LP + Affine IP Framework

• The paper introduces a unified LP + Affine IP framework that integrates linear programming relaxations with affine integer refinement to tackle complex optimization and decoding challenges.
• It employs innovative techniques such as adaptive cut generation, zero-forcing strategies, and block-symmetric polymorphisms to achieve near-optimal performance in coding and robust optimization.
• The methodology unifies algebraic and polyhedral insights to provide scalable, polynomial-time solutions in combinatorial optimization, constraint satisfaction, and matrix inverse eigenvalue problems.

The term "LP + Affine IP algorithm" designates a family of relaxation and separation methods in combinatorial optimization, integer programming, constraint satisfaction, and robust optimization that systematically combine basic linear programming relaxations (LP) with affine-structured integer programming (IP)—often via cut generation, algebraic minion-based hierarchies, or (in continuous domains) iterative lift-projection or affine-scaling schemes. The paradigm applies across binary coding theory (LP decoding with cutting planes), promise constraint satisfaction (with block-symmetric polymorphisms), VCSPs on infinite domains, robust two-stage optimization with explicit affine policies, and certain matrix inverse eigenvalue problems. This article presents the unifying concepts, archetypal algorithms, algebraic solvability criteria, and notable applications, as documented in the relevant literature (0812.2559, Zhang et al., 2011, Brakensiek et al., 2019, Viola et al., 2020, Ciardo et al., 2022, Housni et al., 2021, Barbara, 2017, Riley et al., 10 Apr 2025).

1. Algebraic and Polyhedral Frameworks

The LP + Affine IP approach builds upon the observation that many hard discrete optimization problems admit two canonical relaxations:

• Basic Linear Programming (LP): This relaxation typically encodes constraints as convex polytopes (e.g., via probability simplex or Sherali–Adams liftings), allowing continuous feasible points in polyhedral domains.
• Affine Integer Programming (IP): An associated affine relaxation encodes constraints as integer linear systems, often leveraging linearity without nonnegativity or with refined zero-forcing polyhedral structure. The integer solutions recover the original (discrete) feasible points.

In the algebraic theory of constraint satisfaction and promise variants (PCSPs, VCSPs), this scheme is formalized using the theory of minion homomorphisms and block-symmetric polymorphisms. Specifically, the BLP + Affine algorithm for promise CSPs first solves the LP relaxation to obtain fractional assignments, then refines a companion affine-IP system by zero-forcing all primal variables corresponding to vanishing LP marginals, and finally tests for integer feasibility. This two-stage refinement is exactly characterized by the existence of block-symmetric polymorphisms of arbitrary arity in the polymorphism minion of the template (Brakensiek et al., 2019, Viola et al., 2020, Ciardo et al., 2022).

2. Cutting-Plane and Hierarchical Decoding in Coding Theory

In the context of decoding linear block codes, the LP + Affine IP paradigm manifests as LP decoding with adaptive cut generation. The IP formulation for optimal (ML) decoding introduces bit variables $x_j \in \{0,1\}$ and auxiliary parity variables $z_i \in \mathbb{Z}_{\ge0}$. Relaxing both to real values yields a basic LP (RIPD), but with possible non-integral pseudocodewords, undermining exactness. The remedy:

• Gomory or forbidden-set cuts: Derived whenever $x^* \in \{0,1\}^n$ but some $z_i^* \notin \mathbb{Z}$. These are efficiently generated forbidden-set inequalities and enforce missing facets of the codeword polytope (0812.2559).
• Redundant parity-check (RPC) cuts: Using row operations on $H$ in $\mathrm{GF}(2)$, new checks are constructed that target LP solutions with fractional support. These allow for the systematic cut-off of infeasible pseudocodewords via parity-based inequalities in rows containing a unique non-integral variable.

The separation algorithm alternates between LP solutions and cut generation, mixing original and redundant cuts until either an integral solution is found or no further cuts can be generated. Specific innovations include order-based RPC row construction and efficient updating, guaranteeing that each LP solution is removed from the feasible region at each iteration (0812.2559, Zhang et al., 2011).

Empirical evidence on LDPC and BCH codes indicates frame error rate (FER) gains of 0.4–2 dB over standard LP decoding and practical convergence in a limited number of LP-resolve iterations (0812.2559, Zhang et al., 2011).

3. Hierarchies, Minion Semantics, and Block-Symmetric Polymorphisms

The LP + Affine IP methodology generalizes to hierarchies for promise CSPs and VCSPs through the framework of linear/conic minion-based relaxations (Ciardo et al., 2022). The semi-direct product minion $(Q \ltimes Z)$ encapsulates the joint structure of the LP and affine IP polytopes, refining level-$k$ Sherali–Adams relaxations and enabling extended consistency (BA$^k$) tests.

At each hierarchy level:

• Rational nonnegative variables model the standard LP constraints (stochastic/marginal consistency).
• Integer affine variables enforce exact marginal consistency.
• Zero-forcing (refinement) ensures the affine system’s support overlaps only the LP-feasible region.

The core algebraic criterion for exact solvability by this hierarchy is the presence of block-symmetric polymorphisms of arbitrary width in the polymorphism minion, formalized as $Q \ltimes Z \to \mathrm{Pol}(A,B)$ (Brakensiek et al., 2019, Ciardo et al., 2022, Viola et al., 2020). This is both necessary and sufficient for the combined relaxation to characterize the solution set under the PCSP or PVCSP promise.

