MaxNAR in Graph Coloring & QKD Networks

• MaxNAR is a dual-context metric that measures the maximum number of satisfied requests in flexible list coloring and the worst-case number of disrupted requests in QKD networks.
• In flexible list coloring, maxNAR captures the achievable bound on satisfied vertex color requests, with lower bounds linked to the Hall ratio and graph degeneracy parameters.
• In QKD networks, maxNAR quantifies the vulnerability to single-point jamming attacks, influencing routing, wavelength assignment, and module usage decisions.

to=shell 天天中彩票公司 code: {"cmd":["bash","-lc","python - <<'PY'\nimport urllib.parse, urllib.request\nquery='all:\"Maximum Number of Affected Requests\" OR all:maxNAR'\nurl='http://export.arxiv.org/api/query?search_query='+urllib.parse.quote(query)+'&start=0&max_results=10'\nprint(url)\nprint(urllib.request.urlopen(url, timeout=20).read().decode('utf-8')[:4000])\nPY"]} to=shell code: {"command":"python - <<'PY'\nimport urllib.parse, urllib.request\nquery='all:\"Maximum Number of Affected Requests\" OR all:maxNAR'\nurl='http://export.arxiv.org/api/query?search_query='+urllib.parse.quote(query)+'&start=0&max_results=10'\nprint(url)\nprint(urllib.request.urlopen(url, timeout=20).read().decode('utf-8')[:4000])\nPY"} to=shell code: {"command":"python - <<'PY'\nimport urllib.parse, urllib.request\nquery='all:\"Maximum Number of Affected Requests\" OR all:maxNAR'\nurl='https://export.arxiv.org/api/query?search_query='+urllib.parse.quote(query)+'&start=0&max_results=5'\nprint(url)\ntry:\n data=urllib.request.urlopen(url, timeout=20).read().decode('utf-8')\n print(data[:4000])\nexcept Exception as e:\n print('ERR',e)\nPY"} to=shell code: {"command":"python - <<'PY'\nimport urllib.parse, urllib.request, sys\nquery='all:\"Maximum Number of Affected Requests\" OR all:maxNAR'\nurl='https://export.arxiv.org/api/query?search_query='+urllib.parse.quote(query)+'&start=0&max_results=5'\nprint(url)\nsys.stdout.flush()\ntry:\n resp=urllib.request.urlopen(url, timeout=20)\n data=resp.read().decode('utf-8')\n print('LEN', len(data))\n print(data[:1000])\nexcept Exception as e:\n print('ERR',repr(e))\nPY"} to=shell code: {"command":"echo hello"} Maximum Number of Affected Requests, abbreviated as maxNAR, is an acronym used in two distinct arXiv research contexts. In flexible list coloring, it denotes the maximum number of requested vertex colors that can be simultaneously realized by a proper list coloring for a fixed instance $(G,L,r)$. In quantum-key-distribution (QKD) network design, it denotes the maximum, over all possible single attack locations in a timeslot, of the number of served requests disrupted by that attack. The two usages share a request-centric extremal viewpoint, but they differ fundamentally in semantics: the graph-theoretic quantity is a feasibility-maximization parameter, whereas the networking quantity is a worst-case vulnerability metric minimized by routing, wavelength assignment, and caching decisions (Kaul et al., 2022, Li et al., 14 Aug 2025).

1. Terminological scope and dual usage

The shared acronym conceals a substantive conceptual split. In flexible list coloring, “requests” are preferred colors assigned to a nonempty subset of vertices, and maxNAR measures how many of those preferences can be satisfied by some proper $L$-coloring. In QKD networks, “requests” are traffic demands for secret-key service, and maxNAR measures how many served demands can be interrupted by a single physical-layer jamming attack. A common source of confusion is therefore terminological rather than mathematical: the same acronym names an extremal count in both settings, but the counted events are opposite in operational meaning—satisfied requests in one literature and affected requests in the other.

