Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2505.12123

Logarithmic Approximations for Fair k-Set Selection

Shi Li·Chenyang Xu·Ruilong Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the fair k-set selection problem where we aim to select $k$ sets from a given set system such that the (weighted) occurrence times that each element appears in these $k$ selected sets are balanced, i.e., the maximum (weighted) occurrence times are minimized. By observing that a set system can be formulated into a bipartite graph $G:=(L\cup R, E)$, our problem is equivalent to selecting $k$ vertices from $R$ such that the maximum total weight of selected neighbors of vertices in $L$ is minimized. The problem arises in a wide range of applications in various fields, such as machine learning, artificial intelligence, and operations research. We first prove that the problem is NP-hard even if the maximum degree $Δ$ of the input bipartite graph is $3$, and the problem is in P when $Δ=2$. We then show that the problem is also in P when the input set system forms a laminar family. Based on intuitive linear programming, we show that a dependent rounding algorithm achieves $O(\frac{\log n}{\log \log n})$-approximation on general bipartite graphs, and an independent rounding algorithm achieves $O(\logΔ)$-approximation on bipartite graphs with a maximum degree $Δ$. We demonstrate that our analysis is almost tight by providing a hard instance for this linear programming. Finally, we extend all our algorithms to the weighted case and prove that all approximations are preserved.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shi Li, Chenyang Xu, Ruilong Zhang. 2025-05-17. Logarithmic Approximations for Fair k-Set Selection. https://arxiv.org/abs/2505.12123

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Path Enumeration by Position-Visit Counts in Recombining Trinomial Trees

Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree contains $3^D$ raw trajectories, making exhaustive enumeration rapidly infeasible. However, when each node's value depends only on its position, a raw trajectory's aggregate is determined by its position-visit counts. We call these count vectors cardinality tuples and decompose the admissible tuples into weak-composition mass layers. Leveraging these structures, we introduce a mass-shifting enumeration algorithm that slides integer ``masses'' through cardinality tuples to generate exactly one representative of each path-equivalence class, while the accompanying weak-composition bijections yield exact counting formulas for the generated families. This suppresses redundant raw-path orderings a priori rather than enumerating and deduplicating them afterward. For the full-tuple implementation, we prove an output-sensitive running-time bound at each fixed endpoint, together with a uniform worst-case upper bound $\mathscr{O}(D2^D)$ and an exact worst-case exponential growth base of $2$, compared with base $3$ for exhaustive raw-path enumeration. Thus the construction achieves a provable exponential reduction in the enumeration space, up to polynomial factors. The same framework also recovers the information compressed by the equivalence classes: we derive an exact degeneracy formula for the number of raw paths represented by every cardinality tuple. We further prove that the nonnegative return specialization is exactly the classical Motzkin family, recover its recursive and generating-function structure and the Dyck specialization, and derive a multivariate occupation-profile $J$-fraction whose coefficients recover the corresponding cardinality-tuple degeneracies.

cs.DS↗
arXiv2026

The Longest Common Bitonic Subsequence: Match-Sensitive Algorithms and Conditional Hardness

The longest common bitonic subsequence problem asks for a longest common subsequence of two ordered sequences whose values strictly increase and then strictly decrease; either phase may be empty. We formulate the problem through increasing and decreasing endpoint values at matching position pairs. This gives a constructive quadratic baseline and a matchsensitive algorithm based on two standard dominance-maximum passes. Its time is the sum of an input-sorting term and the number of matches times a squared logarithmic factor. We state the endpoint interface that permits reuse of increasing subsequence algorithms, and distinguish this specialization from new range searching machinery. A linear-size padding reduction transfers the conditional strongly subquadratic lower bound for longest common increasing subsequence to the bitonic problem. Reproducible implementations, exhaustive small-instance checks, and newly measured synthetic experiments document correctness and the practical tradeoff between sparse and dense processing.

cs.DS↗
arXiv2026

Locally Approximating the Top Eigenvector of Bounded Entry Matrices

We provide a local computation algorithm to approximate the top eigenvector $x \in \mathbb{R}^n$ of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries between $-1$ and $1$, building on the work of Swartworth and Woodruff [SODA 25] who show how to approximate the eigenvalues up to additive-$\varepsilon n$ error using $\tilde{O}(1/\varepsilon^4)$ queries. Our local computation algorithm has a preprocessing complexity of $\tilde{O}(1/\varepsilon^4)$ and per-coordinate query complexity of $\tilde{O}(1/\varepsilon^2)$ for an additive-$\varepsilon n$ approximation whenever {$|λ_{\min}(A)| = O(λ_{\max}(A))$. When $λ_{\min}(A)$ greatly exceeds $λ_{\max}(A)$, our complexity degrades to at most $\tilde{O}(1/\varepsilon^{6.\overline{6}})$ in preprocessing and $\tilde{O}(1/\varepsilon^{3.\overline{3}})$ per query. Furthermore, we show a lower bound of $Ω(n/\varepsilon^2)$ on the total number of queries needed to output an approximately top eigenvector (implying that the per-coordinate query complexity of $Ω(1/\varepsilon^2)$ is necessary). As an application, we use our algorithm to provide local computation algorithms for the sparsest-cut and max-cut problems in the dense graph model of Goldreich, Goldwasser, Ron [JACM 98]. By accessing the top eigenvectors (of an approximate normalized adjacency), we implement local versions of Cheeger's inequality and Trevisan's algorithm [SICOMP 12] to obtain "square-root-opt" approximations in polynomial time (as opposed to exponential-in-$\text{poly}(1/\varepsilon)$ time which is incurred in Goldreich, Goldwasser, Ron.

cs.DS↗