Radius of Robust Feasibility (RRF)

• RRF is defined as the supremum perturbation radius for which all admissible data changes preserve the feasibility of a constraint system.
• It connects geometric properties, such as the largest inscribed Euclidean ball in random polytopes, with precise thresholds (R₀ and R₁) that indicate phase transitions in integer feasibility.
• Constructive algorithms like Round-IP leverage discrepancy minimization to efficiently certify robust feasibility, linking theoretical guarantees with practical rounding methods.

The Radius of Robust Feasibility (RRF) is a quantitative measure in optimization and feasibility analysis that characterizes the largest admissible perturbation to problem data under which feasibility—often integer feasibility or robust feasibility with respect to given constraints—remains guaranteed. Across a variety of models, ranging from polytopes with random or uncertain constraints to nonlinear and combinatorial systems, the RRF serves as a critical metric for understanding and certifying the resilience of feasible regions to data uncertainty.

1. Formal Definition and Theoretical Motivation

The RRF is defined as the supremum of the uncertainty radius $\alpha \geq 0$ such that the robust counterpart of a system of constraints remains feasible—meaning the system admits at least one solution under all admissible data perturbations of magnitude $\alpha$. Let $P(A, b) := \{x : Ax \leq b\}$ denote a polytope defined by a constraint matrix $A$ and right-hand side $b$, subject to uncertainty sets $U_\alpha$ parameterized by the radius $\alpha$. The RRF is then

$$\rho = \sup \left\{ \alpha \geq 0 : P(A, b) \text{ is feasible for all } (A, b) \in U_\alpha \right\}.$$

In the integer programming context, a related geometric formulation emerges: the radius of the largest Euclidean ball inscribed in $P(A, b)$. If this radius exceeds a certain threshold, integer feasibility is ensured with high probability for random polytopes (Chandrasekaran et al., 2011).

The RRF’s importance is multifaceted:

• It establishes a clear, quantifiable safety margin against uncertainties in constraint data.
• It connects geometric and combinatorial criteria (such as the presence of an inscribed ball or bounds on linear discrepancy) to probabilistic or worst-case guarantees of feasibility.
• In random and high-dimensional regimes, the RRF reveals phase-transition phenomena, tightly characterizing feasibility thresholds.

2. Mathematical Criteria and Central Results

In the setting of random polytopes $P(n, m, x_0, R)$, the RRF is concretely instantiated as the radius $R$ of the largest Euclidean ball centered at $x_0$ and contained in the polytope. The paper (Chandrasekaran et al., 2011) establishes two explicit thresholds:

• Lower threshold $R_0$ (integer infeasibility):

$$R_0 = \sqrt{\frac{1}{6} \log\left(\frac{m}{n}\right)}$$

If the inscribed ball centered at $x_0 = (\frac{1}{2}, \dots, \frac{1}{2})$ has radius at most $R_0$, then, with high probability, the polytope contains no integer points.

• Upper threshold $R_1$ (integer feasibility):

$$R_1 = 960 \left( \sqrt{\log\left(\frac{m}{n}\right)} + \sqrt{\frac{\log m \cdot \log(mn) \cdot \log(m/\log m)}{n}} \right)$$

If the inscribed ball's radius is at least $R_1$, the polytope is integer feasible with high probability for $m = 2^{O(\sqrt{n})}$.

The transition from infeasibility to feasibility occurs within a constant factor interval for $R$. The underlying mechanism relates to the linear discrepancy of the constraint matrix $A$. For $A$ with unit-norm rows, the linear discrepancy is defined as

$$\text{lin-disc}(A) = \max_{x_0 \in [0,1]^n} \min_{x \in \{0,1\}^n} \|A(x - x_0)\|_\infty.$$

If the right-hand side $b$ satisfies $b_i \geq \text{lin-disc}(A)$ for all $i$, integer feasibility is certified. Thus, the RRF is tightly coupled to the matrix’s linear discrepancy, effectively capturing the worst-case rounding error from fractional to integer solutions.

