Assuming that you are trying to maximize the seating preferences, this problem is NP-Hard =(.

Specifically, consider the decision version of this problem: Given a matrix of preferences, is there some way to assign people to seats such that the total score (sum of resulting preferences of professors to their nearby neighbors) obtained is at or above some $k$?

This problem is NP-Complete.

First, clearly this decision problem is in NP, as if one can provide you with such an assignment then you can compute the total score and verify that their solution works.

Next, note that this problem is essentially equivalent to finding two distinct highest-cost paths that go through the same amount of nodes, and together go through every node. Using this idea, we will solve the hamiltonian path problem.

Given two hamiltonian path instances $G_1=(V_1, E_1)$ $G_2=(V_2, E_2)$ of the same size $n$, consider the combined graph $G=(V_1\cup V_2, E_1 \cup E_2)$, and then create an instance of our seating arrangement problem where every edge is a $1$ if it occurs in our combined graph, and a $0$ otherwise. Because these graphs are undirected, the preferences will be the same in both directions.

The question "is there some way to assign seating positions to professors such that the total score obtained is greater than or equal to $\vert V_1 \vert + \vert V_2 \vert$" is NP-Complete, as it is only possible to do this if a hamiltonian path exists in both graphs.