Factor and Policy Revealing LPs

• Factor- and policy-revealing LPs are specialized linear program frameworks that characterize worst-case performance and construct explicit algorithmic policies for complex optimization problems.
• They employ techniques like LP relaxations, randomized rounding, and duality to derive tight approximation and competitive ratio bounds for packing, covering, and online selection challenges.
• Recent advancements extend these LPs using variational-calculus and continuous limits, enabling robust optimization and the design of feasible affine recourse strategies.

Factor-revealing and policy-revealing linear programs (LPs) are central frameworks in the theoretical analysis and algorithm design for combinatorial optimization, approximation algorithms, online algorithms, and robust optimization under uncertainty. These LP formulations are engineered to characterize, bound, and sometimes construct worst-case algorithmic performance guarantees, including approximation ratios or competitive ratios in online and robust settings. Their evolution includes powerful LP-based relaxations, randomized rounding, analytical limit techniques, and variational-calculus-based generalizations, making them indispensable in modern operations research and theoretical computer science.

1. Fundamental Concepts and Definitions

Factor-revealing and policy-revealing LPs provide two closely related methodologies:

• Factor-revealing LPs: LP relaxations explicitly constructed so their optimal value either matches or bounds the worst-case performance of some algorithmic primitive. Typically, these LPs “reveal” the achievable approximation or competitive ratio by exposing a tight worst-case input, a structural randomized rounding scheme, or both. They are particularly prevalent in the analyses of packing, covering, and matching problems.
• Policy-revealing LPs: LPs whose form or dual structure not only expose the worst-case factor, but also allow explicit construction of algorithmic policies (e.g., affine or threshold policies) whose performance provably matches the LP value. Such constructions are crucial in two-stage adjustable, online, or secretary-type settings, where the optimal policy must adapt to adversarial or revealed information.

Both types of LPs often form a family parameterized by problem size, whose analysis aims to resolve their asymptotic behavior as the instance size grows to infinity (Housni et al., 2021, Xu, 19 Mar 2025).

2. LP-Based Approximation for Disjoint Bilinear and Adjustable Robust Problems

The disjoint bilinear packing problem (PDB) exemplifies the use of factor-revealing LPs. The canonical PDB problem is

$z_{\rm PDB}=\max_{x\in\X,\,y\in\Y}\;\{\,x^T y\}$

where $\X=\{x\in\R^n_+\,|\,P x \le p\}$ and $\Y=\{y\in\R^n_+\,|\,Q y \le q\}$ are packing polytopes. The factor-revealing LP relaxation

$({\rm LP\mbox{-}PDB})\quad \begin{aligned} &z_{\rm LP}=\max_{\omega\ge0}\sum_{i=1}^n \theta_i\gamma_i \omega_i\ &\text{s.t.}\quad \sum_{i=1}^n\theta_i P_i \omega_i \le p,\ \sum_{i=1}^n \gamma_i Q_i \omega_i \le q \end{aligned}$

with $\theta_i=\max_{x\in\X} x_i$ and $\gamma_i=\max_{y\in\Y} y_i$, exposes the structure for randomized rounding guarantees.

The two-stage adjustable robust covering problem (AR) has objective

$\min_{x\in\X}\bigl\{c^T x+\Q(x)\bigr\}, \quad \Q(x) =\max_{h\in\U}\min_{y\ge0}\{d^Ty\mid Ax+By\ge h\}$

with uncertainty set $\U=\{h\ge0\mid R h\le r\}$. Here, the factor-revealing LP and its restriction yield both approximation bounds and explicit affine policies.

These LPs encode both the combinatorial structure of packing/covering and allow construction of randomized or affine recourse strategies with provable guarantees (Housni et al., 2021).

3. Structural Lemmas, Rounding, and Approximation Guarantees

The core technical elements in factor- and policy-revealing LPs are structural lemmas—usually involving randomized rounding and scaling—guaranteeing that the LP optimum can be matched to within a multiplicative factor by a feasible solution or algorithm.

