Past exam of the computer science course of the University of Oxford / 2019 / Artificial intelligence / 2 / d

A useful local search state is any occupied-square subset $S\subseteq Q$, allowing temporary attacks. Let $C(S)$ count unordered attacking pairs and $N=n^2$. MaximizeACTIONS can offer insertion at an empty square, deletion of an occupied square, and relocation of one knight to an empty square. RESULT is respectively $S\cup \{q\}$, $S\setminus \{p\}$, and $(S\setminus \{p\})\cup \{q\}$. Insertions and deletions alone connect the whole state space; relocation is a useful additional proposal.

Every conflicting state has $F(S)\le N-(N+1)<0$, whereas the empty board has score zero. Every conflict-free state has score exactly its knight count. Hence every global maximizer is a maximum-size nonattacking placement. If a knight participates in $r\ge 1$ conflicts, deleting it changes the score by $-1+(N+1)r>0$, so a positive-conflict local stopping point cannot persist when improving deletions are considered.

From a random board, choose score-improving moves, with random restarts or occasional non-improving proposals to escape poor conflict-free local optima. A maximal placement is not necessarily a maximum placement: accepting only legal insertions can get stuck too early. This objective function defines the intended optimization precisely; a finite heuristic run still gives no general global-optimality certificate.