Dynamic Level Maintenance

• Dynamic Level Maintenance is the process of updating graph levels or core metrics after localized changes, ensuring maintained consistency and performance.
• Algorithms utilize bounded propagation and selective updates, leading to dramatic speedups in applications like logic synthesis and dynamic graph analytics.
• Methods employ parallelization techniques, such as matching-based peeling and distributed MPC, to efficiently manage system updates in real-world environments.

The dynamic level maintenance problem encompasses the algorithmic and modeling challenges associated with updating, scheduling, or maintaining graph-theoretic “level” or “core” functions and related constraints under sequences of local, dynamic modifications. Occurring in contexts ranging from large-scale logic synthesis and reliability engineering to infrastructure scheduling and distributed manufacturing, this family of problems is characterized by the need for efficiently updating node or system-level attributes subject to application-specific constraints, often in the presence of locality or parallelism opportunities.

1. Formal Modeling of Dynamic Level Maintenance

Dynamic level maintenance arises in heterogeneous domains but consistently features a structural object (typically a graph or system decomposition), a recursively defined node-level or health metric, and sequences of localized updates.

• Logic Circuit Level Maintenance: In EDA, the problem is formalized on a DAG $G=(V,E)$ representing the circuit, with node-level $\mathcal{L}(v)$ for each $v\in V$ defined recursively as

$\mathcal{L}(v) = \begin{cases} 0 & \text{if } v \text{ is a primary input} \[2pt] 1 + \max\{\mathcal{L}(u): (u\to v)\in E\} & \text{otherwise} \end{cases}$

with localized graph edits $\Delta G_i$ imposing the need to maintain valid levels after each operation (Liu et al., 14 Dec 2025).

• Core Number in Dynamic Graphs: In k-core decomposition, given undirected $G=(V,E)$, the core number $\kappa_G(u)$ of a vertex $u$ is the largest $k$ such that $u$ is in the $k$-core. Insertions/deletions of edges demand efficient recomputation or update of $\kappa_G(u)$ (Wang et al., 2017).
• Redundancy/Reliability Systems: In reliability engineering, dynamic maintenance combines sequential system upgrade/repair policies with evolving system states, modeled explicitly via state-based decision processes (e.g., CTMDP), requiring continuous update of system reliability metrics in response to component-level events (Fairley et al., 17 Apr 2025).
• Production/Degradation Scheduling: In manufacturing, dynamic maintenance addresses scheduling of repairs and production in response to evolving degradation states $x_n(t)$ with decisions $z_n(t)$ (maintenance triggers), where the state function obeys a controlled Markov process (Rokhforoz et al., 2020).

2. Theoretical Foundations, Key Properties, and Locality

A central property for algorithm design is the locality of influence—whether and how far a local operation propagates changes in the system metric.

• Bounded Propagation: For certain update classes, the effect of a local change (edge addition, subgraph replacement) can be shown to impact only nodes within a bounded topological region or “cone,” permitting selective updating with $O(\Delta)$ cost, where $\Delta$ is the maximal affected neighborhood (Liu et al., 14 Dec 2025).
• Core Number Locality: In dynamic k-core maintenance, a single edge insertion/deletion can raise or lower a vertex core number by at most 1. If all updates form a matching, these changes are strictly localized and mutually non-interfering, underpinning efficient parallelization (Wang et al., 2017).
• Order-Metric Equivalence: Node “level” is a monotone function compatible with partial topological order; any valid topological order $\prec$ is a linear extension of the partial order prescribed by level inequalities (if $\mathcal{L}(u)<\mathcal{L}(v)$ then $u\prec v$), enabling efficient propagation schemes (Liu et al., 14 Dec 2025).

3. Algorithmic Methodologies and Parallelization

Dynamic level maintenance algorithms optimize for fast updates using graph analytic, discrete optimization, and distributed computation techniques tailored to locality and application constraints.