For PDB, the existence result states that a randomized polynomial-time algorithm returns $(\tilde x, \tilde y)$ with $\tilde x_i\in\{0, \theta_i/\zeta_1\}$, $\tilde y_i\in\{0, \gamma_i/\zeta_2\}$ for $1\le \zeta_1\le \zeta_1^*$, $1\le \zeta_2\le \zeta_2^*$,

$$\tilde x^T \tilde y \ge \Omega\left(\frac{\ln\ln m_1}{\ln m_1}\frac{\ln\ln m_2}{\ln m_2}\right) z_{\rm PDB}$$

where $m_1$, $m_2$ are the number of constraints (Housni et al., 2021).

In AR, the LP restriction achieves a $O(\frac{\ln n}{\ln\ln n}\frac{\ln L}{\ln\ln L})$-approximation, significantly improving prior bounds on adjustable robustness with right-hand side uncertainty, with $n$ variables and $L$ uncertain constraints.

Constructing feasible solutions proceeds via randomized Bernoulli sampling of LP variables, Chernoff bounds for high-probability constraint satisfaction, and appropriate scaling.

4. Connection to Reformulation-Linearization Technique and Duality

Factor-revealing LPs, particularly those for bilinear and robust problems, often coincide at their first level with classical Reformulation Linearization Technique (RLT) relaxations:

• PDB can be written in epigraph form, with new variables $u_{ij} \le x_i y_j$.
• Multiplying packing constraints by $x_j$, $1-x_j$, etc., and linearizing all resulting monomials produces the RLT LP.
• Higher-level RLT relaxations inherit the same polylogarithmic approximation guarantees, providing a systematic route for tightening relaxations for polynomially constrained bilinear objectives.

Dualizing the LP restriction for AR yields a compact policy-revealing LP, which not only bounds the optimal cost but also constructs a feasible affine recourse policy $(x^*, y^*, \alpha^*)$ that attains the bound, demonstrating the policy-revealing character (Housni et al., 2021).

5. Variational-Calculus Perspective and Continuous Limits

Recent work introduces a variational-calculus framework for analyzing families of factor- and policy-revealing LPs as their instance size $n$ grows (Xu, 19 Mar 2025). In the continuous limit, the LP is recast as an optimization over functions: $$\min_{g:[0,1]\to[0,1]} J[g] = \int_0^1 L(t, g(t), g'(t))\,dt,\quad \text{subject to}\; H_j[g]\le C_j$$ Extremal solutions satisfy Euler–Lagrange equations with Lagrange multipliers enforcing constraints.

This approach:

• Unifies and generalizes the analysis of large LP families.
• Recovers classical limits, such as the $1-1/e$ competitive ratio in secretary or matching-type problems.
• Enables derivation of closed-form ODEs governing the limiting function $g^*$.

Case studies demonstrate variational reformulations for online $b$-matching, online adversarial matching, and the secretary problem, yielding structural uniqueness and faintly stronger insights than discrete LP manipulations (Xu, 19 Mar 2025).

6. Applications, Strengths, and Limitations

Factor- and policy-revealing LPs apply broadly:

• Bilinear packing and matching
• Two-stage and adjustable robust covering or resource allocation
• Online algorithms analyzed via competitive ratios
• Online secretary and selection problems

Their strengths include generalizable approximation frameworks, explicit structural lemmas tied tightly to LP relaxations, and, increasingly, principled analytic tools (variational calculus, ODEs, duality).

However, limitations arise:

• Each complex or high-dimensional LP family requires careful verification that the discrete-to-continuum convergence holds.
• LP relaxations must often be problem-specific and may need strengthening (e.g., RLT, strongly factor-revealing constraints).
• Multi-dimensional instances with stochastic or distributional uncertainty may require advanced analysis beyond classical calculus of variations (Xu, 19 Mar 2025).

7. Summary of Key Theorems

Framework	LP Formulation	Approximation Guarantee
PDB	$(\rm LP-PDB)$	$$O\left(\frac{\ln\ln m_1}{\ln m_1}\frac{\ln\ln m_2}{\ln m_2}\right)$$
AR	$(\rm LP-AR)$	$$O\left(\frac{\ln n}{\ln\ln n}\frac{\ln L}{\ln\ln L}\right)$$
Affine Policy	$(\rm LP-AR)$	Same as above, guarantees feasible affine recourse policy

Both frameworks leverage a structural-lemma + randomized-rounding (factor-revealing), and dualization + explicit recourse (policy-revealing) methodology. This consolidation underpins state-of-the-art approximation and robust optimization algorithms (Housni et al., 2021, Xu, 19 Mar 2025).