Eviction Number in Theory & Applications

• Eviction number is a quantitative measure defining the minimal resource allocation (e.g., guards, cache slots) required to sustain invariant properties in dynamic systems.
• Research shows eviction number relates to combinatorial invariants in graphs and Pareto frontiers in memory systems, offering theoretical bounds and algorithmic guarantees.
• The concept informs optimal strategies in dynamic resource allocation, from cache eviction policies to adaptive mechanisms in machine learning architectures.

An eviction number is a technical parameter that quantifies the efficiency or resource requirements inherent to “eviction” policies in a given system—whether that system is a memory hierarchy, a neural model, a resource allocation problem, or a combinatorial game on graphs. The eviction number encapsulates the minimal level of resources (memory, guards, compute) required to satisfy invariants such as dominance, retention, or operational correctness under ongoing insertions or disturbances. Its precise mathematical definition and operational meaning vary with context; in all cases, it serves as a fundamental complexity measure for dynamic selection processes subject to adversarial or stochastic influence.

1. Eviction Number in Graph Theory

The eviction number, denoted $e^\infty(G)$ for a graph $G$, is defined as the smallest cardinality of sets in an eternal dominating family in the so-called “eviction game.” In this dynamic domination process:

• The family $\mathcal{D}_k = \{D_1, D_2, ..., D_\ell\}$ consists of dominating sets, each with $|D_i| = |D_j|~\forall~i, j$.
• After an external disturbance (an “attack” at some vertex $v$), a guard may move in response. The transition $D_i \to D_j$ (with $D_j = D_i \cup \{w\} \setminus \{v\}$ for some $w$ adjacent to $v$, $w \notin D_i$) must result in a set that is again a dominating set, i.e. $D_j \in \mathcal{D}_k$.
• The family is designed so that for any $D_i$ and any $v \in D_i$, the response mechanism always reestablishes dominion over $G$ via an allowed move.

The eviction number is the minimal $|D_i|$ among families admitting such dynamics, quantifying how many “guards” are needed for indefinite defense (MacGillivray et al., 23 Sep 2025).

Distinct graph invariants interact with $e^\infty(G)$:

• The independence number $\alpha(G)$, the size of the largest set of mutually nonadjacent vertices, does not lower-bound $e^\infty(G)$; there exist graphs (e.g., graphs with two universal vertices) where $e^\infty=1$, but $\alpha$ can be arbitrarily large. Thus, the ratio $\alpha(G)/e^\infty(G)$ is unbounded.
• Conversely, examples show that the reverse ratio $e^\infty(G)/\alpha(G)$ can be close to but exceeds 1 for certain infinite families of connected graphs (e.g., $e^\infty(C_7)=4$, $\alpha(C_7)=3$).

The main bounding result asserts the existence of a function $f(k)$, depending on $k=\alpha(G)$, such that $e^\infty(G)\leq f(k)$ for all graphs of independence number $k$, with $f(k)$ constructed using Ramsey-theoretic arguments. This establishes that while $\alpha$ is not a lower bound, the eviction number is still controlled by the combinatorial structure of independence in $G$ (MacGillivray et al., 23 Sep 2025).

2. Eviction Number in Memory Systems

In memory hierarchy and cache management, the eviction number reflects the mathematical structure underlying optimal eviction strategies and buffer utilization.

The eviction policy is typically modeled as a control process:

• The state at time $t$ is $x_t = (z_t, b_t)$, with $z_t$ representing the trace (e.g., stack of recency orderings in LRU policies), and $b_t$ a vector indicating which items are buffered.
• A disturbance $w_t$ (the address accessed) and control $u_t$ (eviction decision) determine the next state.

Within the context of the Least Recently Used Stack Model (LRUSM):

• The access probability at stack depth $j$ is $s(j)$, and the cumulative $S(j) = \sum_{i=1}^j s(i)$.
• Policies are characterized by thresholds: for a single item “keep until stack depth $k$,” denoted $\mathsf{ev}_k$. The average occupancy cost is $J_{oc}(\mathsf{ev}_k) = k/V$, and the miss cost is $J_{ms}(\mathsf{ev}_k) = (1-S(k))/V$.
• The trade-off curve of achievable (occupancy, miss) pairs forms a Pareto frontier. The optimal point corresponds to the “eviction number” (here, buffer capacity $C$) required to achieve a given miss rate under optimal policy.