Context	Formal object	Operational meaning
Flexible list coloring	$\mathrm{maxNAR}(G,L,r)$	Maximum number of requests satisfied by a proper list coloring
QKD routing and wavelength assignment	$\text{maxNAR}_t$	Worst-case number of served requests affected by one attack in timeslot $t$

This bifurcation also changes the role of uniform guarantees. In list coloring, one studies lower bounds of the form $\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$ that hold for all $k$-assignments and all requests. In QKD, one studies designs that make the per-timeslot worst case as small as possible, typically through isolation of lightpaths, architectural choices between Optical Bypass (OB) and Trusted Relay (TR), and the use of Quantum Key Pools (QKPs).

2. maxNAR in flexible list coloring

Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Let $G$ be a finite, simple graph, let $L:V(G)\to 2^C$ be a list assignment, and let $r$ be a request with non-empty domain $L$0 such that $L$1 for each $L$2. A proper $L$3-coloring is a map $L$4 with $L$5 for all $L$6 and distinct colors on adjacent vertices. The triple $L$7 is $L$8-satisfiable if there exists such an $L$9 satisfying at least an $\mathrm{maxNAR}(G,L,r)$0-fraction of the requests:

$\mathrm{maxNAR}(G,L,r)$1

For a fixed instance, the corresponding maxNAR is

$\mathrm{maxNAR}(G,L,r)$2

A graph $\mathrm{maxNAR}(G,L,r)$3 is $\mathrm{maxNAR}(G,L,r)$4-flexible if $\mathrm{maxNAR}(G,L,r)$5 is $\mathrm{maxNAR}(G,L,r)$6-satisfiable for every $\mathrm{maxNAR}(G,L,r)$7-assignment $\mathrm{maxNAR}(G,L,r)$8 and every request $\mathrm{maxNAR}(G,L,r)$9. Equivalently,

$\text{maxNAR}_t$0

for all $\text{maxNAR}_t$1-assignments $\text{maxNAR}_t$2 and all requests $\text{maxNAR}_t$3. In this formulation, maxNAR is instance-specific, while $\text{maxNAR}_t$4-flexibility supplies uniform lower bounds independent of the particular lists and requests except through $\text{maxNAR}_t$5 (Kaul et al., 2022).

The paper strengthens earlier work in two directions. First, Dvořák–Norin–Postle had shown that if $\text{maxNAR}_t$6 is prime, $\text{maxNAR}_t$7 is $\text{maxNAR}_t$8-degenerate, and $\text{maxNAR}_t$9, then every $t$0-assignment permits the single request to be satisfied. The later paper extends this to all $t$1 for bipartite $t$2-degenerate graphs: for any bipartite $t$3-degenerate graph $t$4, any $t$5-assignment $t$6, and any request $t$7 with $t$8, the instance is $t$9-satisfiable. Second, it improves the general many-request lower bound by proving that every $\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$0-degenerate graph is

$\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$1

Hence for every $\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$2-assignment and every request,

$\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$3

3. Structural bounds, Hall ratio, and the optimal fraction

A central question is how large the guaranteed fraction $\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$4 can be. The paper identifies the precise universal obstruction through the Hall ratio

$\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$5

where $\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$6 is the independence number of $\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$7. It proves that there exists some $\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$8 such that $\mathrm{maxNAR}(G,L,r)\ge \epsilon |D|$9 is $k$0-flexible if and only if

$k$1

Equivalently, $k$2 is not $k$3-flexible for any $k$4 whenever $k$5. The obstruction is immediate when all vertices in $k$6 request the same color, since any proper coloring can satisfy at most $k$7 of them. Conversely, when $k$8, one can color a maximum independent set in $k$9 with the requested color and greedily extend to the remaining vertices, so the bound $G$0 is achievable with sufficiently large lists (Kaul et al., 2022).

