arXiv subjects

Krzysztof Sornat

Publications and source records attributed to Krzysztof Sornat.

At least 19 recordsLinked to original sources

arXiv2026

Outer Diversity of Condorcet Domains

A Condorcet domain is a set of rankings over a given candidate set, such that every election that consists only of (an odd number of) votes from the domain has a transitive majority relation. We study outer diversity of Condorcet domains, i.e., a measure that quantifies expected swap distance from a random vote to a closest one in the domain. We numerically analyze outer diversity for maximal Condorcet domains with few candidates, and then we establish its asymptotic behavior for several special domains, mostly obtaining theoretical results.

cs.GT↗

arXiv2026

Algorithms for Structured Elections under Thiele Voting Rules

We study the computational complexity of winner determination problems in approval-based committee elections under Thiele voting rules. These form a class of rules parameterized by a fixed weight vector that specifies how a voter's satisfaction depends on the number of approved candidates elected. We first analyze the structure of optimal solutions based on the sets of voters who approve each candidate---that is, how voters' approval ballots induce dependencies between candidates---revealing constraints on a winning committee under any fixed Thiele voting rule. Using this, we design FPT algorithms for Proportional Approval Voting (PAV) and other Thiele rules on a natural restricted domain known as the Voter Interval (VI) domain---that is, after a suitable ordering of voters, each candidate is approved by a consecutive interval of voters. In particular, we show that every Thiele rule on VI is FPT with respect to a parameter for which the problem is NP-hard on general instances, even when the parameter takes constant values. Our results advance the understanding of the computational complexity of PAV on Voter Interval instances, which remains one of the central open questions in this area. We further resolve two open questions from the literature on PAV (and other Thiele voting rules) by providing a polynomial-time algorithm for instances where each candidate is approved by at most two voters, and an FPT algorithm parameterized by the total score of a winning committee.

cs.GT↗

arXiv2026

Algorithms for Candidate Control in Greedy Participatory Budgeting Rules

We study the problem of candidate control in participatory budgeting elections. Our focus is on two prominent sequential welfare-based rules---GreedyAV and GreedyCost---which are widely used in practice. Candidate control asks whether we can strategically modify the set of available candidates so as to either ensure that a preferred candidate $p$ is selected or prevent $p$ from being selected. Since all variants of candidate control under the two rules we consider are known to be NP-hard, we analyze the problems through the lens of parameterized complexity and approximability. Under the first lens, we provide a comprehensive classification with respect to natural parameters such as the number of voters, the number of controlled candidates, and the number of distinct costs, as well as their combinations. Within the second perspective, we establish a tight approximability bound.

cs.GT↗

arXiv2026

Participatory Budgeting with Project Groups

We study a generalization of the standard approval-based model of participatory budgeting (PB), in which voters are providing approval ballots over a set of predefined projects and---in addition to a global budget limit, there are several groupings of the projects, each group with its own budget limit. We study the computational complexity of identifying project bundles that maximize voter satisfaction while respecting all budget limits. We show that the problem is generally intractable and describe efficient exact algorithms for several special cases, including instances with only few groups and instances where the group structure is close to be hierarchical, as well as efficient approximation algorithms. Our results could allow, e.g., municipalities to hold richer PB processes that are thematically and geographically inclusive.

cs.GT↗

arXiv2026

Agreement, Diversity, and Polarization Indices for Approval Elections

An index is a function that given an election outputs a value between 0 and 1, indicating the extent to which this election has a particular feature. We seek indices that capture agreement, diversity, and polarization among voters in approval elections, and that are normalized with respect to saturation. By the latter we mean that if two elections differ by the fraction of candidates approved by an average voter, but otherwise are of similar nature, then they should have similar index values. We propose several indices, analyze their properties, and use them to (a) derive a new map of approval elections, and (b) show similarities and differences between various real-life elections from Pabulib, Preflib and other sources.

cs.GT↗

arXiv2026

Polynomial-Time Algorithm for Thiele Voting Rules with Voter Interval Preferences

We present a polynomial-time algorithm for computing an optimal committee of size $k$ under any given Thiele voting rule for elections on the Voter Interval domain (i.e., when voters can be ordered so that each candidate is approved by a consecutive voters). Our result extends to the Generalized Thiele rule, in which each voter has an individual weight (scoring) sequence. This resolves a 10-year-old open problem that was originally posed for Proportional Approval Voting and later extended to every Thiele rule (Elkind and Lackner, IJCAI 2015; Peters, AAAI 2018). Our main technical ingredient is a new structural result -- a concavity theorem for families of intervals. It shows that, given two solutions of different sizes, one can construct a solution of any intermediate size whose score is at least the corresponding linear interpolation of the two scores. As a consequence, on Voter Interval profiles, the optimal total Thiele score is a concave function of the committee size. We exploit this concavity within an optimization framework based on a Lagrangian relaxation of a natural integer linear program formulation, obtained by moving the cardinality constraint into the objective. On Voter Interval profiles, the resulting constraint matrix is totally unimodular, so it can be solved in polynomial time. Our main algorithm and its proof were obtained via human--AI collaboration. In particular, a slightly simplified version of the main structural theorem used by the algorithm was obtained in a single call to Gemini Deep Think.

