Exact random covers of metric trees: balanced rounding, duality, and sharp thresholds

Abstract: Norin and Turcotte's asymptotically sharp bound for graph burning [J. Combin. Theory Ser. B 168 (2024), 208--235] led them to an exact random-cover conjecture for finite metric trees. Let $U[0,r]$ be the uniform probability measure on $[0,r]$. They conjectured that every finite metric tree $T$ of length $L\ge2r$ admits a probability measure on $0$-good ball covers whose expected radius measure is at most $(L/r)U[0,r]$. We prove the conjecture for every finite metric tree. We recast the bootstrapping calculation of Norin and Turcotte as a zero-error replacement certificate. The resulting local scale reduction, together with a three-piece decomposition and a macro-recursion, produces a fractional marked-ball cover with the exact radius budget. We then pass from the fractional cover to random finite covers by a compact rounding argument. For metric-tree balls, Tamir's balancedness theorem and standard balanced-matrix ideality provide the finite-dimensional integrality input. We also prove an arbitrary-budget duality criterion. If $0<R\le L$ and $β$ is a finite positive Borel measure on $[0,R]$, then $β$ dominates the expected radius measure of a random $0$-good cover if and only if $σ(T)\le\int_{[0,R]}\max_{v\in T}σ(B_T(v,s))\,dβ(s)$ for every finite positive Borel measure $σ$ on $T$; it is enough to test finite atomic measures. We use this criterion to extend the uniform range to every $r\le L-\operatorname{diam}(T)/2$, determine the exact range for equal-arm metric stars, and derive deterministic bounds, interval rigidity, and a diameter-defect stability estimate.

• The paper proves that every finite metric tree of length L ≥ 2r admits a probability measure on 0-good covers whose expected radius measure is exactly dominated by (L/r)U[0,r], eliminating the approximation, slack, and length losses in earlier results.
• The paper combines zero-error replacement certificates, geometric recursion, balanced set-cover rounding, and compactness to convert fractional marked-ball covers into random finite covers while preserving coordinatewise budget bounds.
• The paper establishes a budgeted duality criterion and sharp structural consequences, including diameter-dependent thresholds, exact bounds for equal-arm stars, interval rigidity, deterministic extraction, and stability estimates for near-interval trees.

Background and the conjecture

Graph burning models the spread of contagion in discrete rounds: each round a new source ignites while earlier fires advance one edge per round. The burning number $b(G)$ is the minimum number of rounds needed to burn all of $G$. Bonato, Janssen, and Roshanbin established $b(P_n)=\lceil\sqrt n\rceil$ and the spanning-tree reduction $b(G)=\min\{b(T)\}$ over spanning trees, which together motivate the Burning Number Conjecture: $b(G)\le\lceil\sqrt n\rceil$ for every connected graph on $n$ vertices. Norin and Turcotte proved the asymptotically sharp bound $b(G)\le(1+o(1))\sqrt n$, and their proof passes through a continuous relaxation on metric trees in which radii are randomized.

In that continuous setting, a tuple of radii $(r_1,\dots,r_m)$ covers a finite metric tree $T$ of length $L$ if suitable centers make the balls cover $G$0; it is $G$1-good if $G$2. For a probability measure $G$3 on covers, the expected radius measure $G$4 records how often each radius value appears across samples. Norin and Turcotte obtained, for $G$5, a measure on $G$6-covers with $G$7, carrying three losses: the $G$8 factor, the slack allowance, and the length hypothesis. They conjectured that for $G$9 the exact target $b(P_n)=\lceil\sqrt n\rceil$0 is always achievable — the only obstruction being intervals of length below $b(P_n)=\lceil\sqrt n\rceil$1, where an interval of length at most $b(P_n)=\lceil\sqrt n\rceil$2 forces the first moment to be at least $b(P_n)=\lceil\sqrt n\rceil$3, matching the target's first moment exactly.