3. Constructive Algorithm and Practical Computation

To operationalize the RRF threshold, a constructive, randomized polynomial-time algorithm—termed Round-IP—is developed (Chandrasekaran et al., 2011). The Round-IP algorithm uses a partial coloring method based on discrepancy minimization (specifically, an adaptation of the Lovett–Meka partial coloring approach), proceeding in phases:

• In each phase, at least half of the variables are moved closer to either $0$ or $1$ via a controlled random walk (Edge-Walk) that ensures deviation in each constraint is small.
• After $O(\log n)$ phases, all but $O(1)$ variables are essentially integral.
• A final randomized rounding assigns the remaining fractional variables to $0$ or $1$ while incurring minimal additional error.

This method ensures that any point in a sufficiently large ball can be efficiently rounded to an integer point without violating the constraints, provided the RRF is above the threshold $R_1$. The algorithm not only substantiates existence results but also provides efficient certificates of feasibility when the geometric conditions are met.

4. Connection to Discrepancy Theory and Phase Transitions

The RRF’s threshold is fundamentally linked to the geometric and combinatorial notion of discrepancy. The core insight is that the feasibility of an integer solution in a random polytope is governed by the polytope's ability to absorb the rounding error, as measured by the linear discrepancy of $A$. This connection leads to nearly sharp thresholds and is reminiscent of phase transitions in random constraint satisfaction problems (such as random $k$-SAT), where feasibility changes abruptly as problem parameters cross critical values.

Table: Summary of RRF Characterizations from (Chandrasekaran et al., 2011)

Threshold	Expression	Interpretation
$R_0$	$\sqrt{(1/6) \log(m/n)}$	Below this, no integer point (w.h.p., for $x_0=1/2$)
$R_1$	$960 \{\sqrt{\log(m/n)} + \dots \}$	Above this, integer feasible (w.h.p.)
lin-disc$(A)$	$\max_{x_0}\min_x \|A(x-x_0)\|_\infty$	Governs sufficiency for integer feasibility

($w.h.p.$ denotes with high probability.)

5. Implications for Optimization and Applications

These results have immediate significance for integer programming, robust optimization, and the analysis of random combinatorial structures:

• Robust Integer Programming: Ensuring a polytope contains a ball of radius exceeding RRF guarantees integer feasibility even under perturbations or rounding, making solution methods more resilient.
• Algorithmic Design: The connection between linear discrepancy and feasibility thresholds provides not only existence proofs but also efficient algorithms that can be implemented in practice for large-scale problems with random or uncertain data.
• Phase-Transition Analysis: The sharpness of the RRF threshold advances the understanding of how feasibility arises or vanishes in random structures, paralleling important results in random SAT and related fields.

A plausible implication is that these geometric-combinatorial connections may inform the design of new discrepancy-minimization–based rounding algorithms for other classes of robust optimization problems.

6. Comparison with Previous Approaches

Prior to these results, integer feasibility of high-dimensional polytopes, especially under random constraints, typically relied on techniques with worst-case exponential complexity or lacked precise, phase-transition–type thresholds. The methodology of (Chandrasekaran et al., 2011) leverages advanced tools from discrepancy theory—particularly algorithmic techniques following Spencer and the Lovett–Meka algorithm—to provide both constructive guarantees and nearly tight characterizations of the RRF. This represents a marked advance, producing results with concrete, computable constants and efficient randomized algorithms that bridge theoretical optimality and practical computability.

7. Broader Research Context and Extensions

The analytical framework for the RRF as developed in (Chandrasekaran et al., 2011) has motivated further research connecting feasibility, robustness, and geometric discrepancy across random and uncertain systems. Later works have extended these ideas to other forms of discrepancy (hereditary discrepancy), more general uncertainty models, and broader classes of constraints, reinforcing the centrality of the RRF as a universal (albeit context-dependent) criterion for robust feasibility in integer programming and related fields. The phase-transition behavior and algorithmic construction presented stand as paradigms for interpreting and quantifying robustness in random and high-dimensional optimization landscapes.