• Bounded Level Maintenance (boundLM): The boundLM algorithm maintains correct levels and reverse levels after localized DAG updates through dynamic partial order maintenance (dynTO), selective forward (dynLev) and backward (dynRL) level propagation. The core insight is that only $\Delta$-sized regions (where $\Delta$ is the max window size of $\Delta G_i$) need to be updated per operation. The total runtime is $O(|V|\Delta\log\Delta)$ across $|V|$ updates (Liu et al., 14 Dec 2025).
• Parallel Core Maintenance via Matchings: The insertion or deletion set is decomposed via proper edge-coloring into $\Delta_I+1$ matchings, and each matching is then processed in parallel. The K-MatchingInsert (or Delete) subroutine propagates core number changes in the affected k-core subgraph. Per-batch runtime is $$O(m_I \Delta_I + \Delta_I \cdot \max_k\{|E[V^k]| + n^k L^k\})$$ (Wang et al., 2017).
• Distributed Dynamic Maintenance – MPC and Benders: In manufacturing systems, joint dynamic maintenance and production scheduling is formulated as a large-scale MIQP. The algorithm integrates a master–subproblem decomposition (Benders), with global binary maintenance variables decided in the master, and distributed Model Predictive Control (MPC) with dual decomposition among agents/units in the subproblems. The approach yields convergence within few outer iterations and near-optimal distributed schedules (Rokhforoz et al., 2020).
• Bi-objective CTMDP Heuristics: When the maintenance design space is combinatorially large, an Approximate Pareto Population (APP) heuristic first solves a static design problem, then polishes candidate designs via dynamic MDP scalarization, enabling near-complete Pareto fronts within seconds (Fairley et al., 17 Apr 2025).

Context	Key Structure	Update Operation	Update Method
Logic circuits	DAG, level/𝓛(·)	Subgraph replacement ∆Gᵢ	Partial order + bounded cone fix
Graph analytics	Undirected, k-core	Edge addition/deletion (matching)	Parallel matching-based peeling
Reliability	Parallel system, CTMDP	Repair policy change	2-stage, MDP + knapsack MILP
Manufacturing	Agent Markov model	Degradation-based maintenance	Distributed MPC + Benders

4. Complexity Analysis and Scalability

Complexity bounds and scalability are essential for practical deployment in large designs or systems.

• Logic Optimization: The boundLM algorithm achieves $O(|V|\Delta\log\Delta)$ total runtime, breaking the previous $O(|V|^2)$ barrier in dynamic logic synthesis. This holds when each local transformation is confined (small $\Delta$); for unbounded updates, the practical gain is reduced but remains significant (Liu et al., 14 Dec 2025).
• Dynamic Graph Cores: Matching-based parallelization reduces the number of sequential iterations from $O(m_I)$ (edgewise) to $O(\Delta_I)$, where $\Delta_I$ is the peak insert degree per vertex in the update set. Empirically, this yields 3–4 orders of magnitude speedup on real and synthetic data sets; speedup is linear in the number of threads or tasks (Wang et al., 2017).
• Reliability and Manufacturing Systems: Multi-objective MILPs are computationally intractable for large system sizes. Partitioned or distributed heuristics (e.g., APP, Benders-MPC) retain near-complete optimal solution sets at orders-of-magnitude lower computation time (sub-seconds to seconds vs. full MILP exceeding 5 minutes), with optimality gaps typically $<3\%$ and full preservation of privacy and modularity (Fairley et al., 17 Apr 2025, Rokhforoz et al., 2020).