The primary theoretical outcome is the characterization of policies (e.g., K–L and Least Profit Rate, LPR) that achieve optimality for fixed or average occupancy. The LPR policy, which selects for eviction the item with the lowest "profit rate"

$\xi(i) = \max_j \frac{\sum_{k=i}^j s(k)}{j-i+1},$

delivers near-optimal miss rates with logarithmic competitive ratio overhead relative to the offline optimal policy (Bilardi et al., 2011).

3. Algorithmic and Optimization Frameworks

The pursuit of minimal eviction number encompasses optimal control, multiobjective optimization, and algorithmic simulation.

• In stochastic address trace models (e.g., HMRM), the control problem is solved via the BeLLMan equation,

$\lambda \mathbf{1} + h = T h$

for average (“gain optimal”) or transient (“bias optimal”) cost criteria.

• In stack-based traces, efficient computation of the number of misses across all capacities is achieved using stack distance and inclusion properties. With appropriate data structures, per-access computations are $O(\log V)$.
• For multiobjective settings, randomization (Randomized Mixtures of Pure Policies, RMoPs) is used to achieve intermediate occupancy/miss trade-offs not attainable by deterministic thresholding alone.
• Resource allocation for multi-item eviction numbers is formulated as a marginal gain assignment problem.

These techniques ground the operational meaning of the eviction number in tractable, analyzable algorithms that also yield performance guarantees.

4. Resource Allocation, Multiobjective Formulations, and Practical Consequences

When the resource (buffer occupancy, number of guards, cache slots) is treated as a cost, not a constraint, the eviction number connects to Pareto efficiency:

• Pareto frontiers enumerate supported efficient points (SEPs), demarcating the minimal resource required for each attainable level of system performance.
• Randomized strategies strategically interleave policy choices to tune the amortized resource usage.
• This approach allows adaptive adjustment: e.g., changing the eviction number dynamically in response to miss or occupancy requirements.

In practical contexts, this framework underlies the deployment of adaptive cache policies, dynamic resource provisioning in computer systems, and robust search/outreach plans in intervention programs (e.g., active geospatial search to mitigate evictions in cities (Sarkar et al., 19 Dec 2024)).

5. Eviction Number in Learning and Information Extraction Systems

Eviction number concepts appear by analogy in machine learning and knowledge extraction:

• In “In-context KV-Cache Eviction for LLMs via Attention-Gate” (Zeng et al., 15 Oct 2024), the effective in-context eviction number corresponds to the minimal subset of cached KV states (“tokens”) required to preserve accuracy in autoregressive generation.
• The adaptive gating mechanism acts as an eviction policy, dynamically controlling the number of retained states per layer and attention head—a contemporary instantiation of eviction number as a mechanism for efficient neural context compression.

Similar ideas guide scalable, annotation-efficient extraction pipelines (e.g., SynthEHR-Eviction (Yao et al., 10 Jul 2025)), where the minimal core dataset necessary for reliable classifier performance can be interpreted as an informational “eviction number.” This is furthered by multi-stage algorithmic frameworks combining data augmentation, prompt optimization, and hierarchical filtering.

6. Broader Theoretical and Application Implications

The eviction number emerges as a central object in the analysis of dynamic, adversarial, or stochastic systems subject to ongoing resource constraints and selection pressure. Key properties:

• It encodes the tight bound on resources for sustainable operation under worst-case or stochastic sequences.
• Its relationships to classical invariants (e.g., independence number in graphs) are non-monotonic and subtle, necessitating Ramsey-theoretic and probabilistic tools for sharp characterization (MacGillivray et al., 23 Sep 2025).
• In algorithmic memory systems, the eviction number reflects not just resource sufficiency but also the optimal policy structure and its computational realizability.
• In application areas such as reinforcement learning-driven search and structure-aware data annotation, the eviction number informs the efficiency–coverage trade-off underlying all sequential decision problems.

A plausible implication is that further refinement of eviction number bounds—especially in combinatorial domains—will enable improved guarantees for online algorithms, resource allocation protocols, and adaptive defense mechanisms in large-scale networks and distributed systems. The diverse contexts in which eviction number arises point to its unifying role in the paper of dynamic resource-constrained decision processes.