{"paper":{"title":"Ulam Rank Aggregation Is Hard to Approximate for Four Rankings","arxiv_id":"2609.05874","primary_cat":"cs.CC","submitted_at":"2026-09-05T04:31:48+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.54,"id":"s0","own":true,"role":"contribution","text":"We study the approximability of rank aggregation under the Ulam metric.","hedge":{"p":0.66,"label":"asserted"},"own_p":0.81,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.46,"limitation":0.0,"future_work":0.0,"contribution":0.54,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"background","text":"In the \\emph{Ulam median} problem, the goal is to find a ranking (permutation) minimizing the sum of its Ulam distances to the input rankings, while in the \\emph{Ulam center} problem, the objective is to minimize the maximum such distance.","hedge":{"p":0.97,"label":"asserted"},"own_p":0.11,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.99,"id":"s2","own":true,"role":"result","text":"We prove that, for every $0<\\varepsilon< 1/34$, it is $\\mathrm{NP}$-hard to approximate either Ulam median or Ulam center within a factor of $35/34-\\varepsilon$, even when the input consists of only four rankings.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.91,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":3,"p":0.98,"id":"s3","own":true,"role":"result","text":"We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.15,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.98,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.01}},{"i":4,"p":0.84,"id":"s4","own":false,"role":"gap","text":"Prior to our work, only the exact versions of both problems were known to be $\\mathrm{NP}$-hard, and that too only when the number of input rankings is unbounded [Fischer et al., ESA'25 and Bachmaier et al., J. of Discrete Algorithms'15].","hedge":{"p":1.0,"label":"asserted"},"own_p":0.06,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.57,"role_probabilities":{"gap":0.84,"method":0.0,"result":0.0,"background":0.16,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":0.44,"id":"s5","own":false,"role":"result","text":"Furthermore, our inapproximability results are optimal in terms of the number of input rankings since for three inputs it is already known to be polynomial-time solvable [Chakraborty, Das, Krauthgamer, SODA'21].","hedge":{"p":1.0,"label":"asserted"},"own_p":0.21,"main_claim_p":0.11,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.02,"method":0.0,"result":0.44,"background":0.05,"limitation":0.01,"future_work":0.0,"contribution":0.07,"interpretation":0.41}},{"i":6,"p":0.67,"id":"s6","own":true,"role":"method","text":"En route, we introduce a new general framework for reducing Boolean constraint satisfaction problems (CSP) to the Ulam median with only four inputs.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.77,"role_probabilities":{"gap":0.0,"method":0.67,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.33,"interpretation":0.0}},{"i":7,"p":0.82,"id":"s7","own":true,"role":"method","text":"As a specific instantiation of the reduction framework, we obtain our hardness-of-approximation results.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.94,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.82,"result":0.07,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.11,"interpretation":0.0}},{"i":8,"p":0.89,"id":"s8","own":true,"role":"method","text":"The corresponding hardness for the Ulam center follows from a reduction from the Ulam median.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.89,"result":0.05,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":1.0,"text":"We prove that, for every $0<\\varepsilon< 1/34$, it is $\\mathrm{NP}$-hard to approximate either Ulam median or Ulam center within a factor of $35/34-\\varepsilon$, even when the input consists of only four 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