arXiv:2605.24845v1 Announce Type: new Abstract: Combinatorial counting problems pervade artificial intelligence, statistics, and discrete mathematics. Whether the task is enumerating subsets, multisets, permutations, partitions, or compositions under structural and arithmetic constraints, solving it remains a stubbornly manual exercise. Closed-form derivations are powerful but brittle, while naive encodings to propositional model counting or constraint satisfaction destroy the exchangeability that makes counting tractable in the first place. We present Cofola (COmbinatorial counting LAnguage with First-Order logic), a typed declarative language whose primitives are the combinatorial objects that recur in everyday counting questions, including sets, bags, tuples, sequences, circles, partitions, and compositions, together with natural relational and arithmetic constraints over them. A denotational semantics maps every Cofola program to a well-defined combinatorial counting problem, and a three-phase compilation pipeline (preprocessing, decomposition, and symmetry-preserving encoding) reduces this problem to a weighted first-order model counting (WFOMC) instance augmented with coefficient-extraction constraints. To stay inside known domain-liftable fragments whenever possible, the encoding groups indistinguishable entities, breaks the symmetry of unordered groupings lexicographically, and encodes sequences and circles via order axioms. On a suite of representative combinatorial counting problems, ranging from textbook math problems to multi-object scenarios that the closest prior framework cannot express, Cofola produces concise specifications and a uniform solving pipeline that is practical end-to-end.

arXiv:2605.23966v1 Announce Type: cross Abstract: Optimization modeling serves as the pivotal bridge between natural-language problem descriptions and optimization solvers, and remains a cornerstone for bringing operations research (OR) into real-world decision making. Recent advances in large language models (LLMs) have driven significant progress in automatic optimization modeling. However, existing methods still lack explicit validation during the modeling process, allowing errors introduced in earlier stages to carry through the pipeline and ultimately reduce final modeling accuracy. To address this challenge, we introduce TriVAL, a tri-validation framework that performs explicit validation at three stages of automatic optimization modeling: semantic specification, mathematical formulation, and code generation. At each stage, TriVAL follows a construct-validate-revise loop that assesses the current result against stage-specific criteria and revises it when needed. This design helps identify and correct errors before they accumulate across stages, helping preserve faithfulness throughout the modeling process. To evaluate automatic optimization modeling on more challenging combinatorial problems, we further introduce NL4COP, a benchmark of 150 instances across 50 diverse problem types with more complex decision logic, more tightly coupled constraints, and more demanding modeling requirements than existing benchmarks. Experiments on NL4COP and established benchmarks show that TriVAL consistently outperforms state-ofthe-art methods, with the largest gains on the most challenging problems.

arXiv:2605.24779v1 Announce Type: cross Abstract: Submodular optimization has become a fundamental paradigm for data selection, retrieval, summarization, and representation learning due to its ability to model coverage, diversity, and representativeness. However, classical submodular objectives optimize only the selected subset and do not explicitly preserve structural information between the selected subset and the remaining data. In many modern machine learning applications, including train/validation/test splitting, benchmark construction, and robust subset selection, the quality of a selection depends critically on preserving balanced structure across both the selected subset and its complement. In this work, we introduce Complement Submodular Information (CSI), a new class of complement-aware submodular objectives that quantify shared structural information between a subset and its complement. Our framework induces complement-aware variants of several classical submodular functions including Facility Location, Graph Cut, LogDet, Saturated Coverage, Set Cover, Probabilistic Set Cover, and Feature Based Functions. We analyze the theoretical properties of CSI objectives and show that they exhibit approximate monotonicity under bounded curvature conditions, leading to near-$(1-1/e)$ greedy approximation guarantees. Empirically, CSI objectives consistently outperform standard submodular objectives on robust hidden-slice-aware subset selection. In particular, CSI objectives significantly improve preservation of coherent rare/tail semantic structure while simultaneously suppressing noisy and isolated outliers, leading to substantially improved downstream predictive performance. Synthetic experiments further illustrate how different CSI instantiations capture complementary notions of representativeness, diversity, connectivity, and balanced neighborhood preservation.