Gerrymandering on graphs: Computational complexity and parameterized algorithms

💡 Research Summary

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The paper investigates the computational complexity of a graph‑theoretic formulation of gerrymandering, extending earlier models to allow each voter (graph vertex) to assign a non‑negative integer weight to every candidate. This generalized problem, called Weighted Gerrymandering (W‑GM), captures realistic electoral settings such as blanket primaries where multiple candidates receive votes from the same locality. The authors first settle an open question left by Ito et al. (AAMAS 2019) by proving that the original Gerrymandering problem (GM) remains NP‑complete even when the underlying graph is a simple path and the number of candidates is unbounded. The reduction is from the Rainbow Matching problem on paths, which is known to be NP‑complete, thereby showing that the difficulty does not disappear on linear topologies once candidate preferences become diverse.

Having established hardness, the paper turns to parameterized complexity. Three natural parameters are considered: the number of districts k (the required number of connected subgraphs in the partition), the number of voters n (the size of the graph), and the number of candidates m. Prior work already ruled out FPT algorithms parameterized solely by k + m, because the problem is NP‑complete on complete bipartite graphs K_{2,n} even for k = 2 and m = 2. Consequently, any FPT approach must either focus on n or restrict the input to forests.

Parameter k on paths.
For path graphs the authors design both deterministic and randomized fixed‑parameter tractable (FPT) algorithms with respect to k. The key insight is that any district on a path is uniquely identified by its leftmost and rightmost vertices. By enumerating all O(n²) possible districts they construct an auxiliary graph H whose vertices correspond to districts and whose edges connect districts that appear consecutively along the original path. The task of finding a feasible partition then becomes the problem of locating a length‑k path in H that covers all original vertices and maximizes the number of districts won by the distinguished candidate p. Using representative‑set techniques and known FPT algorithms for k‑length path detection (e.g., color‑coding, fast subset convolution), they obtain a deterministic algorithm running in time 2^{O(k)}·n^{O(1)} and a randomized version running in 2^{k}·n^{O(1)} with one‑third error probability. The algorithms assume a tie‑breaking rule that selects a single winner per district; however, the authors note that the methods can be adapted to handle multiple winners.

Parameter n on arbitrary graphs.
When the number of voters n is taken as the parameter, the authors present an algorithm with running time 2^{n}·(n+m)^{O(1)}. The approach hinges on encoding each candidate’s winning districts as a family of subsets F_c. For each subset S they create a monomial y^{χ(S)} where χ(S) is the n‑bit characteristic vector of S. The disjointness of districts translates into a simple condition on binary addition of characteristic vectors, allowing the problem to be reduced to a series of polynomial multiplications. By employing Fast Fourier Transform (FFT) based polynomial multiplication, the authors efficiently combine the families for all candidates, ultimately checking whether a selection of k districts exists in which p wins strictly more districts than any other candidate. This algebraic reduction yields the claimed 2^{n}·(n+m)^{O(1)} bound.

Open problems and limitations.
The paper notes that extending the k‑parameterized algorithm beyond paths is unlikely, as the problem remains NP‑complete on graphs of pathwidth 2 (e.g., K_{2,n}) and on graphs with feedback‑vertex‑set size 1. For trees, a simple O(n·k^{k‑1}) algorithm exists by guessing the k − 1 edges whose removal yields the districts, but an FPT algorithm parameterized solely by k on trees remains open.

Technical contributions.
The work introduces two sophisticated tools to the study of social‑choice problems: (1) representative‑set families, which compress the exponential search space for district selections while preserving optimality, and (2) FFT‑based polynomial multiplication, which transforms a combinatorial disjointness check into an algebraic convolution. Both techniques are applied here for the first time to gerrymandering‑type problems, suggesting broader applicability to other election‑control or coalition‑formation settings.

Practical implications.
From a policy perspective, the algorithms provide exact decision procedures for small‑scale elections where either the number of districts or the electorate size is modest. The weighted model accommodates multi‑candidate ballots, making the results relevant for primary elections, proportional representation systems, or any scenario where voters allocate varying numbers of votes to multiple contenders. While the exponential dependence on k or n limits scalability, the existence of FPT algorithms delineates the boundary between tractable and intractable instances, informing both theoreticians and practitioners about which parameter regimes are amenable to exact computation.