This leads to the list flexibility number

$G$1

the smallest list size that guarantees the optimal universal fraction. The paper establishes the inequalities

$G$2

It also records exact values for several families: $G$3 These identities locate maxNAR within a broader hierarchy involving the chromatic number, list chromatic number, degeneracy, and list packing number. Proposition 14 further shows that every graph $G$4 is $G$5-flexible, where $G$6 is the list packing number.

The paper also shows that $G$7 need not remain within an additive constant of $G$8: there is no universal constant $G$9 such that

$L:V(G)\to 2^C$0

for all graphs. This separates optimal-fraction flexibility from ordinary list colorability.

4. Constructive methods and representative graph classes

Several constructive mechanisms underlie the lower bounds. The proof of the general degeneracy theorem uses a randomized process on a degeneracy ordering $L:V(G)\to 2^C$1. When vertex $L:V(G)\to 2^C$2 is processed, the algorithm chooses uniformly from the first two colors in $L:V(G)\to 2^C$3 that are unused among its at most $L:V(G)\to 2^C$4 earlier neighbors. If $L:V(G)\to 2^C$5 has requested color $L:V(G)\to 2^C$6, then the probability that $L:V(G)\to 2^C$7 survives among the available colors is at least $L:V(G)\to 2^C$8, and conditional on survival it is chosen with probability $L:V(G)\to 2^C$9. Therefore each request is satisfied with probability at least $r$0, and linearity of expectation yields a coloring satisfying at least $r$1 requests. The exposition notes that derandomization can be pursued by the method of conditional expectations (Kaul et al., 2022).

For single-request realizability in bipartite $r$2-degenerate graphs, the proof uses a tailored orientation together with the Alon–Tarsi Theorem. The argument constructs a directed version with bounded out-degrees and uses edge-disjoint paths, via Menger’s Theorem, to reverse orientations so that the requested color at the chosen vertex becomes realizable. The required edge-disjoint paths can be found by max-flow, giving a polynomial-time route to the orientation. A separate constructive device, expressed in Proposition 5, uses a coloring of $r$3: if $r$4 is $r$5-choosable, then $r$6 is $r$7-flexible. Corollary 6 specializes this via a degeneracy bound on the square graph.

The paper supplies several graph-class consequences. For $r$8-degenerate graphs with $r$9, one always has

$L$00

Thus for $L$01, $L$02, and $L$03, the guaranteed fractions are $L$04, $L$05, and $L$06, respectively. Trees attain the optimal Hall-ratio fraction because $L$07 and $L$08. For grids $L$09, the paper states $L$10 and proves $L$11-flexibility; for ladders $L$12, the optimal fraction $L$13 is achieved with $L$14, so $L$15. Complete graphs give the opposite extreme: since $L$16, one has $L$17, and achieving the optimal universal fraction $L$18 requires $L$19.

5. maxNAR in QKD networks and attack-radius minimization

In QKD networking, the physical network is modeled as a directed, weighted graph

$L$20

with quantum channels indexed by $L$21 and timeslots $L$22. Requests are key demands $L$23, each with a source–destination pair and required key rate $L$24. The paper introduces maxNAR as a security metric for a single high-power jamming attack per timeslot. The attack model depends on architecture. Under TR, quantum signals are regenerated at each trusted relay hop, so an attack on a physical link disrupts only requests traversing that same directed physical link. Under OB, signals pass optically through intermediate nodes without regeneration, so an attack on a link along an OB lightpath can propagate downstream within the same bypass chain and disrupt other requests that use any link in that OB lightpath. Requests served purely from QKP caches are not affected by physical-layer jamming during that timeslot because no quantum transmission occurs on the fiber (Li et al., 14 Aug 2025).

