In this work, we provide the first practical evaluation of the structural rounding framework for approximation algorithms. Structural rounding works by first editing to a wellstructured class, efficiently solving the edited instance, and "lifting" the partial solution to recover an approximation on the input. We focus on the well-studied Vertex Cover problem, and edit to the class of bipartite graphs (where Vertex Cover has an exact polynomial time algorithm). In addition to the naïve lifting strategy for Vertex Cover described by Demaine et al. in the paper describing structural rounding, we introduce a suite of new lifting strategies and measure their effectiveness on a large corpus of synthetic graphs. We find that in this setting, structural rounding significantly outperforms standard 2-approximations. Further, simpler lifting strategies are extremely competitive with the more sophisticated approaches. The implementations are available as an open-source Python package, and all experiments are replicable. *

The generalized coloring numbers of Kierstead and Yang [7] offer an algorithmically useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters. First, we show that it is NP-hard to compute the weak 2-coloring number (answering an open question of Grohe et al. [5]). We then complete the picture by proving that the r-coloring number is also NP-hard to compute for all r ≥ 2. Finally, we give an approximation algorithm for the r-coloring number which improves both the runtime and approximation factor of the existing approach of Dvořák [3]. Our algorithm greedily orders vertices with small enough i-reach for every i ≤ r and achieves an O(C r−1 k r−1 )-approximation, where C j is the jth Catalan number.

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In a representative democracy, elections involve partitioning geographical space into districts which each elect a single representative. Legislation is then determined by votes from these representatives, and thus political parties are incentivized to win as many districts as possible (ideally a plurality). Gerrymandering is the process by which these districts' boundaries are manipulated to give favor to a certain candidate or party. Cohen-Zemach et al. (AAMAS 2018) proposed Gerrymandering as a formalization of this problem on graphs (as opposed to Euclidean space) where districts partition vertices into connected subgraphs. More recently, Gupta et al. (SAGT 2021) studied its parameterized complexity and gave an FPT algorithm for paths with respect to the number of districts k. We prove that Gerrymandering is W[2]-hard on trees (even when the depth is two) with respect to k, answering an open question of Gupta et al. Moreover, we prove that Gerrymandering remains W[2]-hard in trees with ℓ leaves with respect to the combined parameter k + ℓ. To complement this result, we provide an algorithm to solve Gerrymandering that is FPT in k when ℓ is a fixed constant.