Projection-width as a structural parameter for discrete separable optimization

While several classes of integer linear optimization problems are known to be solvable in polynomial time, far fewer tractability results exist for integer nonlinear optimization. In this work, we narrow this gap by identifying a broad class of discrete nonlinear optimization problems that admit polynomial-time algorithms. Central to our approach is the notion of projection-width, a structural parameter for systems of separable constraints, defined via branch decompositions of variables and constraints. We show that several fundamental discrete optimization and counting problems can be solved in polynomial time when the projection-width is polynomially bounded, including optimization, counting, top-k, and weighted constraint violation problems. Our results subsume and generalize some of the strongest known tractability results across multiple research areas: integer linear optimization, binary polynomial optimization, and Boolean satisfiability. Although these results originated independently within different communities and for seemingly distinct problem classes, our framework unifies and significantly generalizes them under a single structural perspective.

💡 Research Summary

The paper introduces a novel structural parameter called projection‑width for discrete separable optimization problems and demonstrates that a wide range of integer nonlinear problems become polynomial‑time solvable when this parameter is bounded by a polynomial.
A separable system is defined as a quadruple ⟨X, D, C⁽=⁾, C⁽≥⁾⟩ where X is a finite set of variables, D a finite domain, and C⁽=⁾ and C⁽≥⁾ are collections of equality and inequality constraints, respectively. Each constraint is a sum of unary functions f₍cₓ₎: D → ℤ, one per variable. By subtracting the minimum value of each f₍cₓ₎, the authors obtain non‑negative functions g₍cₓ₎ with minimum zero, and rewrite constraints in terms of sets Γ_c (for equalities) and thresholds γ_c (for inequalities).

The core notion of a projection is defined for a partial assignment τ: X′ → D. For each constraint c, the constraint bound cb_c(τ) = Σ_{x∈X′} g₍cₓ₎(τ(x)) is computed, and the projection of c under τ is min(cb_c(τ), γ_c). Collecting these values for a set of constraints C′ yields a map C′/τ.

To capture the interaction between variables and constraints, the authors use a branch decomposition T of the bipartite set X ∪ C (a rooted binary tree whose leaves correspond one‑to‑one to variables or constraints). For any node v, let X_v and C_v be the variables and constraints appearing in the leaves of the subtree rooted at v, and let X̄_v, C̄_v denote the complements. Two families of projections are defined:

• proj(v) = { C_v/τ | τ: X_v → D }
• proj⁻(v) = { C̄_v/τ | τ: X̄_v → D }.

The projection‑width of a decomposition is the maximum, over all nodes v, of |proj(v)| and |proj⁻(v)|. The projection‑width of the separable system is the minimum such value over all possible branch decompositions.

A series of lemmas (2‑5) establishes precise composition rules for these projection sets across parent and child nodes. In particular, for a parent node v with children v₁ and v₂, any Φ ∈ proj(v) can be uniquely expressed via Φ¹ ∈ proj(v₁) and Φ² ∈ proj(v₂) by the rule Φ_c = min{Φ¹_c + Φ²_c, γ_c} for each constraint c in C_v. Symmetric relations (L2) and (L3) connect proj(v) with proj⁻(v₁) and proj⁻(v₂). These identities enable a bottom‑up dynamic programming algorithm that constructs all projection sets in time polynomial in the size of the sets, rather than exponential in |X| or |C|.

Using this machinery, the authors design algorithms for four fundamental problems defined on a separable system:

1. Optimization – find a feasible assignment maximizing a separable objective.
2. Counting – compute the number of feasible assignments.
3. Top‑k – enumerate the k best feasible assignments.
4. Weighted constraint violation – minimize a weighted sum of violated inequality constraints.

For each problem they obtain a running time of the form

 O( w_proj³·(|X|+|C|)·|C| + w_proj·|C⁽=⁾|·Λ + |X||C||D|·log|D| ),

where w_proj is the projection‑width and Λ is the time needed to test membership of an integer in a set Δ_c (the data structure for the equality‑constraint value sets). The Top‑k algorithm incurs an additional O(k·log k) term, while the weighted‑violation algorithm simplifies because only inequality constraints are relevant. All bounds are expressed under the standard arithmetic model, assuming unit‑cost for basic operations.

The paper then shows that projection‑width subsumes several well‑known structural parameters:

• PS‑width for CNF formulas (used in weighted MaxSAT and #SAT algorithms).
• Incidence treewidth for integer linear programs (bounded incidence treewidth yields polynomial‑time solvability).
• Hypergraph incidence treewidth for binary polynomial optimization.

Consequently, the authors recover as corollaries the polynomial‑time algorithms for ILP instances with bounded incidence treewidth (Gorodkin‑Ostrowski‑Rothvoß 2017), binary polynomial optimization on hypergraphs of bounded incidence treewidth (Cunningham‑de Groot‑Peters‑Davis‑Gottlob 2024), and weighted MaxSAT/#SAT on CNF formulas of bounded PS‑width (Sæther‑Telle‑Vatshelle 2014). In each case, the earlier results correspond to the special situation where all functions are linear, domains are binary, and only inequality constraints appear.

Finally, the authors discuss broader implications: projection‑width provides a unified lens for understanding tractability across disparate domains, suggests new avenues for constructing compact extended formulations (via the approach of Martin‑Rothvoß‑Cunningham 1990), and enables translation of separable constraints into deterministic DNNF representations (as in Capelli 2016). They acknowledge that estimating projection‑width efficiently remains an open problem and that practical performance will depend on the structure of the specific application.

In summary, the paper delivers a powerful, general framework that extends the frontier of polynomial‑time solvable integer nonlinear problems by introducing projection‑width, rigorously analyzing its combinatorial properties, and demonstrating its ability to unify and generalize several landmark tractability results.