Voting rules aggregate individual preferences into collective decisions, but the rankings they receive contain only ordinal information. The metric distortion framework studies ordinal voting rules in settings where voters and candidates are embedded in an unknown metric space. Deterministic rules have optimal worst-case distortion $3$, while recent randomized rules break the $3$ barrier. We study whether such improvements can coexist with low worst-case sensitivity with respect to the Wasserstein distance of lotteries under one-voter deletion and approximate differential privacy under one-voter replacement. On the sensitivity side, we give a randomized rule with distortion at most $3-\varepsilon$ for an absolute constant $\varepsilon>0$ and, for $m$ candidates and $n$ voters, a worst-case sensitivity bound of $O((\log m+1)/n)$. On the privacy side, for every $\delta\in(0,1)$ and all $n$ above an absolute constant, we construct a variant rule whose mechanism releasing a single sampled winner has distortion at most $3-\varepsilon$ and is $(O((\log m+\log(1/\delta)+1)/n),\delta)$-differentially private. Both constructions use the same family of Gibbs distributions over constant-size candidate lists, with only the temperature parameter differing between the sensitivity and differential-privacy guarantees. Our analysis builds on the biased-metric viewpoint behind the recent improvement over the $3$ barrier and proves a stability property for the biased-metric ratio.
Shinsaku Sakaue, K. Fujii, Soh Kumabe et al.· arXiv.org· 0 citations
We study restricted-link augmentation to $2$-vertex-connectivity. An instance consists of a graph $G$, possibly disconnected, a set $L$ of admissible links on its vertices, integer link costs in $\{1,\dots,W\}$, and an integer $k$; the task is to add at most $k$ links of minimum total cost so that the resulting multigraph is $2$-vertex-connected. Recent work gives $O^*(k^{O(k)})$-time algorithms for unweighted $\lambda$-vertex-connectivity augmentation for every $\lambda\leq 4$ [Carmesin and Ramanujan, SODA 2026], and an $O^*((k+\lambda)^{O(k)})$-time algorithm for arbitrary $\lambda$ [Korhonen and Thorup, arXiv 2026]. We give a deterministic algorithm with running time $O^*(36^kW)$. Thus, for $\lambda=2$, the unweighted running time improves from $O^*(k^{O(k)})$ to $O^*(36^k)$, and the algorithm also handles link costs with pseudo-polynomial dependence on $W$. We reduce the problem to a boundary-pair variant of $2$-vertex-connected spanning subgraph, where each vertex is assigned a pair of incident edges with an associated pair cost. We solve this variant using a cancellation identity, inspired by Cut&Count [Cygan et al., TALG 2022], obtained by applying M\"obius inversion to decompositions along cut vertices: the identity cancels every connected spanning graph with more than one block and keeps exactly the $2$-vertex-connected spanning graphs.
We study the parameterized complexity of Induced Subgraph Isomorphism (ISI) and Maximum Common Induced Subgraph (MCIS) with respect to the cluster vertex deletion number $k$. For ISI, we give a randomized $O^*(k^{O(k)})$-time algorithm, showing that ISI is fixed-parameter tractable under this parameter and resolving an open question of Hanaka et al. [WALCOM 2026]. Our algorithm is optimal under the Exponential Time Hypothesis (ETH), and is based on a reduction to Exact Multicolored Matching solvable via algebraic techniques. For MCIS, we present a randomized $O^*(2^{O(k^2)})$-time algorithm via a reduction to a weighted variant of Exact Multicolored Matching, and we prove a matching ETH-based lower bound by showing that a $k$-by-$k$ binary matrix feasibility problem with list-constrained rows and columns admits no $O^*(2^{o(k^2)})$-time algorithm, which may be of independent interest. These results reveal that, in this setting, MCIS is strictly harder than ISI. Finally, for the three-graph variant 3-MCIS, we show that it becomes NP-hard already when each input graph has cluster vertex deletion number 2.
Tomohiro Koana, Soh Kumabe, Y. Otachi· 0 citations
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