Cake cutting is a classic model for studying fair division of a heterogeneous, divisible resource among agents with individual preferences. Addressing cake division under a typical requirement that each agent must receive a connected piece of the cake, we develop approximation algorithms for finding envy-free (fair) cake divisions. In particular, this work improves the state-of-the-art additive approximation bound for this fundamental problem. Our results hold for general cake division instances in which the agents' valuations satisfy basic assumptions and are normalized (to have value $1$ for the cake). Furthermore, the developed algorithms execute in polynomial time under the standard Robertson-Webb query model. Prior work has shown that one can efficiently compute a cake division (with connected pieces) in which the additive envy of any agent is at most $1/3$. An efficient algorithm is also known for finding connected cake divisions that are (almost) $1/2$-multiplicatively envy-free. Improving the additive approximation guarantee and maintaining the multiplicative one, we develop a polynomial-time algorithm that computes a connected cake division that is both $\left(\frac{1}{4} +o(1) \right)$-additively envy-free and $\left(\frac{1}{2} - o(1) \right)$-multiplicatively envy-free. Our algorithm is based on the ideas of interval growing and envy-cycle-elimination. In addition, we study cake division instances in which the number of distinct valuations across the agents is parametrically bounded. We show that such cake division instances admit a fully polynomial-time approximation scheme for connected envy-free cake division.

翻译：蛋糕分割是研究在具有个体偏好的主体之间公平分配异质可分割资源的经典模型。针对每个主体必须获得一块相连蛋糕这一典型要求下的蛋糕分割问题，我们开发了寻找无嫉妒（公平）蛋糕分割的近似算法。特别地，本研究改进了该基本问题的顶级加性近似界。我们的结果适用于一般蛋糕分割实例，其中主体估值满足基本假设且已归一化（对蛋糕总价值为$1$）。此外，所开发的算法在标准Robertson-Webb查询模型下能以多项式时间执行。先前研究表明，可有效计算任意主体加性嫉妒至多为$1/3$的（相连块）蛋糕分割。已知存在有效算法用于寻找（几乎）$1/2$乘性无嫉妒的相连蛋糕分割。在改进加性近似保证并保持乘性近似的同时，我们开发了一种多项式时间算法，可计算出同时满足$\left(\frac{1}{4} +o(1) \right)$-加性无嫉妒和$\left(\frac{1}{2} - o(1) \right)$-乘性无嫉妒的相连蛋糕分割。我们的算法基于区间增长和嫉妒环消除的思想。此外，我们研究了主体间不同估值数量受参数约束的蛋糕分割实例。结果表明，此类实例存在针对相连无嫉妒蛋糕分割的完全多项式时间近似方案。