Optimal Sequential Flows

We provide a new algebraic technique to solve the sequential flow problem in polynomial space. The task is to maximise the flow through a graph where edge capacities can be changed over time by choosing a sequence of capacity labelings from a given finite set. Our method is based on a novel factorization theorem for finite semigroups that, applied to a suitable flow semigroup, allows to derive small witnesses. This generalises to multiple in/output vertices, as well as regular constraints.

💡 Research Summary

The paper studies the “sequential flow” problem, a dynamic variant of the classic max‑flow problem where edge capacities are not fixed but can be chosen at each discrete time step from a finite set A of capacity labelings. Each labeling a∈A assigns a capacity (a natural number or ω for unbounded) to every edge of a directed graph G=(V,E) with distinguished source s and sink t. A word w=a₁…a_ℓ∈A* defines a pipeline of ℓ+1 layers: at time i the edge (v→v′) has capacity a_i(v,v′). A sequential flow is a sequence of ordinary flows f₁,…,f_ℓ that respect the capacity of the corresponding layer and satisfy flow‑conservation across consecutive layers. The value of such a flow is the amount exiting s at time 1 (equivalently entering t at time ℓ). The objective is to compute optSeqFlow = sup{|f| : f is a sequential flow}. The length ℓ is unrestricted, so the supremum may be infinite (denoted ω).

Previous work showed PSPACE‑hardness of the decision version and gave an exponential‑space upper bound via reduction to distance automata. The authors improve this dramatically: they present a polynomial‑space algorithm that computes the exact optimum, matching the known lower bound and extending to several generalisations (regular language constraints on the capacity word, and “fair” sequential flows where several edges must receive equal flow).

The solution proceeds in two stages.

Stage 1 – Unboundedness detection.
The authors abstract each capacity matrix to the three‑valued max‑min semiring M = ({0,1,ω}, max, min), mapping every finite positive number to 1 while preserving 0 and ω. Matrix multiplication over M captures whether a path between two vertices has finite, unbounded, or zero capacity after a sequence of labels. The set of such matrices forms a finite semigroup F (the “flow semigroup”). Elements of F can be enumerated using a nondeterministic polynomial‑space search (Algorithm 1). Crucially, they study idempotent elements e (e²=e). Lemma 5 classifies the possible long‑term flow behaviour of an edge (v,v′) in the pipeline eⁿ into three cases: (i) immediate ω‑capacity, (ii) linear growth (unstable 1‑edges that can be “amplified” via intermediate ω‑edges), and (iii) bounded growth (stable 1‑edges). To capture case (ii) they introduce a new operation e♯ that replaces the iterated effect of e by ω on unstable pairs, while leaving stable pairs unchanged. Theorem 26 proves that any finite semigroup admits a “small summary” – a compact representation of its elements together with the ♯‑operation – of polynomial size. Using this, Algorithm 1 checks whether some element’s ♯‑iteration yields ω between s and t; if so, optSeqFlow = ω.

Stage 2 – Computing the finite optimum.
If unboundedness is ruled out, the authors show (Theorem 16) that the supremum is at most exponential in |V|, because each iteration of an idempotent can increase flow by at most a linear factor in the number of vertices. They then design Algorithm 2, which works entirely with the compact summaries from Stage 1. It performs a dynamic‑programming style search over the semigroup, using matrix multiplication over M and the ♯‑operation to combine prefixes, and keeps track of the best finite flow value reachable from s to t. All intermediate data structures are of polynomial size, so the algorithm runs in polynomial space.

Generalizations.

1. Regular constraints. If the capacity word must belong to a regular language L, the same framework applies by taking the product of the flow semigroup with the syntactic monoid of L. Theorem 37 shows that the polynomial‑space algorithm still computes the optimum under this restriction, thereby subsuming the classic max‑flow (L = Σ*) and the bounded‑time flow model of Akrida et al.
2. Fair sequential flow. The “fair” variant requires that a prescribed set of edges each carry the same amount of flow. By augmenting the semigroup with additional counters and applying the same factorisation technique, Theorem 35 yields a polynomial‑space solution.

Technical contributions.

• A novel factorisation theorem for arbitrary finite semigroups (Theorem 26) that yields small summaries and enables efficient manipulation of idempotent iterations.
• The definition of the ♯‑operation on idempotent matrices, which abstracts the effect of arbitrarily many repetitions and turns certain linear growth patterns into an ω‑value.
• A tight PSPACE upper bound for the sequential flow problem, matching the known PSPACE‑hardness, and extending it to richer settings (regular constraints, fairness).
• Insightful examples (Figures 1‑4) illustrating how unbounded flow can arise only through intricate nesting of capacity words, and how the algebraic machinery captures these phenomena.

Impact and future work.
By linking dynamic network optimisation to algebraic properties of finite semigroups, the paper opens a new avenue for tackling problems where the control sequence is unbounded. The factorisation technique may find applications in verification of infinite‑state systems, analysis of distance automata, and synthesis problems for Markov decision processes. Future directions include refining the unboundedness test to lower complexity classes, extending the approach to stochastic or continuous‑time capacities, and exploring connections with tropical algebra and max‑plus semirings.