Online Service with Per-Batch Maximum Delay

Abstract: We study online service with one maximum-waiting-time charge per service batch. Requests arrive at points of a finite metric, and a mobile server pays for its movement and, for each service walk, the maximum waiting time among the requests served by that walk. We distinguish elective service, where an encountered request may be left pending, from automatic service, where every encounter serves it. Although the two semantics have different optimal schedule structures, we prove that their offline optimal values are equal. On a finite line and on an explicitly represented weighted tree, the common offline value is computable by polynomial-time dynamic programming, whereas exact optimization on arbitrary finite metrics is NP-hard. For the online problem, we prove a metric-independent group-trajectory certificate lemma that charges spatially separated request groups to two parity classes of time windows. It yields deterministic polynomial-time competitive ratios 10 on a line, 12 on a weighted tree, and 20 on an arbitrary finite metric, under both service semantics. With an exact metric-Steiner-tree oracle, the general-metric ratio improves to 12. The polynomial algorithm uses a half-scaled running maximum of terminal-MST weights; the running maximum is necessary because terminal MST weight is not monotone under new arrivals. A fixed two-point line gives a deterministic visible-service lower bound of 3 for every metric class above. Finally, when request locations are hidden until visited, dyadic exploration is 84-competitive on a known finite line. This phenomenon is line-specific: one hidden request gives deterministic and randomized lower bounds 3 and 2 on a line, while a d-leaf unit star gives lower bounds 2d-1 and d.

• The paper introduces a mobile-server model that charges each service batch by route length plus its maximum request delay, capturing tail-latency objectives while breaking the additive structure of classical service-with-delay problems.
• The paper proves exact polynomial-time offline algorithms on finite lines and weighted trees, establishes NP-hardness on general finite metrics, and shows that elective and automatic service have identical offline optimum values despite different feasible schedules.
• The paper develops group-trajectory certificates yielding deterministic visible-online ratios of 10 on lines, 12 on trees, and 20 on general metrics, while spatially blind service achieves ratio 84 on known finite lines but suffers linear lower bounds on high-degree stars.

This paper introduces and analyzes an online routing problem in which a mobile server on a finite metric space serves requests that arrive over time, and each nonempty service batch is charged once with the maximum waiting time among its participants, rather than the sum of individual delays. The cost of a service walk executed at time $t$ serving batch $B$ is $len(P)+\max_{q\in B}(t-a(q))$, and the objective sums this quantity over all service events. This per-batch maximum-delay objective models tail-latency and incident-based accounting regimes and destroys the request-additive structure underlying classical online service with delay (2608.18577).

Offline tractability and semantic equivalence

The paper first establishes two metric-independent structural facts. Under elective service—where an encountered request may be left pending—every feasible schedule can be transformed into one of no greater cost whose service batches are consecutive blocks of complete arrival epochs, each served at its latest arrival time. On a finite line this yields an endpoint-indexed dynamic program running in $O(Q+m^2N)$ time (with $Q$ request occurrences, $m$ arrival epochs, $N$ metric points), using a line distance transform to evaluate block-transition costs in constant time. On an explicitly represented weighted tree, a tree route kernel based on the identity $w(T_{H,x,y})=w(H)+(a+b+c)/2$ gives an exact $O(Q+mn+m^2n^2)$ dynamic program. By contrast, exact offline optimization on arbitrary finite metrics is NP-hard under both semantics, even when every request arrives at time zero, via a reduction from Hamiltonian Path on shortest-path metrics.

A structurally striking result is that automatic service—where every encounter forces a serve—does not admit the consecutive-block normal form: a two-point counterexample shows optimal automatic schedules can be nonconsecutive while every consecutive-block schedule pays strictly more movement. Nevertheless, the offline optimal values coincide on every finite metric: any normalized elective schedule can be replayed automatically, since premature stationary hits have zero delay and can only shrink later batches. Consequently the elective line DP computes the exact automatic optimum as well. The proof relies on instantaneous movement, arrival-before-action ordering, zero-cost zero-age co-located service, and deletion monotonicity of a batch maximum; it does not extend to positive travel time or request-specific waiting rates.

A universal lower bound

On the two-point line $\{0,D\}$, every deterministic visible online algorithm has competitive ratio at least $B$0, robust to additive constants, under either service semantics. Since the same instance is simultaneously a line, a tree, and an arbitrary finite metric, this lower bound applies to every visible setting considered. It isolates the irreducible decision of whether to move now or preserve the current endpoint.