cs.GT↗

arXiv2026

Outer Diversity of Structured Domains

An ordinal preference domain is a subset of preference orders that the voters are allowed to cast in an election. We introduce and study the notion of outer diversity of a domain and evaluate its value for a number of well-known structured domains, such as the single-peaked, single-crossing, group-separable, and Euclidean ones.

cs.GT↗

arXiv2026

How Similar Are Two Elections?

We introduce and study isomorphic distances between ordinal elections (with the same numbers of candidates and voters). The main feature of these distances is that they are invariant to renaming the candidates and voters, and two elections are at distance zero if and only if they are isomorphic. Specifically, we consider isomorphic extensions of distances between preference orders: Given such a distance d, we extend it to distance d-ID between elections by unifying candidate names and finding a matching between the votes, so that the sum of the d-distances between the matched votes is as small as possible. We show that testing isomorphism of two elections can be done in polynomial time so, in principle, such distances can be tractable. Yet, we show that two very natural isomorphic distances are NP-complete and hard to approximate. We attempt to rectify the situation by showing FPT algorithms for several natural parameterizations.

cs.GT↗

arXiv2026

Participatory Budgeting Project Strength via Candidate Control

We study the complexity of candidate control in participatory budgeting elections. The goal of constructive candidate control is to ensure that a given candidate wins by either adding or deleting candidates from the election (in the destructive setting, the goal is to prevent a given candidate from winning). We show that such control problems are NP-hard to solve for many participatory budgeting voting rules, including Phragmén and Method of Equal Shares, but there are natural cases with polynomial-time algorithms (e.g., for the GreedyAV rule and projects with costs encoded in unary). We also argue that control by deleting candidates is a useful tool for assessing the performance (or, strength) of initially losing projects, and we support this view with experiments.

cs.GT↗

arXiv2025

Diversity of Structured Domains via k-Kemeny Scores

In the k-Kemeny problem, we are given an ordinal election, i.e., a collection of votes ranking the candidates from best to worst, and we seek the smallest number of swaps of adjacent candidates that ensure that the election has at most k different rankings. We study this problem for a number of structured domains, including the single-peaked, single-crossing, group-separable, and Euclidean ones. We obtain two kinds of results: (1) We show that k-Kemeny remains intractable under most of these domains, even for k=2, and (2) we use k-Kemeny to rank these domains in terms of their diversity.

cs.GT↗

arXiv2025

Robust Committee Voting, or The Other Side of Representation

We study approval-based committee voting from a novel perspective. While extant work largely centers around proportional representation of the voters, we shift our focus to the candidates while preserving proportionality. Intuitively, candidates supported by similar voter groups should receive comparable representation. Since deterministic voting rules cannot achieve this ideal, we develop randomized voting rules that satisfy ex-ante neutrality, monotonicity, and continuity, while maintaining strong ex-post proportionality guarantees. Continuity of the candidate selection probabilities proves to be the most demanding of our ex-ante desiderata. We provide it via voting rules that are algorithmically stable, a stronger notion of robustness which captures the continuity of the committee distribution under small changes. First, we introduce Softmax-GJCR, a randomized variant of the Greedy Justified Candidate Rule (GJCR) [Brill and Peters, 2023], which carefully leverages slack in GJCR to satisfy our ex-ante properties. This polynomial-time algorithm satisfies EJR+ ex post, assures ex-ante monotonicity and neutrality, and provides $O(k^3/n)$-stability (ignoring $\log$ factors). Building on our techniques for Softmax-GJCR, we further show that stronger stability guarantees can be attained by (i) allowing exponential running time, (ii) relaxing EJR+ to an approximate $α$-EJR+, and (iii) relaxing EJR+ to JR. We finally demonstrate the utility of stable voting rules in other settings. In online dynamic committee voting, we show that stable voting rules imply dynamic voting rules with low expected recourse, and illustrate this reduction for Softmax-GJCR. Our voting rules also satisfy a stronger form of stability that coincides with differential privacy, suggesting their applicability in privacy-sensitive domains.

cs.GT↗

arXiv2024

An Experimental Comparison of Multiwinner Voting Rules on Approval Elections

In this paper, we experimentally compare major approval-based multiwinner voting rules. To this end, we define a measure of similarity between two equal-sized committees subject to a given election. Using synthetic elections coming from several distributions, we analyze how similar are the committees provided by prominent voting rules. Our results can be visualized as ``maps of voting rules'', which provide a counterpoint to a purely axiomatic classification of voting rules. The strength of our proposed method is its independence from preimposed classifications (such as the satisfaction of concrete axioms), and that it indeed offers a much finer distinction than the current state of axiomatic analysis.