At timeslot $L$25, the candidate attack locations are the active routes

$L$26

For a request $L$27 and an attack location $L$28, the indicator of disruption is

$L$29

The per-timeslot maxNAR is then

$L$30

The optimization objective in Routing and Wavelength Assignment with Minimal Attack Radius (RWA-MAR) is

$L$31

The paper also gives an equivalent link-sharing interpretation: maxNAR is “the maximum number of requests any one request is link-sharing with,” where link-sharing means traversal of at least one common directed physical link. This equivalence emphasizes that the vulnerability metric is induced by overlap structure in the deployed routing plan. A common misconception is to read maxNAR as an average-case damage measure; in fact it is explicitly a worst-case single-attack count, evaluated per timeslot.

6. ILP formulation, heuristic design, and reported performance

The RWA-MAR model is formulated as an ILP with variables for auxiliary-link flow $L$32, in-flight quantum realization $L$33, QKP realization $L$34, route selection $L$35, QKD-path usage $L$36, key generation $L$37, QKP storage $L$38, cache-provided key rate $L$39, physical-link usage $L$40, route activity $L$41, attack-effect indicators $L$42, and service indicators $L$43. The objective is $L$44, subject to flow conservation, flow composition, module capacity, route selection and exclusivity, key-rate consistency, QKP storage dynamics, and a linearization of the maxNAR computation. In particular, the model enforces

$L$45

so minimization drives $L$46 to the maximum affected-request count over active routes (Li et al., 14 Aug 2025).

The proposed scalable heuristic, denoted Min-maxNAR, operates per timeslot by building shortest-path initial solutions, using a tunable parameter $L$47 to control the initial OB-versus-TR preference, allocating channels and modules, updating QKP levels, and then applying Tabu-search rerouting. Smaller $L$48 favors OB, which uses fewer modules but may raise maxNAR; larger $L$49 favors TR, which usually lowers maxNAR but consumes more modules. Opportunistic caching routes additional lightpaths, while resources are available, to store keys for future timeslots. The heuristic then iterates a fixed number $L$50 of times, rerouting randomly selected lightpaths over $L$51-shortest alternatives and choosing the neighbor with the lowest current maxNAR, with ties broken by avgNAR or resource usage.

The reported evaluation uses a PoliQi ring topology and an NSF topology. On PoliQi, both the ILP and the heuristic achieve optimal $L$52 for OBTR and TR, while OB yields the highest maxNAR; the ILP requires more than 10 hours, whereas the heuristic matches the optimum in approximately 5 seconds. On the NSF topology with $L$53, the heuristic reduces maxNAR by approximately $L$54 versus the baseline for OB and OBTR0. TR achieves the lowest maxNAR but exhausts resources after about 100 requests because of its higher module consumption. For avgNAR, the heuristic improves by up to $L$55 versus the baseline in OBTR. When $L$56 is raised from 0 to 80, OBTR80 reduces maxNAR by $L$57 versus OBTR0 and by $L$58 versus OB; OBTR0 reduces maxNAR by $L$59 versus OB. In the same comparison, OBTR80 improves avgNAR by $L$60 versus OBTR0, and TR remains the lowest-avgNAR architecture, reported as $L$61 better than OBTR80. Resource trade-offs remain explicit throughout: TR exhausts resources, OBTR80 uses $L$62 fewer modules than TR but $L$63 more than OBTR0, and OBTR0 uses $L$64 more than OB while achieving better maxNAR and avgNAR. In a multi-timeslot experiment with OBTR0, 145 requests, and 5 timeslots, the heuristic lowers maxNAR in timeslot 1 from 37 to 32 and then benefits from stored keys in timeslots 2–4, where maxNAR stabilizes around 1–2 before a slight rise in timeslot 5 as caches deplete.

These results identify the main design tension in the QKD interpretation of maxNAR. OB is module-efficient but enlarges the attack footprint because a jamming event can propagate along an optical bypass chain. TR localizes failures but is module-intensive. QKP decouples service from active quantum transmission in the current timeslot and therefore shrinks the set of requests that can be affected. The resulting metric is not merely descriptive: it is embedded directly into routing, wavelength assignment, storage planning, and architecture selection as the security objective minimized by the system designer.