4. Robust and Bilinear Optimization: LP + Affine Policy Construction

A variant of the LP + Affine framework applies to two-stage adjustable robust optimization and packing disjoint bilinear programs. Here, the paradigm is to:

• Construct an LP relaxation (e.g., for $$\max \{ x^T y : x \in \mathcal{X}, y \in \mathcal{Y} \}$$ with $\mathcal{X},\mathcal{Y}$ packing polytopes).
• Obtain a near-optimal solution to the LP and use randomized rounding to produce "near-integral" solutions, with explicit bounds $$\Omega( \frac{\log\log m_1}{\log m_1}\frac{\log\log m_2}{\log m_2})$$ relative to the true optimum (Housni et al., 2021).
• For adjustable two-stage covering with right-hand-side uncertainty, an LP restriction is constructed that yields an explicit feasible affine recourse policy matching the LP solution value, providing sharp approximation bounds $$O(\frac{\log n}{\log\log n} \frac{\log L}{\log\log L})$$ and improving iteratively on prior state-of-the-art (Housni et al., 2021).

5. Affine Scaling, Continuity, and Lift-and-Project Algorithms

LP + Affine IP concepts pervade certain continuous and matrix optimization domains:

• Affine scaling methods: Differential-barrier-based affine-scaling interior-point algorithms generalize the classical log-barrier, tuning the shape of the interior region and enforcing affine scaling steps with respect to a strictly convex gauge. Rigorous convergence guarantees for the primal sequence, dual iterates' analytic center, and strict complementarity arise under mild step-size conditions and penalty parameter choices (Barbara, 2017).
• Lift-and-Projection (LP) for Inverse Eigenvalue Problems: The lift-and-project method for least-squares inverse eigenvalue problems alternates between projection onto a spectral constraint manifold ("lift") and minimization over an affine subspace. This is shown to be equivalent to a Riemannian gradient descent with respect to the Frobenius metric induced on the subspace. Each iteration—comprising partial eigendecompositions and Gram solve updates—guarantees a descent inequality and efficient convergence, notably improving practical runtime in large-scale sparse settings (Riley et al., 10 Apr 2025).

6. Algorithmic Structure and Complexity

Across domains, LP + Affine IP algorithms exhibit a canonical structure:

1. Initial LP relaxation: Solve for fractional solutions under relaxed convex constraints (often polytime).
2. Affine (integer) refinement or policy: Solve or search within an affine (integer) system, frequently under a zero-forcing or cut-generating update from the LP marginals.
3. Separation and iteration: If violated constraints or non-integrality persists, generate additional cuts or apply randomized rounding/affine policy extraction.
4. Termination and extraction: Upon integrality or absence of further valid cuts, extract an admissible discrete solution (codeword, integer assignment, explicit affine policy, or decision "NO").

Complexity is dictated by LP and structured IP subproblems: solving the LP is polynomial-time, as is checking feasibility of the refined affine system under block structure (Kannan–Lenstra style arguments). For many instances (fixed domain/arity, bounded code length, bounded hierarchy level), the overall procedure runs in polynomial time (Brakensiek et al., 2019, 0812.2559, Viola et al., 2020, Ciardo et al., 2022). In robust optimization, explicit affine policies and randomization enable strongly polynomial implementations under mild assumptions (Housni et al., 2021).

7. Relations to Hierarchy, Practical Impact, and Limitations

The LP + Affine IP methodology provides a robust, unifying framework across coding, CSP, optimization, and matrix analysis. Its main practical consequences include:

• Achieving near-optimal performance in ML decoding for codes where standard LP fails, narrowing the gap to intractable ML (0812.2559, Zhang et al., 2011).
• Providing a characterizable algebraic dichotomy in PCSP/VCSP tractability (Brakensiek et al., 2019, Viola et al., 2020, Ciardo et al., 2022).
• Enabling randomized and affine policies in robust two-stage covering with provable, optimal polylogarithmic approximation ratios (Housni et al., 2021).
• Admitting highly efficient and robust implementations of affine-scaling and lift–and–project algorithms in large-scale continuous settings (Barbara, 2017, Riley et al., 10 Apr 2025).

However, tight solvability is contingent on the existence of block-symmetric polymorphisms (for CSPs/VCSPs), or the feasibility of the affine refinement. Instances lacking these structures (e.g., directed cycle unions without nontrivial block-symmetric polymorphisms) fail exactness of the LP + Affine IP method; for generic IPs, cut or separation generation may require exponential iterations in the worst case (Brakensiek et al., 2019, 0812.2559). Nevertheless, in most structured settings, the combined relaxation and affine refinement scheme guarantees sharp performance and tractable computation.

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Citations:

• "A Separation Algorithm for Improved LP-Decoding of Linear Block Codes" (0812.2559)
• "Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes" (Zhang et al., 2011)
• "The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs" (Brakensiek et al., 2019)
• "The combined basic LP and affine IP relaxation for promise VCSPs on infinite domains" (Viola et al., 2020)
• "Hierarchies of Minion Tests for PCSPs through Tensors" (Ciardo et al., 2022)
• "LP-based Approximations for Disjoint Bilinear and Two-Stage Adjustable Robust Optimization" (Housni et al., 2021)
• "An affine scaling method using a class of differential barrier functions" (Barbara, 2017)
• "A Riemannian Gradient Descent Method for the Least Squares Inverse Eigenvalue Problem" (Riley et al., 10 Apr 2025)