假设ETH，则没有次指数算法。作为推论，我们获得了有关同态问题复杂度的新结果。$P_{Ť}$-无图。