Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.28658

Goal Staying Makes Sum-of-Costs Anonymous Multi-Agent Path Finding NP-Hard

Hang Ma
View original source ↗Read source document ↗Export metadata ↓

Abstract

Anonymous Multi-Agent Path Finding (AMAPF) admits polynomial-time network-flow algorithms for several objectives, including makespan, total distance, and sum-of-costs (SoC) when agents disappear upon reaching goals. We show that standard goal-staying AMAPF is fundamentally different. We first formulate SoC minimization by augmenting the standard time-expanded flow model with goal-settlement constraints and show that the resulting linear programming relaxation is non-integral. We then prove that minimizing SoC in goal-staying AMAPF is NP-hard via a reduction from 3-SAT. Together with the polynomial-time result for the disappearing variant, this establishes a sharp complexity boundary determined by whether completed agents remain at their goals.

Explore related subjects

cs.MAcs.CCcs.RO

Keep this discovery

BibTeXRIS

Hang Ma. 2026-08-21. Goal Staying Makes Sum-of-Costs Anonymous Multi-Agent Path Finding NP-Hard. https://arxiv.org/abs/2608.28658

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

Discover connections

cs.CC subjectcs.MA subjectcs.RO subject

Connections use source metadata and explicit phrase matches, not verified experimental comparisons.

KEEP EXPLORING

Related papers

arXiv2026

Post-Training Language Models for Gold-Medal Performance in Coding Competitions

Competitive programming has become a key test of large language model reasoning, with international competitions such as IOI and ICPC representing its most challenging settings. We present an end-to-end specialization pipeline combining large-scale problem curation, synthetic reasoning traces, supervised fine-tuning (SFT), and reinforcement learning (RL). Using 22,000 curated problems, we train Nemotron-3-Nano-CC (30B-A3B) with SFT and RL and Nemotron-3-Ultra-CC (550B-A55B) with SFT alone. We further introduce GenCorrect, a feedback-driven test-time compute strategy that iteratively generates, evaluates, and refines diverse solutions. On IOI 2025, Nano-CC improves from 130 points to 291 after post-training and to 468 with GenCorrect, exceeding the gold threshold of 438.3 while Ultra-CC reaches 502. Guided by these results, we develop a competition-specific Ultra-CC system and evaluate it prospectively during IOI 2026. Under the same time, internet-access, and submission constraints as human contestants, it scores 535.4 out of 600, exceeding both the gold threshold of 361.12 and the top human score of 498.27. To our knowledge, this is the first AI system to outscore the highest-scoring human contestant on an IOI problem set.

cs.LG↗
arXiv2026

Quantum Blind Rotation for Fast Functional Bootstrapping

Fully homomorphic encryption (FHE) enables privacy-preserving cloud computation, but its efficiency is often limited by the cost of bootstrapping. In particular, existing functional bootstrapping techniques have complexity exponential in the plaintext size. In this work, we show that employing a single quantum server can reduce this dependence. We propose a quantum functional bootstrapping algorithm that allows to evaluate any efficiently computable function in time polynomial in the plaintext size. For general functional bootstrapping over $l$-bit plaintexts, we obtain a time--space tradeoff: poly($l$)-time evaluation can be achieved with O$(2^l)$ qubits, while reducing the space complexity increases the time complexity. Technically, we extend a key classical cryptographic operation, known as \emph{blind rotation}, to the quantum setting by replacing polynomial-exponent encoding with quantum phase encoding. Underlying our extension are insights for the quantum extension of polynomial-based cryptographic tools that may gain dramatic speedups.

quant-ph↗
arXiv2026

Quantum Query Complexity of Finding a Tarski Fixed Point on a High-Dimensional Grid

The Knaster-Tarski fixed-point theorem states that every monotone function over a complete lattice has a fixed point. Beyond its fundamental role in order theory, the theorem and its algorithmic variants have found broad applications in areas such as economics, game theory, and programming languages. While the query complexity of finding a Tarski fixed point has been extensively studied in classical models, comparatively little is known in the quantum setting. We prove an $Ω(k\log n)$ quantum query lower bound for finding a fixed point of a monotone function on $[n]^k$, using the nonnegative spectral adversary method. In the two extremal regimes $n = 2$ and $k = 1$, our quantum lower bound matches the previous classical lower bounds $Ω(k)$ and $Ω(\log n)$, respectively. For $n, k\geq 2$, our bound improves the best previous classical lower bound when $n < k$ and is within a factor of $\log n / \log k$ compared to the known classical lower bound when $n \geq k$. To construct the adversary matrix, we develop the Tree--Filtration Adversary Method. Besides yielding our lower bound, the method offers a more transparent combinatorial interpretation of the nonnegative spectral adversary method. When the hard instances of a problem admit a tree-like organization and suggest an intuition analogous to classical decision-tree lower bounds, our method provide a promising approach to establishing quantum complexity lower bounds.

quant-ph↗