A 2.37332-Competitive Algorithm for Online Square Packing with Gravity
We consider online packing of axis-parallel squares into a unit-width strip under the Tetris and gravity constraints: An incoming square must be lowered from above along a monotonic downwards path until it reaches support from below. Fekete, Kamphans, and Schweer [Algorithmica, 2014] gave an algorithm with asymptotic competitive ratio $34/13\approx2.6154$ in this model. We present $\mathrm{AsymmetricSlots}$, a recursive algorithm based on splitting each slot into a wide and narrow subslot. The proof uses a local charging argument: squares that are large relative to its associated slot pay for the height they create with their own area, while smaller squares are balanced between the two subslots and may use a bounded temporary credit. For a suitable split parameter $p^\star$, we prove $\mathrm{AsymmetricSlots}{p^\star}(σ)\le 2.37332 \operatorname{OPT} (σ)+O(1)$ for every input sequence $σ$. Additionally, we show that the same framework gives an algorithm with asymptotic competitive ratio $O(κ)$ for rectangles of aspect ratio at most $κ$, and a matching $ Ω(κ)$ lower bound shows that the dependence on $κ$ is asymptotically optimal. For the square algorithm, we give a lower bound of $2$ on its asymptotic competitive ratio.