The paper under review proves this conjecture in full: every finite metric tree of length $b(P_n)=\lceil\sqrt n\rceil$4 admits a probability measure on $b(P_n)=\lceil\sqrt n\rceil$5-good covers whose expected radius measure is dominated by $b(P_n)=\lceil\sqrt n\rceil$6 (2608.18967).

Proof architecture

The argument proceeds in three layers.

Trimming. A metric analogue of the Land–Lu sum-plus-maximum lemma shows that any radius multiset with $b(P_n)=\lceil\sqrt n\rceil$7 covers $b(P_n)=\lceil\sqrt n\rceil$8. Combined with a largest-first trimming rule, this extracts a Borel map that converts any random cover with radii bounded by $b(P_n)=\lceil\sqrt n\rceil$9 into a random $b(G)=\min\{b(T)\}$0-good cover without increasing the expected radius measure. This removes the slack allowance pointwise rather than in expectation.

Compact rounding. The paper separates rounding from tree geometry via an abstract theorem: for compact demand space $b(G)=\min\{b(T)\}$1, compact object space $b(G)=\min\{b(T)\}$2, and a closed ideal incidence relation $b(G)=\min\{b(T)\}$3, every finite fractional cover can be rounded to a probability measure on finite counting measures that almost surely covers $b(G)=\min\{b(T)\}$4 with intensity dominated by the fractional cover. Ideality means every extreme point of the set-covering polyhedron is integral. For metric-tree balls, Tamir's balancedness theorem (intersection matrices of neighborhood subtrees of a tree are balanced) plus Cornuéjols' balanced-matrix ideality supply the integrality input; tightness and weak convergence handle the passage from finitely many demands to all of $b(G)=\min\{b(T)\}$5.

Fractional construction. The core new contribution recasts Norin–Turcotte's bootstrapping algebra as a zero-error replacement certificate: given a random cover with expected radius measure $b(G)=\min\{b(T)\}$6 and moment budget $b(G)=\min\{b(T)\}$7, one obtains a certificate resolving part of the coverage while leaving unresolved scales among the $b(G)=\min\{b(T)\}$8, with exact budget equality. A local analysis using Norin–Turcotte's two-ball family shows every unresolved scale satisfies $b(G)=\min\{b(T)\}$9, where $b(G)\le\lceil\sqrt n\rceil$0 is the optimal two-piece split parameter ($b(G)\le\lceil\sqrt n\rceil$1). A three-piece decomposition (refining their branch-transfer argument) handles non-balanced splits by reducing to pairwise unions with fewer leaves; balanced splits halve the length. Strong induction on leaf count yields a macro-recursion in which unresolved subtrees shrink geometrically, so total unresolved weight tends to zero and the limiting measure is an exact fractional marked-ball cover. Compact rounding then produces the random finite cover.

A notable quantitative feature: at finite depth $b(G)\le\lceil\sqrt n\rceil$2 the construction already gives a fractional cover with expected radius measure at most $b(G)\le\lceil\sqrt n\rceil$3 once $b(G)\le\lceil\sqrt n\rceil$4, recovering Norin–Turcotte's approximate statement as an explicit geometrically convergent approximation to the exact result.

Budgeted duality

Beyond the uniform case, the paper establishes an exact duality criterion for arbitrary radius budgets. For a finite positive Borel measure $b(G)\le\lceil\sqrt n\rceil$5 on $b(G)\le\lceil\sqrt n\rceil$6, the following are equivalent:

1. There is a probability measure $b(G)\le\lceil\sqrt n\rceil$7 on $b(G)\le\lceil\sqrt n\rceil$8-good covers with $b(G)\le\lceil\sqrt n\rceil$9.
2. There is a fractional marked-ball cover $n$0 with push-forward radius measure $n$1.
3. Every finite positive Borel measure $n$2 on $n$3 satisfies the concentration inequality

$n$4

It suffices to test finite atomic measures $n$5. The proof combines a lifting lemma (random covers lift to fractional ones via Borel center selection), weak compactness of the budget-feasible set, a support-function computation for atomic demands, and Hahn–Banach separation with the finite-intersection property. This criterion is the workhorse for all subsequent sharpness results.

