4 problems

Kawamura–Soejima's conjecture. The maximum length $L_k$ diverges as $k\to\infty$.

Czyzowicz et al.'s optimality conjecture. The strategy above is optimal.

Kawamura–Soejima's conjecture. The best possible efficiency satisfies $\alpha\leq\frac{2}{3}$ for every set of agent speeds.

Let $k$ agents patrol a circle of unit length, with maximum speeds $v_1,dots,v_k>0$, and require all agents to move clockwise. Relabel them so that $v_1\geq v_2\geq\cdots\geq v_k$.…