Guaranteeing Half-Maximin Shares Under Cardinality Constraints

We study the problem of fair allocation of a set of indivisible items among agents with additive valuations, under cardinality constraints. In this setting the items are partitioned into categories, each with its own limit on the number of items it may contribute to any bundle. One example of such a problem is allocating seats in a multitrack conference. We consider the fairness measure known as the maximin share (MMS) guarantee, and propose a novel polynomial-time algorithm for finding $1/2$-approximate MMS allocations. We extend the notions and algorithms related to ordered and reduced instances to work with cardinality constraints, and combine these with a bag filling style procedure. Our algorithm improves on that of Biswas and Barman (IJCAI-18), with its approximation ratio of $1/3$. We also present an optimizing algorithm, which for each instance, instead of fixing $\alpha = 1/2$, uses bisection to find the largest $\alpha$ for which our algorithm obtains a valid $\alpha$-approximate MMS allocation. Numerical tests show that our algorithm finds strictly better approximations than the guarantee of $1/2$ for most instances, in many cases surpassing $3/5$. The optimizing version of the algorithm produces MMS allocations in a comparable number of instances to that of Biswas and Barman's algorithm, on average achieving a better approximation when MMS is not obtained.