Sharp thresholds and rigidity consequences

Diameter extension. Writing $n$6 and $n$7, the uniform conclusion holds for all $n$8. Since non-interval trees satisfy $n$9, this strictly extends the range $b(G)\le(1+o(1))\sqrt n$0 guaranteed by the main theorem whenever $b(G)\le(1+o(1))\sqrt n$1 is not an interval.

Equal-arm stars. For the star $b(G)\le(1+o(1))\sqrt n$2 ($b(G)\le(1+o(1))\sqrt n$3 arms of length $b(G)\le(1+o(1))\sqrt n$4), the admissible range is exactly $b(G)\le(1+o(1))\sqrt n$5: necessity follows from a packing argument on arm endpoints (a ball of radius $b(G)\le(1+o(1))\sqrt n$6 contains at most one endpoint), showing the diameter range is best possible for stars with arbitrarily many leaves.

Interval rigidity. On a metric interval, domination by $b(G)\le(1+o(1))\sqrt n$7 forces equality of measures: almost every sampled cover has all positive radii summing to exactly $b(G)\le(1+o(1))\sqrt n$8, and the covering balls have pairwise disjoint interiors tiling the interval. This strengthens the known threshold $b(G)\le(1+o(1))\sqrt n$9 into a full structural characterization.

Deterministic extraction and stability. Any random cover with $(r_1,\dots,r_m)$0 yields deterministic covers optimizing Borel cost functions: in particular, $(r_1,\dots,r_m)$1-good covers with at most $(r_1,\dots,r_m)$2 balls, or with $(r_1,\dots,r_m)$3. A diameter-defect stability theorem bounds the deficit $(r_1,\dots,r_m)$4 by the diameter defect $(r_1,\dots,r_m)$5: its first moment is at most $(r_1,\dots,r_m)$6, its tail satisfies $(r_1,\dots,r_m)$7, and geometric overlap/excess quantities satisfy Markov-type tails $(r_1,\dots,r_m)$8. Thus near-interval trees admit nearly rigid covers.

Finally, the admissible set $(r_1,\dots,r_m)$9 always lies in $T$0 and contains $T$1, with exact descriptions for intervals and equal-arm stars.

Limitations and open questions

The authors are explicit that the main theorem does not prove the Burning Number Conjecture. In transferring from a metric tree to a discrete tree, centers may lie inside edges, and Norin–Turcotte's discretization enlarges radii; the continuous loss is removed but the final discrete step remains open. The compact rounding theorem is presented as a self-contained formulation tailored to coordinatewise intensity domination, not a new general theory of infinite covering — broader frameworks remain those of Aharoni–Holzman and Rademacher–Toriello–Vielma. The paper also leaves open the determination of $T$2 for general trees beyond intervals and equal-arm stars; the packing obstruction gives necessary conditions but the exact threshold for arbitrary trees is unresolved.

Conclusion

This paper resolves the Norin–Turcotte exact random-cover conjecture for all finite metric trees, replacing three losses in the asymptotic theory (the $T$3 factor, the slack allowance, and the length restriction) with an exact statement. Methodologically, it contributes a zero-error replacement-certificate formalism, a uniform local scale reduction bounded by half the optimal-split parameter, a three-piece macro-recursion, and a compact balanced-rounding bridge from fractional to random covers grounded in Tamir's balancedness theorem. The accompanying budgeted duality criterion, tested on atomic measures alone, yields sharp thresholds for stars, a diameter-dependent existence range shown to be tight, deterministic extraction bounds, and a rigidity/stability theory quantifying deviation from the interval case. The result completes the continuous layer of the burning-number program while leaving the discretization step as the remaining obstacle to the full conjecture.