Homomorphism Extension

We define the Homomorphism Extension (HomExt) problem: given a group G, a subgroup M ≤ G and a homomorphism φ : M → H , decide whether or not there exists a homomorphism φ̃ : G → H extending φ, i.e., φ̃|M = φ. This problem arose in the context of list-decoding homomorphism codes but is also of independent interest, both as a problem in computational group theory and as a new and natural problem in NP of unsettled complexity status. We consider the case H = Sm (the symmetric group of degree m), i.e., φ : G → H is a G-action on a set of m elements. We assume G ≤ Sn is given as a permutation group by a list of generators. We characterize the equivalence classes of extensions in terms of a multidimensional oracle subset-sum problem. From this we infer that for bounded G the HomExt problem can be solved in polynomial time. Our main result concerns the case G = An (the alternating group of degree n) for variable n under the assumption that the index of M in G is bounded by poly(n). We solve this case in polynomial time for all m < 2/ √ n. This is the case with direct relevance to homomorphism codes (Babai, Black, and Wuu, arXiv 2018); it is used as a component of one of the main algorithms in that paper.