Computing Thiele Rules on Interval Elections and their Generalizations

Researchers resolve a long-standing open question in computational social choice by proving that Thiele rules, particularly Proportional Approval Voting, can be computed in polynomial time on the voter interval domain despite the constraint matrix not being totally unimodular. The breakthrough extends to more general domains (VCI and LC) and establishes that the LC domain strictly contains the VCI domain.

This theoretical computer science research addresses a fundamental computational challenge in approval-based committee voting systems. Thiele rules have attracted significant academic attention for their desirable properties—proportional representation, Pareto optimality, and support monotonicity—yet their NP-hardness in general settings has limited practical application. The paper demonstrates that structured preference domains enable efficient computation, a pattern increasingly recognized in algorithmic social choice.

The significance lies in bridging a gap between theory and practice. While Thiele rules were previously computable in polynomial time only on the candidate interval domain via linear programming, the resolution for the voter interval domain remained unknown for years. By proving an optimal integral solution exists despite the non-totally unimodular constraint matrix, the authors unlock computational tractability for a broader class of preference structures. This represents intellectual progress in understanding which restrictions on voter preferences make hard problems tractable.

The extension to voter-candidate interval and linearly consistent domains demonstrates the generality of the approach. The paper's graph-theoretic analysis establishing that LC strictly contains VCI provides new structural insights into these domains' relationships, previously unstudied. The alternative LC definition with natural interpretation in approval elections suggests these theoretical constructs have practical grounding.

The finding that tree-based generalizations of VCI render Thiele rules NP-hard again illustrates the boundary conditions of computational tractability. This research strengthens the theoretical foundations of social choice algorithms and may inform the design of voting systems where computational efficiency matters—whether in academic contexts, organizational decision-making, or decentralized governance protocols.

• →Thiele rules become polynomial-time computable on voter interval domains despite constraint matrices lacking total unimodularity
• →The LC domain strictly contains the VCI domain, establishing previously unknown structural relationships between preference domains
• →An alternative LC definition provides natural interpretation in approval elections and offers independent theoretical interest
• →Tree-based generalizations of VCI preserve NP-hardness, identifying boundaries of computational tractability
• →The breakthrough resolves a repeatedly posed open question in computational social choice theory