5. Practical Applications and Empirical Evaluation

Dynamic level maintenance underpins several critical operations:

• EDA/Logic Synthesis: Maintaining node-levels in DAGs guides constraint-driven synthesis passes (e.g., rewrite, refactor, resubstitution) while ensuring no invalid timing depth is introduced. In EPFL and IWLS’05 benchmarks (up to $4.2\times10^7$ AND-nodes), boundLM delivered $1074.8\times$ faster level maintenance than O($|V|^2$ baselines and preserved solution quality (Liu et al., 14 Dec 2025).
• Graph Analytics: Core maintenance algorithms update k-core decompositions in evolving social, web, or biological graphs, supporting real-time community or anomaly detection. Experiments on LiveJournal, YouTube, and synthetic graphs up to $2^{21}$ nodes report sustained <20 ms per-update latency and >$100\times$ parallel speedup (Wang et al., 2017).
• Networked Infrastructure: In interdependent electric-gas grid scheduling, bilevel dynamic maintenance with game-theoretic equilibrium selection quantifies the impact of maintenance resource limits, order-of-move, and flow-model fidelity on operator payoffs and system reliability (Wang et al., 2017).
• Reliability and Maintenance Policy Design: For parallel systems, integrating dynamic maintenance into design (BO-IDDMP) produces Pareto fronts strictly dominating static designs, with minimal loss of solution coverage even for highly heterogenous systems (Fairley et al., 17 Apr 2025).
• Distributed Manufacturing: MPC+Benders algorithms enable large-scale, agentwise scheduling under degradation constraints, with practical implementations converging in a few global iterations and maintaining optimality gaps below 3% (Rokhforoz et al., 2020).

6. Limitations, Extensions, and Cross-Domain Relevance

The main limitations relate to the locality assumptions and the scale of individual updates.

• Locality Dependency: Algorithms with bounded propagation critically assume that each update modifies a small neighborhood (small $\Delta$). For global transformations, bounds degrade to quadratic or worse and selective propagation loses efficacy (Liu et al., 14 Dec 2025).
• Resubstitution and Unbounded Forward Updates: In logic optimization, resubstitution steps may unbind the region of influence, necessitating augmented scheduling (e.g., using priority queues for forward propagation) (Liu et al., 14 Dec 2025).
• Refinements and Learning Integration: Incorporating learned heuristics can further reduce unnecessary updates in logic flows (e.g., skipping low-gain windows). Efficient partial order lists (e.g., Bender–Cole) may lower practical constants (Liu et al., 14 Dec 2025).
• Applicability to Other Domains: The bounded locality paradigm extends to task/dataflow scheduling, incremental longest-path computation, and generic dependency-graph analysis under locally bounded edits (Liu et al., 14 Dec 2025).
• Game-theoretic and Multi-agent Dynamics: As system interaction increases, dynamic maintenance must incorporate anticipatory and strategic behavior (e.g., Nash or Stackelberg equilibrium computation, coupled MILPs), which can significantly affect the solution structure and computational tractability (Wang et al., 2017).

7. Summary Table: Algorithmic Landscape

Paper (arXiv)	Domain	Level/Core Metric	Dynamic Update Type	Complexity	Empirical Speedup
(Liu et al., 14 Dec 2025)	Logic Synthesis	DAG level	Local neighborhood edits	$O(|V|\Delta\log\Delta)$	$1074.8\times$ vs. $O(|V|^2)$ (level)
(Wang et al., 2017)	Graph Analytics	Core number	Edge (batch, matching)	$O(\Delta_I)$ iterations	Up to $10^4\times$ (parallel)
(Rokhforoz et al., 2020)	Manufacturing	Degradation level	Maintenance/production	Distributed via MPC+Benders	$<3\%$ gap, large scale
(Fairley et al., 17 Apr 2025)	Reliability	Failure indicator	Repair action	2-stage heuristic	$>100\times$ vs. MILP (Pareto)
(Wang et al., 2017)	Infrastructure	Utility/revenue	Maintenance schedule	Bilevel game+MILP	Illustrative for interdependent grids

Dynamic level maintenance thus represents a foundational problem class intersecting combinatorial optimization, dynamic graph algorithms, reliability theory, and distributed systems, with recent advances focusing on locality, bounded-update regimes, and parallel or distributed scheduling for large-scale, dynamically evolving networks and infrastructures.