Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2511.00184

Scheduling Problems with Constrained Rejections

Sami Davies·Venkatesan Guruswami·Xuandi Ren
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study bicriteria versions of Makespan Minimization on Unrelated Machines and Santa Claus by allowing a constrained number of rejections. Given an instance of Makespan Minimization on Unrelated Machines where the optimal makespan for scheduling $n$ jobs on $m$ unrelated machines is $T$, (Feige and Vondrák, 2006) gave an algorithm that schedules a $(1-1/e+10^{-180})$ fraction of jobs in time $T$. We show the ratio can be improved to $0.6533>1-1/e+0.02$ if we allow makespan $3T/2$. To the best our knowledge, this is the first result examining the tradeoff between makespan and the fraction of scheduled jobs when the makespan is not $T$ or $2T$. For the Santa Claus problem (the Max-Min version of Makespan Minimization), the analogous bicriteria objective was studied by (Golovin, 2005), who gave an algorithm providing an allocation so a $(1-1/k)$ fraction of agents receive value at least $T/k$, for any $k \in \mathbb{Z}^+$ and $T$ being the optimal minimum value every agent can receive. We provide the first hardness result by showing there are constants $δ,\varepsilon>0$ such that it is NP-hard to find an allocation where a $(1-δ)$ fraction of agents receive value at least $(1-\varepsilon) T$. To prove this hardness result, we introduce a bicriteria version of Set Packing, which may be of independent interest, and prove some algorithmic and hardness results for it. Overall, we believe these bicriteria scheduling problems warrant further study as they provide an interesting lens to understand how robust the difficulty of the original optimization goal might be.

Explore related subjects

cs.DScs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sami Davies, Venkatesan Guruswami, Xuandi Ren. 2025-10-31. Scheduling Problems with Constrained Rejections. https://arxiv.org/abs/2511.00184

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

Polynomial Time Algorithms for the Kadison-Singer Problem

Marcus, Spielman, and Srivastava [MSS15] established the existence of Kadison--Singer partitions. We provide polynomial-time algorithms for the Kadison--Singer problem. For Hermitian matrices $A_1,\ldots,A_m\in\mathbb C^{n\times n}$ of rank at most one, we give two algorithms that find signs $σ\in\{\pm1\}^m$ satisfying $\|\sum_i σ_iA_i\|\le C\|\sum_i A_i^2\|^{1/2}$. The deterministic algorithm achieves $C=3.3443$ using $\widetilde O(mn^2+n^{4.75})$ arithmetic operations. The randomized algorithm achieves $C=4.8628$ using $\widetilde O(mn^2+n^{3.58})$ arithmetic operations in expectation. For vectors satisfying $\sum_i a_ia_i^*=I$ and $\|a_i\|^2\leα$, the algorithms yield partitions $[m]=I_1\cup I_2$ satisfying $\|\sum_{i\in I_j}a_ia_i^*-I/2\|\le (C/2)\sqrtα$ for $j=1,2$.

cs.DS↗