Parameterized Complexity of Temporal Agony

Abstract: Real-world networks are often organized in several layers forming a hierarchy which determines the interaction between the individual components. In order to discover such hierarchies in temporal networks, Tatti [ECML PKDD 2018] introduced the temporal agony problem Seg-Agony. Here, the goal is to assign each vertex a certain rank (from 1 to $k$) such that arcs only point from lower ranks to higher ranks. Backward arcs are penalized depending on the difference between the corresponding ranks. Since arcs may change over time, each vertex is allowed to change its rank $\ell\ge 1$ times in order to minimize the overall penalty $α$ (called temporal agony). We study the parameterized complexity of Seg-Agony with a special focus on the number $k$ of possible ranks for which we identify the precise complexity border. We show that the problem is polynomial-time solvable for $k=2$, NP-hard for $k=3$ and $\ell=1$ but polynomial-time solvable for constant $α$, and NP-hard for $k=4$ and $\ell=1$ even for $α=0$. We further show a polynomial-time algorithm for a constant number $n$ of vertices and fixed-parameter tractability for the combined parameter $n+\ell$.

• The paper establishes a sharp rank-based trichotomy: two ranks are polynomial-time solvable, three ranks are NP-hard but XP in agony, and four ranks remain NP-hard even with zero agony on DAGs.
• The paper uses per-vertex dynamic programming for two ranks, a 2-SAT reduction and bounded-violation branching for three ranks, and SAT-based gadget reductions to prove higher-rank hardness.
• The paper gives an FPT algorithm parameterized by the number of vertices plus allowed changes, while leaving FPT status for agony at three ranks and several temporal-DAG cases open.

Background and problem definition

The paper studies the computational complexity of discovering hierarchies in temporal directed networks, building on the agony framework introduced by Gupte et al. for static digraphs [GSLMI11] and extended to temporal networks by Tatti [Tatti18]. In the static setting, one assigns each vertex a rank from $[k]$ so that arcs point from lower to higher ranks as much as possible; violations are penalized via a penalty function $p$, with the standard choice $p_l(x) = \max(0, x+1)$.

The temporal variant, Seg-Agony, allows each vertex to change its rank $\ell$ times over $\tau$ discrete time steps. Formally, a rank assignment is a rank $\ell$-segmentation built from $\ell+1$ static rank functions per vertex with monotone change points, and the objective is to minimize the total penalty

$$q(r,G,p_l) \coloneqq \sum_{t\in[\tau]}\sum_{(u,v)\in A_t} w_t(u,v)\cdot p_l(r(u,t)-r(v,t)).$$

The decision problem asks whether a rank $\ell$-segmentation with temporal agony at most $\alpha$ exists. Prior work established NP-hardness for $p$0, $p$1, $p$2, but without pinning down the number of ranks (a close inspection shows the reduction requires $p$3). The present work delineates the precise complexity border in terms of the number of ranks $p$4, which is the natural parameter since practical hierarchies have few levels.

Main results: a complete classification for small $p$5

The central contribution is a trichotomy over the number of ranks:

Ranks	Complexity	Key conditions
$p$6	Polynomial time	$p$7
$p$8	NP-hard ($p$9), but XP in $p_l(x) = \max(0, x+1)$0	$p_l(x) = \max(0, x+1)$1
$p_l(x) = \max(0, x+1)$2	NP-hard even for $p_l(x) = \max(0, x+1)$3, $p_l(x) = \max(0, x+1)$4, DAGs	Excludes XP for constant $p_l(x) = \max(0, x+1)$5

Two ranks. For $p_l(x) = \max(0, x+1)$6 and any penalty with $p_l(x) = \max(0, x+1)$7, the agony decomposes into per-vertex contributions that depend only on each vertex's own rank — specifically, on the weights of incoming arcs of rank-1 vertices and outgoing arcs of rank-2 vertices. This decoupling enables an independent dynamic program per vertex over time steps and change-point budgets, yielding the stated polynomial running time. Notably, this tractability is fragile: the authors also prove that minimizing $p_l(x) = \max(0, x+1)$8 with two ranks becomes NP-hard on unweighted static digraphs whenever $p_l(x) = \max(0, x+1)$9, via a reduction from Maximum Cut, and even on unweighted DAGs when $\ell$0. This strengthens Tatti's earlier concave-penalty hardness, which required four ranks and weighted arcs.

