Automatic Generation of Polynomial Symmetry Breaking Constraints

Symmetry in integer programming causes redundant search and is often handled with symmetry breaking constraints that remove as many equivalent solutions as possible. We propose an algebraic method which allows to generate a random family of polynomial inequalities which can be used as symmetry breakers. The method requires as input an arbitrary base polynomial and a group of permutations which is specific to the integer program. The computations can be easily carried out in any major symbolic computation software. In order to test our approach, we describe a case study on near half-capacity 0-1 bin packing instances which exhibit substantial symmetries. We statically generate random quadratic breakers and add them to a baseline integer programming problem which we then solve with Gurobi. It turns out that simple symmetry breakers, especially combining few variables and permutations, most consistently reduce work time.

💡 Research Summary

The paper addresses the pervasive problem of symmetry in integer programming (IP), which causes redundant exploration of equivalent solutions during branch‑and‑bound search. While most existing symmetry‑breaking techniques rely on linear constraints—such as lex‑leader ordering, orbitopes, or constraints derived from stabilizer chains—the authors propose a novel, algebraic approach that automatically generates families of polynomial (potentially non‑linear) inequality constraints to break symmetry.

The core idea is simple yet powerful: given any polynomial h(x) in the decision variables and a permutation P belonging to the problem’s symmetry group G, the inequality h(Px) − h(x) ≤ 0 is a valid symmetry breaker. The authors prove (Theorem 2.3) that for any choice of h and any P∈G, there exists at least one optimal solution that satisfies this inequality, guaranteeing that adding the constraint does not eliminate all optimal solutions. The proof exploits the finiteness of the orbit Gx* of an optimal solution x* and selects a point in the orbit where h attains its maximum; at that point the inequality holds for every group element.

A further theoretical contribution (Theorem 2.8) shows that if for every non‑identity permutation P the strict inequality h(Px) − h(x) < 0 defines a non‑empty open set L_P, then the intersection of all such sets forms a fundamental region for G. Thus, the polynomial‑based constraints partition the feasible space just as linear symmetry breakers do, but they can describe far richer geometric shapes.

To evaluate the method, the authors focus on a challenging class of 0‑1 bin‑packing instances where item sizes cluster around half the bin capacity (near half‑capacity regime). In this setting, many items are indistinguishable and bin labels can be permuted arbitrarily, leading to a huge symmetry group composed of item‑exchange and bin‑label permutations. Synthetic benchmark families are generated with up to four million binary variables and roughly four thousand constraints, yielding baseline solve times of about five minutes on an Apple M2 Pro.

For each benchmark, a “base polynomial” h is drawn from a small template pool (e.g., x·y, x² + y², x³ − 3x). A random subset of permutations from G is selected (typically 2–3 permutations), and the corresponding inequalities h(Px) − h(x) ≤ 0 are added simultaneously to the model. The experiments use Gurobi 12.0.3 with a limit of 1800 work‑units (a deterministic measure of computational effort). Results show:

1. Quadratic breakers consistently outperform linear ones, reducing work‑units by 10–30 % on average.
2. The most reliable gains arise from small‑scale quadratic breaker sets (few variables, few permutations). These constraints add modest LP‑relaxation overhead while still eliminating large portions of symmetric sub‑trees.
3. The automatically generated polynomial breakers outperform Gurobi’s built‑in symmetry‑breaking mechanisms.
4. Larger sets of breakers (≥5) introduce overhead that can outweigh the benefits, leading to more variable performance.

The authors conclude that non‑linear symmetry breakers are not only theoretically sound but also practically effective for IPs with known symmetry groups. They highlight several avenues for future work: handling cases where the symmetry group is only partially known, exploring higher‑degree polynomials, and applying the technique to other combinatorial optimization problems such as graph coloring, scheduling, and network design where symmetry is a major obstacle.

Overall, the paper makes a solid contribution by extending symmetry‑breaking from the linear domain into the polynomial realm, providing both rigorous proofs and convincing empirical evidence that simple quadratic constraints can substantially accelerate the solution of symmetric integer programs.