Group-trajectory certificates

The central analytical device is a weighted group-certificate lemma. For ordered, interior-disjoint time windows containing pairwise identity-disjoint request groups $B$1, any weights $B$2 bounded by both the window length and the ambient Steiner-tree weight $B$3 satisfy $B$4 for every feasible elective schedule $B$5. Strictly interior groups are paid by the temporal span of the offline batch containing them; boundary groups induce trajectory connectors whose arclength congestion is at most two, because each offline batch has only two boundary slots and each interbatch cut splits at most one contiguous group. The coefficient two is tight. Combined with a parity split of consecutive online transitions—which makes same-color certificate groups identity-disjoint—each color class forms a genuine extracted subinstance lower bound, yielding $B$6.

Visible upper bounds across metric classes

Applying this machinery gives deterministic polynomial-time guarantees summarized below; all hold under both elective and automatic service.

Setting	Offline	Visible online
Finite line	exact polynomial	LB 3 det.; UB 10
Weighted tree	exact polynomial	LB 3 det.; UB 12
Finite metric	NP-hard	LB 3 det.; UB 20 polynomial, 12 with oracle

The line algorithm uses a global-hull rule that triggers when the oldest pending request has paid for the pending hull diameter plus its separation from the previous hull; pair certificates over two parity classes give ratio 10. On trees, the exact minimal subtree is monotone and explicit, giving ratio 12 via a subtree-trigger algorithm with open-DFS routes. On general metrics, ambient Steiner weight is monotone but hard to compute; an exact-Steiner oracle yields ratio 12 as a structural benchmark. For polynomial time, terminal-MST weight is within factor two of Steiner weight but is not monotone—a new terminal can become a useful connector and decrease it, as demonstrated on a star metric. The remedy is a phase-local running maximum scaled by one half: monotone, causal, and still a valid spatial certificate. This yields ratio 20 in time $B$7. A general transfer principle shows any $B$8-approximate terminal-spanning structure gives ratio $B$9.

Spatially blind service

The paper also studies a variant orthogonal to standard non-clairvoyance: release times, identities, and waiting clocks are visible, but locations remain hidden until visited. On a known finite line, dyadic anchor exploration with closed sweeps at radii $len(P)+\max_{q\in B}(t-a(q))$0, followed by a cleanup batch in the same atomic action, achieves ratio 84 for both elective service and automatic service (the latter via persistent virtual ghosts). The phase bound rests on the scale relation $len(P)+\max_{q\in B}(t-a(q))$1 between the farthest served request and the successful sweep radius.

This constant-factor behavior on lines is topology-specific and does not persist elsewhere. Even with a single hidden request, a unit star with $len(P)+\max_{q\in B}(t-a(q))$2 leaves forces deterministic ratio $len(P)+\max_{q\in B}(t-a(q))$3 and randomized ratio $len(P)+\max_{q\in B}(t-a(q))$4 against an oblivious adversary, while on a three-point line the corresponding bounds are 3 and 2. The blind-line analysis also depends substantively on the finite known metric, a positive minimum scale $len(P)+\max_{q\in B}(t-a(q))$5, and the convention that one instantaneous sweep is one atomic service event; interleaving arrivals inside sweeps would break the exploration argument.

Limitations and open questions

Several gaps and assumptions are explicit. Quantitatively, the visible deterministic ratio lies between 3 and the bounds 10/12/20, and no randomized improvement is known; the parity certificate does not yet expose a distribution over phase offsets analogous to global-shift arguments from static aggregation. Whether the randomized gain in two-sided search survives dynamic batching and parity certificates on the blind line is unresolved. Computationally, since exact offline optimization is NP-hard even for simultaneous arrivals, the natural question is whether route approximation can be combined with endpoint-indexed schedule approximation without compounding errors across batches, and whether structures better than the factor-two terminal MST improve the polynomial ratio 20. Model extensions—request-specific rates, general delay functions, positive travel times, stochastic arrivals—are outside the scope of the current certificates; several natural threshold rules fail already on a two-point line under such generalizations.

Conclusion

The paper shows that replacing additive delay with one per-batch maximum fundamentally restructures mobile-service problems while leaving them tractable in strong senses. Elective and automatic semantics differ in feasible event structure but share equal offline values; offline optimization is exactly polynomial on lines and trees but NP-hard on arbitrary metrics; and a single group-trajectory certificate lemma yields constant competitive ratios of 10, 12, and 20 across these classes, with blindness costing only a constant on lines but forcing linear loss on high-degree stars.