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This paper introduces a mathematical framework to quantify how well word or gesture orders minimize swap distance within a permutohedron, a graph representing all possible permutations of a sequence. They demonstrate that crosslinguistic gestures are surprisingly optimal, achieving at least 77% optimality in minimizing swap distance. The authors further propose framing various linguistic optimization problems, including swap distance minimization, under the quadratic assignment problem (QAP) to establish a general principle of optimal assignment.
Gestures across languages are surprisingly efficient, achieving at least 77% optimality in minimizing the number of adjacent swaps needed to reach their observed order.
The structure of all the permutations of a sequence can be represented as a permutohedron, a graph where vertices are permutations and two vertices are linked if a swap of adjacent elements in the permutation of one of the vertices produces the permutation of the other vertex. It has been hypothesized that word orders in languages minimize the swap distance in the permutohedron: given a source order, word orders that are closer in the permutohedron should be less costly and thus more likely. Here we explain how to measure the degree of optimality of word order variation with respect to swap distance minimization. We illustrate the power of our novel mathematical framework by showing that crosslinguistic gestures are at least $77\%$ optimal. It is unlikely that the multiple times where crosslinguistic gestures hit optimality are due to chance. We establish the theoretical foundations for research on the optimality of word or gesture order with respect to swap distance minimization in communication systems. Finally, we introduce the quadratic assignment problem (QAP) into language research as an umbrella for multiple optimization problems and, accordingly, postulate a general principle of optimal assignment that unifies various linguistic principles including swap distance minimization.