cs.GT↗

arXiv2023

Diversity, Agreement, and Polarization in Elections

We consider the notions of agreement, diversity, and polarization in ordinal elections (that is, in elections where voters rank the candidates). While (computational) social choice offers good measures of agreement between the voters, such measures for the other two notions are lacking. We attempt to rectify this issue by designing appropriate measures, providing means of their (approximate) computation, and arguing that they, indeed, capture diversity and polarization well. In particular, we present "maps of preference orders" that highlight relations between the votes in a given election and which help in making arguments about their nature.

cs.GT↗

arXiv2022

Near-Tight Algorithms for the Chamberlin-Courant and Thiele Voting Rules

We present an almost optimal algorithm for the classic Chamberlin-Courant multiwinner voting rule (CC) on single-peaked preference profiles. Given $n$ voters and $m$ candidates, it runs in almost linear time in the input size, improving the previous best $O(nm^2)$ time algorithm of Betzler et al. (2013). We also study multiwinner voting rules on nearly single-peaked preference profiles in terms of the candidate-deletion operation. We show a polynomial-time algorithm for CC where a given candidate-deletion set $D$ has logarithmic size. Actually, our algorithm runs in $2^{|D|} \cdot poly(n,m)$ time and the base of the power cannot be improved under the Strong Exponential Time Hypothesis. We also adapt these results to all non-constant Thiele rules which generalize CC with approval ballots.

cs.GT↗

arXiv2022

An O(loglog n)-Approximation for Submodular Facility Location

In the Submodular Facility Location problem (SFL) we are given a collection of $n$ clients and $m$ facilities in a metric space. A feasible solution consists of an assignment of each client to some facility. For each client, one has to pay the distance to the associated facility. Furthermore, for each facility $f$ to which we assign the subset of clients $S^f$, one has to pay the opening cost $g(S^f)$, where $g(\cdot)$ is a monotone submodular function with $g(\emptyset)=0$. SFL is APX-hard since it includes the classical (metric uncapacitated) Facility Location problem (with uniform facility costs) as a special case. Svitkina and Tardos [SODA'06] gave the current-best $O(\log n)$ approximation algorithm for SFL. The same authors pose the open problem whether SFL admits a constant approximation and provide such an approximation for a very restricted special case of the problem. We make some progress towards the solution of the above open problem by presenting an $O(\log\log n)$ approximation. Our approach is rather flexible and can be easily extended to generalizations and variants of SFL. In more detail, we achieve the same approximation factor for the practically relevant generalizations of SFL where the opening cost of each facility $f$ is of the form $p_f+g(S^f)$ or $w_f\cdot g(S^f)$, where $p_f,w_f \geq 0$ are input values. We also obtain an improved approximation algorithm for the related Universal Stochastic Facility Location problem. In this problem one is given a classical (metric) facility location instance and has to a priori assign each client to some facility. Then a subset of active clients is sampled from some given distribution, and one has to pay (a posteriori) only the connection and opening costs induced by the active clients. The expected opening cost of each facility $f$ can be modelled with a submodular function of the set of clients assigned to $f$.

cs.DS↗

arXiv2022

Fine-Grained Liquid Democracy for Cumulative Ballots

We investigate efficient ways for the incorporation of liquid democracy into election settings in which voters submit cumulative ballots, i.e., when each voter is assigned a virtual coin that she can then distribute as she wishes among the available election options. In particular, we are interested in fine-grained liquid democracy, meaning that voters are able to designate a partial coin to a set of election options and delegate the decision on how to further split this partial coin among those election options to another voter of her choice. The fact that we wish such delegations to be transitive -- combined with our aim at fully respecting such delegations -- means that inconsistencies and cycles can occur, thus we set to find computationally-efficient ways of resolving voter delegations. To this aim we develop a theory based fixed-point theorems and mathematical programming techniques and we show that for various variants of definitions regarding how to resolve such transitive delegations, there is always a feasible resolution; and we identify under which conditions such solutions are efficiently computable.

cs.GT↗

arXiv2022

How to Sample Approval Elections?

We study the multifaceted question of how to sample approval elections in a meaningful way. Our analysis aims to discern the properties of various statistical cultures (both established and new ones). Based on the map-of-elections framework by Szufa et al. [2020], we graphically represent statistical cultures; and, by that, provide an intuitive understanding of their differences and properties.

cs.GT↗

arXiv2021

The Complexity of Subelection Isomorphism Problems

We study extensions of the Election Isomorphism problem, focused on the existence of isomorphic subelections. Specifically, we propose the Subelection Isomorphism and the Maximum Common Subelection problems and study their computational complexity and approximability. Using our problems in experiments, we provide some insights into the nature of several statistical models of elections.

cs.GT↗