Three ranks. The zero-agony case with $\ell$1 is solvable in polynomial time by a structural preprocessing argument combined with a reduction to 2-SAT: vertices appearing in directed paths of length three have forced ranks, remaining vertices admit exactly two candidate rank schedules encoded as Boolean variables, and arc constraints translate into 2-CNF clauses solvable via Aspvall–Shiloach-style implication graphs. Branching on at most $\ell$2 violating arcs then yields membership in XP for parameter $\ell$3: Seg-Agony with $\ell$4, $\ell$5 is solvable in $\ell$6 time. Both results hold for any penalty function positive on nonnegative arguments.

In contrast, the general case with $\ell$7 and $\ell$8 is NP-complete, via a reduction from Max 2-SAT (with variables appearing exactly three times positively and negatively). The construction uses $\ell$9 layers: early layers force variable vertices into binary rank choices encoding truth assignments, intermediate layers propagate these choices to literal vertices, and a final layer penalizes unsatisfied clauses. The budget arithmetic is tight — forced penalties contribute exactly $\tau$0, leaving slack $\tau$1 corresponding precisely to unsatisfied clauses. The authors note the lifetime can be made constant if weighted instances are allowed, and remark that both the algorithm and hardness proof exploit the specific structure of $\tau$2 where applicable.

Four ranks. The complexity jump is sharp: Seg-Agony with $\tau$3, $\tau$4, $\tau$5, and $\tau$6 is NP-hard even when the underlying digraph is a DAG, via a reduction from Monotone 3-SAT. Anchor vertices fix all four ranks across the lifetime; layer 6 forces each variable vertex into rank 2, creating a binary choice of whether it occupies rank 2 before or after that layer; clause gadgets planted in layers 2–5 (negative clauses) and 8–11 (positive clauses) are satisfiable exactly when at least one literal vertex retains freedom in its rank. This strictly strengthens Tatti's prior hardness result and rules out any XP algorithm parameterized by $\tau$7, $\tau$8, or $\tau$9 individually at constant values.

Fixed-parameter tractability for $\ell$0

On the positive side, the authors give a dynamic program over all pairs of change-count functions $\ell$1 and rank functions $\ell$2, exploiting the observation that $\ell$3 in any optimal segmentation. With compatibility transitions between consecutive layers, this yields an algorithm running in $\ell$4 time, establishing FPT membership for the combined parameter $\ell$5 and XP membership for $\ell$6 alone. Correctness follows by induction over time steps against the optimum restricted to fixed $\ell$7 and $\ell$8.

Limitations and open questions

Several boundaries of the analysis remain open. The most prominent is whether Seg-Agony with $\ell$9 lies in FPT for parameter $\ell+1$0: the analogy between zero-agony instances and $\ell+1$1-SAT suggests a connection to Almost-2-SAT, which is FPT [RazgonO09], but the authors note stronger tools may be needed, possibly drawing on Chain-SAT techniques [KimKPW25]. It is not even known whether parameterizing by $\ell+1$2 yields FPT. Further open questions include the complexity for $\ell+1$3 with $\ell+1$4 change points per vertex, whether the problem is in FPT for $\ell+1$5 alone (open already for $\ell+1$6), and whether analogous rank-based classifications hold under a global budget on total rank changes rather than per-vertex budgets. Additionally, the NP-hardness for $\ell+1$7 relies on underlying digraphs containing directed cycles; whether hardness persists on temporal DAGs with three ranks is unresolved.

Conclusion

This work completes the complexity classification of Seg-Agony with respect to the number of ranks, exhibiting sharp phase transitions at $\ell+1$8, $\ell+1$9, and $$q(r,G,p_l) \coloneqq \sum_{t\in[\tau]}\sum_{(u,v)\in A_t} w_t(u,v)\cdot p_l(r(u,t)-r(v,t)).$$0 that mirror the satisfiability hierarchy: zero-agony feasibility behaves like 2-SAT for three ranks and like 3-SAT for four, while bounded-agony optimization corresponds to Max-SAT-type objectives. Combined with the FPT result for $$q(r,G,p_l) \coloneqq \sum_{t\in[\tau]}\sum_{(u,v)\in A_t} w_t(u,v)\cdot p_l(r(u,t)-r(v,t)).$$1, the paper provides a fine-grained map of tractability for temporal hierarchy discovery and isolates the parameterization by target agony at three ranks as the principal algorithmic question left unanswered.