[Paper Review] Applying Predicate Detection to the Constrained Optimization Problems

Vijay K. Garg|arXiv (Cornell University)|Dec 26, 2018

Game Theory and Voting SystemsEconomics, Econometrics and Finance15 references3 citations

TL;DR

This paper introduces a unified lattice-linear predicate detection framework that solves constrained combinatorial optimization problems by modeling feasible solution spaces as distributive lattices. By casting problems like stable marriage, shortest path, and market clearing prices as predicate detection over lattices, the method derives classical algorithms (Gale-Shapley, Dijkstra, Demange-Gale-Sotomayor) as special cases and generalizes them to handle lattice-linear constraints, enabling efficient parallel and constructive enumeration of constrained optimal solutions.

ABSTRACT

We present a method to design parallel algorithms for constrained combinatorial optimization problems. Our method solves and generalizes many classical combinatorial optimization problems including the stable marriage problem, the shortest path problem and the market clearing price problem. These three problems are solved in the literature using Gale-Shapley algorithm, Dijkstra's algorithm, and Demange, Gale, Sotomayor algorithm. Our method solves all these problems by casting them as searching for an element that satisfies an appropriate predicate in a distributive lattice. Moreover, it solves generalizations of all these problems - namely finding the optimal solution satisfying additional constraints called {\em lattice-linear} predicates. For stable marriage problems, an example of such a constraint is that Peter's regret is less than that of Paul. For shortest path problems, an example of such a constraint is that cost of reaching vertex $v_1$ is at least the cost of reaching vertex $v_2$. For the market clearing price problem, an example of such a constraint is that $item_1$ is priced at least as much as $item_2$. In addition to finding the optimal solution, our method is useful in enumerating all constrained stable matchings, and all constrained market clearing price vectors.

Motivation & Objective

• To unify the solution of fundamental combinatorial optimization problems—stable marriage, shortest path, and market clearing prices—under a single algorithmic framework.
• To generalize classical algorithms by incorporating additional constraints modeled as lattice-linear predicates.
• To enable efficient parallel and distributed computation of optimal solutions in constrained settings.
• To provide a constructive method for enumerating all constrained stable matchings and market clearing price vectors.
• To establish theoretical foundations for sublattice structure and join-irreducible element representation in constrained solution spaces.

Proposed method

• Model the solution space of each problem as a distributive lattice of n-dimensional vectors, where each vector represents an assignment (e.g., man-woman pairings, node costs, item prices).
• Define a boolean feasibility predicate B that checks whether a given vector satisfies problem-specific constraints (e.g., no blocking pairs, valid tree structure, market clearing).
• Prove that the feasibility predicate B is lattice-linear, enabling efficient search via monotonic advancement from the bottom of the lattice.
• Use the lattice-linear predicate detection algorithm to find the infimum (optimal) feasible solution, which corresponds to the man-optimal stable matching, shortest path, or minimum market clearing price.
• Extend the method to enumerate all constrained solutions by computing the slice of join-irreducible elements using a dual lattice-linear predicate and constructive Birkhoff’s theorem.
• Construct the slice of the feasible sublattice by computing J(B,e) for each edge e, using a lattice-linear predicate detection procedure for B_e(G) ≡ B(G) ∧ (e ∈ G).

Experimental results

Research questions

• RQ1 Can a single algorithm framework solve and generalize classical combinatorial optimization problems like stable marriage, shortest path, and market clearing prices?
• RQ2 How can lattice-linear predicates be used to model additional constraints in these problems while preserving algorithmic efficiency?
• RQ3 Can the lattice structure of feasible solutions be exploited to derive known algorithms (e.g., Gale-Shapley, Dijkstra) as special cases?
• RQ4 Is it possible to constructively enumerate all constrained stable matchings and market clearing price vectors using lattice-theoretic methods?
• RQ5 What is the role of dual lattice-linear predicates and reverse-forbidden states in enabling backward search and sublattice characterization?

Key findings

• The lattice-linear predicate detection algorithm generalizes and derives the Gale-Shapley, Dijkstra’s, and Demange-Gale-Sotomayor algorithms as special cases when constraints are absent.
• The method enables efficient parallel computation of optimal solutions by leveraging the lattice structure and monotonic advancement from the bottom element.
• The set of all constrained stable matchings forms a finite distributive lattice, which can be represented via the poset of its join-irreducible elements (the 'slice').
• The slice can be computed constructively by solving a sequence of lattice-linear predicate detection problems for each edge in the graph.
• The feasibility predicate for constrained stable matching is both lattice-linear and post-lattice-linear, ensuring the feasible set is closed under joins and forms a sublattice.
• The method provides a concise, polynomial-time representation of all constrained market clearing price vectors when prices are integral, via the slice construction.