packing arguments

This work addresses long-standing open problems in extremal set theory, such as Chvátal’s conjecture, by introducing a novel paradigm for customized search space partitioning based on solution construction strategies, replacing conventional domain-agnostic lookahead-based methods. By integrating this approach with a proof-generating exact mixed-integer linear programming (MILP) solver, the proposed framework substantially enhances search efficiency. Empirical evaluation demonstrates successful verification of the largest finite instance of Chvátal’s conjecture to date, marking significant progress toward resolving this fundamental problem in combinatorics.

This paper studies covering-type mixed-integer linear programming (CMILP) problems with a fixed number of constraints, encompassing classical models such as multidimensional knapsack covering, facility location, and supplier selection. Methodologically, we leverage polyhedral vertex structure analysis to decompose the problem into a family of multidimensional knapsack covering subproblems—each involving only one continuous variable—and integrate linear programming relaxation with tailored approximation schemes. We further derive a compact, theoretically optimal linear formulation. Our main contributions are the first polynomial-time approximation scheme (PTAS) and fully polynomial-time approximation scheme (FPTAS) for CMILP under a fixed constraint count, breaking the long-standing 2-approximation barrier for the single-constraint case. Notably, our FPTAS for the single-constraint setting achieves both scalability and provable accuracy guarantees, significantly extending the tractable problem size and solution quality.

This paper studies linear combinatorial optimization in the comparison oracle model, where the optimal solution is identified solely via pairwise comparisons ( <, =, > ) of weights of feasible subsets. To address the overly strong assumptions of traditional value oracles, we propose the first global subspace learning framework, integrating inferential dimension analysis, algebraic structural modeling, and discrete integer sorting techniques. Theoretically, we achieve the first separation between information complexity and computational complexity. Algorithmically, our approach attains an $O(nB log(nB))$ query complexity for fundamental combinatorial structures—including minimum cut, shortest path, and bipartite matching—where $n$ denotes the ground set size and $B$ the bit complexity of weights. This establishes a polynomial-time, low-query paradigm for weakly supervised combinatorial optimization, significantly advancing beyond prior value-oracle–dependent methods.

This paper addresses the efficient computation of the $k$ most diverse solutions for combinatorial optimization problems over distributive lattices, where diversity is measured by the sum of pairwise Hamming distances. Method: We propose the first general polynomial-time framework, unified by three structural conditions that characterize problem classes admitting efficient $k$-diverse solution computation. Our approach leverages distributive lattice theory, $s$-$t$ cut modeling, and stable matching techniques; it extends to multiple diversity measures and yields a simplified algorithm for maximum mutually exclusive solution sets. Contribution/Results: We achieve the first polynomial-time algorithms for $k$-diverse solutions on classical problems—including minimum $s$-$t$ cut and stable matching—significantly enhancing both diversity and practical utility of solution sets. The framework provides a unifying theoretical foundation for diverse solution enumeration over distributive lattices, with broad applicability across discrete optimization domains.

This paper studies geometric hitting set and covering problems for families of convex polyhedra parameterized continuously, motivated by modeling two-stage finite-adaptive decisions in robust optimization—particularly nonlinear settings under left-hand-side uncertainty. Methodologically, it establishes a novel paradigm linking continuous parametric hitting set problems to finite-adaptive robust optimization; develops algorithms integrating computational geometry, convex analysis, parametric polyhedral theory, and robust optimization modeling. Contributions include: (i) the first strongly polynomial-time algorithm for the problem when both the polyhedral dimension and the parameter space dimension are constants; (ii) a strongly quadratic-time algorithm for single-parameter families in constant dimension. These results break computational bottlenecks in nonlinear robust optimization and provide the first strongly polynomially solvable framework—with efficient algorithms—for finite-adaptive robust optimization.

This work unifies graph classes defined by forbidden induced subgraphs or induced minors with those characterized by specific tree-decomposition structures by introducing a novel parameter, induced-$\mathcal{H}$-packing treewidth. This parameter measures, for each bag of a tree decomposition, the maximum number of pairwise non-adjacent induced copies of graphs from a family $\mathcal{H}$. It generalizes existing notions such as tree independence number and induced matching treewidth. For various graph families $\mathcal{H}$—including $\{P_3\}$ and all cycles—the paper demonstrates that the Maximum Weight Independent Set (MWIS) problem is solvable in polynomial time on graphs of bounded induced-$\mathcal{H}$-packing treewidth, thereby partially resolving and significantly extending an open question posed by Bodlaender et al. regarding the tractability of MWIS.

This work addresses the limitation in hardness proofs for path-packing problems that rely on randomized weight assignments by introducing the first deterministic variant of the isolation lemma. Combining combinatorial constructions with algebraic techniques, the authors explicitly design deterministic weights and employ formal verification to guarantee their correctness. This approach successfully eliminates probabilistic assumptions from several known hardness results, replacing randomized assignments with fully deterministic ones. Consequently, it achieves complete derandomization of the corresponding complexity lower-bound proofs and significantly broadens the applicability of the isolation lemma within theoretical computer science.

This study systematically evaluates the rigorous proof reasoning and explicit construction capabilities of large language models on Olympiad-level combinatorics problems. To this end, we introduce a benchmark comprising 100 expert-annotated competition problems, categorizing tasks into analytical (proof-oriented) and constructive (implementation-oriented) types. We propose a unified evaluation protocol that integrates rubric-guided proof assessment with deterministic verification of constructions, enhanced by a Best@4 multi-solution sampling strategy. Experimental results show that the strongest model achieves an average score of 65.4% overall (75.3% under Best@4), with markedly divergent performance across the two task types, revealing current limitations in creative mathematical reasoning—particularly on existence and construction problems. This work presents the first fine-grained distinction and joint evaluation of these capabilities, offering a new benchmark and diagnostic framework for mathematical reasoning research.

This work addresses the challenge of “quietly” planting multiple solutions with arbitrary Hamming distances into $k$-SAT instances while preserving computational hardness. The authors introduce a novel planting mechanism that establishes controlled correlations between variable selection and negation patterns, leveraging binary linear $[k, t, r]$ codes to construct distributions satisfying a generalized $(r-1)$-wise uniformity. This approach is the first to enable the planting of an arbitrary number of solutions with prescribed geometric relationships, yielding instances that are statistically indistinguishable from uniformly random ones in the statistical query (SQ) model. Building on this construction, the paper establishes a statistical query lower bound for distinguishing between satisfiable instances with multiple planted solutions and unsatisfiable ones, thereby resolving an open problem posed by Hsieh et al.

This work addresses challenging instances of the NP-hard knapsack problem (KP) and its bounded variant (BKP) by substantially enhancing the COMBO solver. The proposed approach integrates multiplicity reduction, runtime item aggregation, refined dominance fixing rules, and a novel divisibility-based bound, combined with core dynamic programming, weak upper-bound estimation, and surrogate relaxation under cardinality constraints. These innovations significantly strengthen state-space pruning and symmetry-breaking capabilities. Evaluated on multiple standard benchmark sets of hard instances, the method consistently outperforms the current state-of-the-art solvers COMBO and BOUKNAP, often achieving speedups of several orders of magnitude and establishing a new performance benchmark for KP and BKP solving.