Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.04637

SAT Encoding for Set of At-Most-K Cardinality Constraints with Ladder Shape

Hieu Truong Xuan·Tuyen Kieu Van·Khanh To Van
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper presents a novel SAT encoding designed to represent a sequence of At-Most-K cardinality constraints arranged in a ladder-like structure, referred to as the Ladder AMK constraint. These AMK constraints feature a ladder-like arrangement, with shared common variables in the center and two variables differing at the beginning and the end. The proposed encoding first decomposes the sequence of AMK constraints into separated blocks, then encodes each block as a whole rather than encoding each AMK constraint individually. This approach improves the efficiency of constraint solving in SAT solvers by significantly reducing the size of the encoding, precisely the number of variables, and the generated clauses required. To achieve this, the encoding employs Sequential Counter to encode each block and constructs clauses that link adjacent blocks, ensuring equivalence with the input constraints. Compared to the previously proposed encoding that uses BDD for Ladder AMO constraint, our encoding requires fewer variables and clauses. Our study is the first study that develops SAT encoding for Ladder AMK constraints with k>1. Experimental results on single Ladder AMO and Ladder AMK constraints indicate that our encoding significantly enhances both the size of the encoding and the efficiency of solving time compared to existing encodings. Furthermore, by evaluating on the two real-world combinatorial problems, namely Anti-bandwidth Problem and Nurse Rostering Problem, we found that the proposed encoding surpasses alternative encodings, as well as more advanced Constraint Programming and Mixed Integer Programming solvers. In summary, the SAT encoding technique presented in this paper provides a more efficient method for encoding sequence constraint, specifically Ladder AMK constraint, which could enhance the effectiveness of SAT solving for various problems involving this type of constraint.

Explore related subjects

cs.AI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hieu Truong Xuan, Tuyen Kieu Van, Khanh To Van. 2026-11-03. SAT Encoding for Set of At-Most-K Cardinality Constraints with Ladder Shape. https://arxiv.org/abs/2611.04637

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

KEEP EXPLORING

Related papers

arXiv2026

Failure-Driven Workflow Refinement

Optimizing LLM-based workflows is typically formulated as a global search, where candidate workflows are evaluated based on a scalar metric. This paradigm, however, suffers from a critical flaw: information collapse. By reducing rich, multi-step execution traces to simple success/failure signals, existing methods are rendered blind to the underlying structure of failures, fundamentally preventing them from modeling the workflow's failure distribution. We reconceptualize this challenge as a distributional problem. We propose a new paradigm where the optimization goal is not to maximize a scalar score, but to directly minimize a workflow's Expected Failure Mass, i.e., the integral of its failure probability density function defined over a high-dimensional Failure Signature Space (FSS). This distributional lens allows us to move from inefficient, zero-order optimization to a principled, gradient-like descent on the failure landscape itself. We introduce CE-Graph, a framework that operationalizes this paradigm through a novel, failure-driven refinement process. CE-Graph approximates the failure distribution from a pool of counterexamples, identifies its densest regions as recurring failure modes, and applies targeted, operator-constrained graph edits via a Propose-and-Verify mechanism to greedily reduce the failure mass. On math, code, and QA benchmarks, our CE-Graph achieves higher robustness at a significantly lower cost than strong baselines. This suggests that a system's reliability emerges not from avoiding failures, but from systematically learning and reshaping the geometric structure of its failure distributions.

cs.AI↗
arXiv2026

Payoff scaling shapes cooperation in LLM agents across languages

Large language models (LLMs) are increasingly deployed as autonomous agents that negotiate, coordinate, and make decisions on behalf of human users. Understanding how they cooperate is therefore important for AI governance and machine behaviour research. We examine two factors: payoff magnitude and interaction language. Classical game theory predicts that multiplying all payoffs by a positive constant should not change behaviour because the underlying strategic incentives remain unchanged. We test this prediction using five widely adopted commercial LLMs playing an iterated prisoner's dilemma across ten payoff scales and five languages (English, French, Vietnamese, Chinese and Arabic). Contrary to theoretical expectations, changing only the numerical scale of payoffs substantially alters cooperation. Models respond differently, often in opposite directions, meaning that averages across models can conceal important behavioural heterogeneity. Sensitivity to payoff scale also depends strongly on prompt language. Moreover, payoff scale modifies the effects of persona instructions: at sub-unit stakes, where no payoff shown exceeds 1, all five models cooperate more under a selfish than a cooperative persona, whereas Claude and Gemini reverse this pattern at larger scales. These findings show that cooperation measures depend critically on payoff scale and language and should not be treated as fixed model traits.

cs.AI↗
arXiv2026

An Alternative Trajectory for Generative AI

The generative artificial intelligence (AI) ecosystem is undergoing rapid transformations that threaten its sustainability. As models transition from research prototypes to high-traffic products, the energetic burden has shifted from one-time training to recurring, unbounded inference. This is exacerbated by reasoning models that inflate compute costs by orders of magnitude per query. The prevailing pursuit of artificial general intelligence through scaling of monolithic models is colliding with hard physical constraints: grid failures, water consumption, and diminishing returns on data scaling. This trajectory yields models with impressive factual recall but struggles in domains requiring in-depth reasoning, possibly due to insufficient abstractions in training data. Current large language models (LLMs) exhibit genuine reasoning depth only in domains like mathematics and coding, where rigorous, pre-existing abstractions provide structural grounding. In other fields, the current approach fails to generalize well. We propose an alternative trajectory based on domain-specific superintelligence (DSS). We argue for first constructing explicit symbolic abstractions (knowledge graphs, ontologies, and formal logic) to underpin synthetic curricula enabling small language models to master domain-specific reasoning without the model collapse problem typical of LLM-based synthetic data methods. Rather than a single generalist giant model, we envision "societies of DSS models": dynamic ecosystems where orchestration agents route tasks to distinct DSS back-ends. This paradigm shift decouples capability from size, enabling intelligence to migrate from energy-intensive data centers to secure, on-device experts. By aligning algorithmic progress with physical constraints, DSS societies move generative AI from an environmental liability to a sustainable force for economic empowerment.

cs.AI↗