lp-based alignment

This work addresses the exponential runtime overhead of the classical A* algorithm when handling long trajectories or those with substantial deviations. The authors reformulate alignment-based consistency checking as a totally unimodular linear programming (LP) problem over the synchronous product reachability graph, leveraging the underlying network flow structure to obtain integer-optimal solutions directly via LP relaxation—thus circumventing combinatorial optimization bottlenecks. This is the first application of totally unimodular LP to this task, revealing a complementary relationship between A* and LP-based approaches and enabling a high-accuracy algorithm selection strategy. Experiments on 2.1 million instances demonstrate that the LP method substantially accelerates computation on challenging trajectories, with a hybrid strategy achieving an average speedup of 38.6% and a selection accuracy of 96%.

Existing soft global constraints exhibit weak communication among variables, while LP-based reconstruction methods suffer from scalability issues, limiting modeling flexibility and lower-bound inference in constraint programming with soft constraints. Method: This paper embeds linear constraints as local cost functions into cost function networks and extends soft arc consistency (SAC) for the first time to enable exact pruning with linear constraints. A novel virtual propagation mechanism is introduced to achieve strong lower-bound inference without explicitly expanding constraints. Contribution/Results: The approach bridges expressive modeling and computational efficiency by preserving the compact structure of linear constraints while enabling tighter bounds. Experimental evaluation on multiple benchmark instances demonstrates significant improvements in lower-bound quality; in several cases, total solving time is reduced by over 30%.

This paper studies the Maximum Weighted co-2-plex finding a maximum-weight vertex subset in a weighted graph such that its induced subgraph has maximum degree at most one. We establish, for the first time, a bijection between feasible co-2-plexes and stable sets in an auxiliary graph, embed the co-2-plex polytope into an extended space, and characterize input-graph contraction-perfectness via total dual integrality. Based on this structural insight, we derive a family of valid inequalities separable in polynomial time, uncovering deep connections between the extended formulation and the integer polytope. The resulting integer linear programming (ILP) model yields tighter linear programming relaxations. Experiments demonstrate that our approach significantly outperforms state-of-the-art methods across sparse to dense graphs, achieving substantial improvements in computational efficiency.

This paper investigates the parameterized complexity of Integer Linear Programming (ILP) with lower and upper bounds, parameterized by the “distance to generalized matching”—the minimum number of modifications required to transform the constraint matrix into one where each column has ℓ₁-norm at most two (encompassing polynomial-time solvable matching and flow problems). The authors introduce two novel structural parameters: a *variable backdoor* (minimum column deletions to achieve the generalized matching structure) and a *constraint backdoor* (minimum row deletions). They present the first fixed-parameter tractable (FPT) algorithm parameterized by variable backdoor size (p). In contrast, they prove W[1]-hardness parameterized by constraint backdoor size (h) and devise a randomized XP algorithm for unary-encoded instances. Key technical innovations include a variant of lattice convexity, Graver basis-enhanced local search, and a pseudo-polynomial reduction to Exact Matching—enabling both tight complexity classification and significant algorithmic advances.

Understanding the interplay between association schemes and Lovász–Schrijver lift-and-project relaxations—particularly SDP-based ones—in combinatorial optimization, especially for stable set polytopes and hypergraph matching. Method: We introduce the notion of *deeply vertex-transitive graphs*, construct *hypermatching pseudo-schemes* (noncommutative association structures), and develop a systematic contraction method to derive commutative sub-schemes. Our approach integrates algebraic combinatorics, spectral graph theory, and SDP analysis. Contribution/Results: We derive Delsarte–Hoffman-type tight bounds on clique and stability numbers; precisely characterize the lift-and-project rank of hypergraph matching relaxations; and establish a general contraction framework from noncommutative pseudo-schemes to commutative sub-schemes. This yields a unified algebraic–convex optimization paradigm for symmetric combinatorial optimization problems.

This work addresses the lack of formal guarantees regarding semantic preservation during problem reformulation and solver correctness in constraint programming. It presents the first end-to-end verified framework implemented in the Lean theorem prover, enabling formal proofs of parameterized equivalence, equisatisfiability, and symmetry-breaking correctness for entire families of problems. The approach combines general, parameterized proofs with instance-level certificate checking, thereby eliminating the need to trust external solvers. Verified certificates are produced via backend transformations, and a single high-level proof suffices for arbitrarily large instances. This methodology achieves dramatic search-space reductions—up to a factor of twenty million—and enables full verification of the largest instances in just a few minutes.

This work addresses the lack of machine-verifiable formalizations of line search methods in nonlinear optimization, which has hindered algorithmic reliability. Within the Lean 4 theorem prover, it presents the first systematic formalization of several classical line search criteria—including Armijo, Goldstein, Wolfe, and their nonmonotone variants—alongside rigorous definitions of gradient descent, descent directions, and backtracking step-size selection. The study fully verifies the Zoutendijk convergence theorem within this framework, thereby establishing a comprehensive formal foundation for line search theory. This contribution significantly enhances the verifiability and trustworthiness of nonlinear optimization algorithms through mechanized mathematical reasoning.

This study addresses the unclear practical efficacy of automatically generated polynomial symmetry-breaking constraints in integer linear programming across different solvers. The authors systematically evaluate the performance of mainstream mathematical programming and SMT solvers when handling such constraints, comparing three strategies: native quadratic handling, internal reformulation, and explicit linearization. Their experiments reveal that the effectiveness of symmetry breaking is highly solver-dependent, advocating for a solver-aware evaluation paradigm. The findings indicate that compact families of quadratic symmetry-breaking constraints generally enhance solver performance, whereas excessive linearization, overly large breaking sets, or inappropriate reformulations often lead to model bloating or search degradation, thereby diminishing or even reversing potential benefits.