Sequence graphs realizations and ambiguity in language models

Several popular language models represent local contexts in an input text as bags of words. Such representations are naturally encoded by a sequence graph whose vertices are the distinct words occurring in x, with edges representing the (ordered) co-occurrence of two words within a sliding window of size w. However, this compressed representation is not generally bijective, and may introduce some degree of ambiguity. Some sequence graphs may admit several realizations as a sequence, while others may not admit any realization. In this paper, we study the realizability and ambiguity of sequence graphs from a combinatorial and computational point of view. We consider the existence and enumeration of realizations of a sequence graph under multiple settings: window size w, presence/absence of graph orientation, and presence/absence of weights (multiplicities). When w = 2, we provide polynomial time algorithms for realizability and enumeration in all cases except the undirected/weighted setting, where we show the #P-hardness of enumeration. For a window of size at least 3, we prove hardness of all variants, even when w is considered as a constant, with the notable exception of the undirected/unweighted case for which we propose an XP algorithms for both (realizability and enumeration) problems, tight due to a corresponding W[1]-hardness result. We conclude with an integer program formulation to solve the realizability problem, and with dynamic programming to solve the enumeration problem. This work leaves open the membership to NP for both problems, a non-trivial question due to the existence of minimum realizations having exponential size on the instance encoding.