We study committee voting rules under ranked preferences, which map the voters' preference relations to a subset of the alternatives of predefined size. In this setting, the compatibility between proportional representation and committee monotonicity is a fundamental open problem that has been mentioned in several works. We address this research question by designing a new committee voting rule called the Solid Coalition Refinement (SCR) rule that simultaneously satisfies committee monotonicity and Dummett's PSC as well as one of its variants called inclusion PSC. This is the first rule known to satisfy both of these properties. Moreover, we show that this is effectively the best that we can hope for as other fairness notions adapted from approval voting are incompatible with committee monotonicity. For truncated preferences, we prove that the SCR rule still satisfies PSC and a property called independence of losing voter blocs, thereby refuting a conjecture of Graham-Squire et al. (2024). Finally, we discuss the consequences of our results in the context of rank aggregation.

Envy-freeness is one of the most important fairness concerns when allocating items. We study envy-free house allocation when agents have uncertain preferences over items and consider several well-studied preference uncertainty models. The central problem that we focus on is computing an allocation that has the highest probability of being envy-free. We show that each model leads to a distinct set of algorithmic and complexity results, including detailed results on (in-)approximability. En route, we consider two related problems of checking whether there exists an allocation that is possibly or necessarily envy-free. We give a complete picture of the computational complexity of these two problems for all the uncertainty models we consider.

Traditional orthogonal range problems allow queries over a static set of points, each with some value. Dynamic variants allow points to be added or removed, one at a time. To support more powerful updates, we introduce the Grid Range class of data structure problems over integer arrays in one or more dimensions. These problems allow range updates (such as filling all cells in a range with a constant) and queries (such as finding the sum or maximum of values in a range). In this work, we consider these operations along with updates that replace each cell in a range with the minimum, maximum, or sum of its existing value, and a constant. In one dimension, it is known that segment trees can be leveraged to facilitate any $n$ of these operations in $\tilde{O}(n)$ time overall. Other than a few specific cases, until now, higher dimensional variants have been largely unexplored. We show that no truly subquadratic time algorithm can support certain pairs of these updates simultaneously without falsifying several popular conjectures. On the positive side, we show that truly subquadratic algorithms can be obtained for variants induced by other subsets. We provide two approaches to designing such algorithms that can be generalised to online and higher dimensional settings. First, we give almost-tight $\tilde{O}(n^{3/2})$ time algorithms for single-update variants where the update operation distributes over the query operation. Second, for other variants, we provide a general framework for reducing to instances with a special geometry. Using this, we show that $O(m^{3/2-ε})$ time algorithms for counting paths and walks of length 2 and 3 between vertex pairs in sparse graphs imply truly subquadratic data structures for certain variants; to this end, we give an $\tilde{O}(m^{(4ω-1)/(2ω+1)}) = O(m^{1.478})$ time algorithm for counting simple 3-paths between vertex pairs.