Non-Trivial Strongly Polynomial Dynamic Algorithm

• The paper introduces a non-trivial strongly polynomial dynamic update method for maintaining minimum mean cycles in directed graphs under edge insertions.
• It leverages a novel parallel SSSP algorithm with subquadratic work, heavy-light decomposition, and near-list structures to surpass trivial recomputation times.
• The method relies solely on addition and comparison operations, eliminating dependency on numerical data scales and setting a new benchmark in dynamic optimization.

A non-trivial strongly polynomial dynamic algorithm refers to an algorithm that maintains or updates solutions to an optimization problem under dynamic changes (such as edge insertions in a graph), with the following properties: (1) the per-update time is asymptotically faster than recomputing the solution from scratch (the "trivial" baseline), and (2) the running time is bounded by a polynomial in the number of entities (e.g., states, edges, or actions) only, completely independent of the numerical magnitudes or encoding lengths of the problem data such as edge costs or transition probabilities. This criterion places the algorithm within the "strongly polynomial" regime, distinguishing it from weakly polynomial approaches whose complexity may depend on bit-length or arithmetic precision.

1. Definitions and Foundational Criteria

A dynamic algorithm supports efficient updates to its output after local changes to the input—such as edge additions or deletions in graphs, or new actions in a Markov Decision Process (MDP). "Non-trivial" signifies that the dynamic update cost is not merely re-executing the static algorithm in full but is polynomially faster per update in the asymptotic regime.

"Strongly polynomial" is a more restrictive criterion in complexity theory: the algorithm's running time (total and per-update) must be polynomial in the basic input dimensions (e.g., number of vertices $n$, number of edges $m$), and must not depend on any numerical data scales (such as logarithms or maximum magnitudes of edge weights, rewards, or transition probabilities). In practical terms, the algorithm must operate using only addition and comparison operations, rather than bit-level arithmetic.

2. Historical Context and Baseline Algorithms

Classic algorithms for problems such as shortest paths (SSSP) and minimum mean cycle typically either recompute the solution from scratch after each input update or employ data structures that require complexity dependent on data magnitudes (weakly polynomial). For minimum mean cycle, the standard recomputation method via Karp's algorithm yields an $O(nm)$ bound per update in the dynamic setting—no asymptotic improvement over static computation existed prior for the general case. For discounted dynamic programming, value iteration is a widely studied baseline, but it fails the strongly polynomial property.

Significant research attention has addressed the gap between strongly polynomial and weakly polynomial algorithms for both static and dynamic settings, with foundational work characterizing feasibility under fixed parameters (e.g., Ye's analysis of policy iteration for fixed discount factor).

3. Algorithmic Advances: Strongly Polynomial Dynamic Minimum Mean Cycle

Recent research (Karczmarz et al., 22 Oct 2025) established the first non-trivial strongly polynomial dynamic algorithm for maintaining the minimum mean cycle in directed graphs under edge insertions. Fundamental to this advance is a new parallel SSSP algorithm with subquadratic work and truly sublinear depth:

For any $t \in [1, n^{1/17}]$, directed SSSP in dense graphs with non-negative weights can be solved within

$$\text{work:}\quad (m + n^{9/5} t^{17/5}),\qquad \text{depth:}\quad (n / t)$$

where $(\cdot)$ notation suppresses polylogarithmic factors.

Combining this result with the parametric search framework (Megiddo), the minimum mean cycle can be dynamically maintained in $O(m n^{1-1/22})$ worst-case time per edge insertion. This complexity is polynomially faster than the standard $O(nm)$ recomputation approach, and relies exclusively on addition and comparison operations—rendering the method strongly polynomial.

Problem	Work/Time Complexity	Remarks
Directed SSSP (dense graphs)	$(m + n^{9/5} t^{17/5})$ work	Sublinear depth, strongly polynomial
Min-mean cycle (dynamic)	$O(m n^{1-1/22})$ per update	First non-trivial strongly polynomial

4. Technical Innovations Enabling Non-Trivial Strongly Polynomiality

Key techniques underpinning these results include:

• Heavy-light vertex decomposition: Splits the graph into vertices with bounded congestion, enabling efficient parallel operations and controlling the combinatorial blowup common in dynamic routines.
• Near-list structures: Linear-sized precomputed neighbor lists facilitate rapid local discoveries, pivotal for subquadratic work and for responsiveness to dynamic changes.
• Auxiliary filtering: Degree filters maintain manageable in/out-degree after updates, supporting persistence and efficient depth reduction.

For the minimum mean cycle problem, these building blocks permit efficient dynamic updates via repeated SSSP computation under parametric search, as each trial value of $\lambda$ in the minimization of the mean cycle corresponds to a specific SSSP instance with transformed edge weights.

5. Conceptual Significance and Implications

The existence of a non-trivial strongly polynomial dynamic algorithm addresses a long-standing open question in combinatorial optimization, shifting the paradigm for dynamic graph algorithms:

• Prior algorithms for dynamic minimum mean cycle either lacked strong polynomiality (dependence on edge weight magnitude) or failed to surpass trivial recomputation.
• The parallel SSSP framework generalizes to a suite of other dynamic optimization problems, among them min-cost flow and assignment, provided their formulations reduce to SSSP with non-negative weights.
• A plausible implication is improved parallel and dynamic algorithms for additional objectives, as the core routines avoid data scale dependencies and provide polynomial, modular update times.

6. Connections to Discounted Dynamic Programming

In contrast, value iteration for discounted dynamic programming, as rigorously demonstrated in (Feinberg et al., 2013), is not strongly polynomial. For MDPs with $m$ actions, the number of required value iteration steps can be exponential in $m$ even with exact arithmetic, precluding strongly polynomial running time. Specifically, in constructed examples, the number of iterations required before the optimum is discovered is at least

$\frac{e^m}{-\ln \beta}$

for $m$ actions, where $\beta$ is the discount factor and $e^m$ derives from the rapid growth in rewards in the constructed family of MDPs.

This negative result underscores that strongly polynomial algorithms in dynamic programming require careful algorithmic design, such as policy iteration or linear programming methods in certain parameter regimes, and cannot be achieved for value iteration generically.

7. Future Directions and Open Questions

Recent developments suggest the possibility of a broader strong polynomiality in dynamic combinatorial optimization, especially in the parallel regimes and via new decomposition and filtering approaches. Open questions include:

• Extending strongly polynomial dynamic algorithms to more general optimization objectives beyond min-mean cycle and SSSP.
• Achieving similar results under deletions or more complex update models.
• Understanding the tradeoffs between depth, work, and update responsiveness in real-world, large-scale graphs.

A plausible implication is that further research may yield dynamic algorithms in additional domains (such as network flow) meeting the dual criteria of non-trivial per-update cost and independence from data encoding size, building on the paradigm illustrated by heavy-light decompositions and near-list primitives.