Fault-Tolerant Matroid Bases

Fedor V. Fomin; Laure Morelle; Matthias Bentert; Petr A. Golovach

REVIEW 3 major objections 3 minor 20 references

Fault-Tolerant Matroid Bases

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves a fixed-parameter tractable algorithm for finding a minimum set of matroid elements that stays spanning after any k deletions, parameterized by k plus the matroid's rank, and shows the parameterization is tight.

desk verdict The k+r FPT algorithm and the rank-based dichotomy are real; Theorem 1 holds, but Lemma 2 needs an explicit spanning invariant. read the letter →

arxiv 2506.22010 v1 pith:CT4VEK5Y submitted 2025-06-27 cs.DS cs.DM

Matthias Bentert , Fedor V. Fomin , Petr A. Golovach , Laure Morelle This is my paper · ORCID

classification cs.DScs.DM MSC 05B3568Q1768Q2568R05

keywords matroidsfaulttolerancefixed-parametertractableh-uniformsetsmatroidrankindependenceoracleNP-hardnessbulk-robustoptimization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces Fault-Tolerant Basis: find a minimum-size set B of matroid elements such that after deleting any k of them, the rest still spans the entire ground set. The main result is a fixed-parameter tractable algorithm parameterized by both k and the matroid's rank r, running in (kr)^{O($kr^{4}$)} · $n^{{O(1)}}$ time on matroids given by independence oracles. This two-variable parameterization is shown to be tight: the problem is NP-hard already for k=1, para-NP-hard for rank at least 3, and polynomial-time solvable for rank at most 2. A sympathetic reader would care because the problem unifies fault-tolerance for vector spaces, graphs, and matching-like structures, so the algorithm and the hardness dichotomy give a single answer across these settings.

What carries the argument

The key machinery is the h-uniform set and the recursive core construction built on it. A set X is h-uniform if rank(X)=|X|=h and every h-element subset of X has rank h; Observation 2 says a set of size k+r is k-fault-tolerant exactly when it is r-uniform. The algorithm Important(M,X) greedily builds a maximal h-uniform set Z inside cl(X); if Z is large enough, Lemma 1 guarantees that a k-fault-tolerant basis can be moved so that its part inside cl(X) lies in Z itself; otherwise the algorithm recurses on all (h−1)-element subsets S of Z and outputs the union of the returned sets, relying on the identity cl(X)=∪_{S⊆Z, |S|=h−1} cl(S) to cover the whole closure. The bounded size of Z, about (h−1)[(k+1)r]^{r−1}+(k+1)r, is what keeps the search space and the recursion depth under control.

What would settle it

Run the algorithm Important(M,X) on a small matroid (for example, a rank-3 matroid with a 3-element circuit and a spanning element) and check after each recursive call whether cl(Y_S) contains S for every chosen (h−1)-subset S; if any call returns Y_S with S not contained in cl(Y_S), the identity cl(X)=∪ cl(Y_S) fails and the proof of Claim 1 collapses, which would mean the FPT algorithm can output a set W that excludes all k-fault-tolerant bases.

Watch

Extended reading notes

Core claim

The central claim is that k-fault-tolerant matroid bases can be found efficiently when both the failure count k and the matroid rank r are small, and that this joint parameterization is the right one. The algorithm constructs a small core set W of elements such that, if any k-fault-tolerant basis exists, one exists entirely inside W; it then brute-forces over candidate subsets of W. The construction is driven by h-uniform sets, sets where every h-element subset has rank h, and uses the fact that a set of size k+r is a k-fault-tolerant basis exactly when it is r-uniform. The paper also proves that fixing k alone or r alone leaves the problem intractable, giving a clean dichotomy that separates the easy rank-2 case from the hard rank-3 and above case.

Load-bearing premise

The recursive core construction assumes that each call Important(M,S) returns a set Y_S whose span still contains S, so that the closures of the (h−1)-element subsets S of Z genuinely cover the closure of X; if that spanning invariant fails for some S, the core set W may miss every k-fault-tolerant basis.

Editorial extensions

If this is right

If the central claim is correct, then Fault-Tolerant Basis is fixed-parameter tractable with parameters k and r, meaning instances where both the number of failures and the matroid rank are small can be solved in time that grows polynomially with the ground set size n.
The tightness results imply that no FPT algorithm exists for the parameter k alone (since the problem is W[1]-hard for k) and no FPT or even polynomial algorithm exists for the parameter r alone for r ≥ 3, unless P = NP.
The rank-2 weighted case being polynomial-time solvable means that in the plane (rank 2 linear matroids), fault-tolerant bases can be computed in O(n^2) time even with weights.
The core-set reduction also shows the problem is in XP when parameterized by k, since after shrinking to W one can brute-force over subsets of size at most (k+1)r.
The equivalence between k-fault-tolerant bases and r-uniform sets of size k+r gives a direct certificate of fault-tolerance that could be checked without enumerating all failure sets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the h-uniform core-shrinking technique may transfer to the broader bulk-robust matroid basis model, where failure scenarios are arbitrary interdiction sets rather than all k-subsets; the recursion only needs the identity cl(X)=∪ cl(S) over (h−1)-subsets, which holds in any matroid, so a structured family of interdiction sets could be handled similarly if the core construction can
A testable extension: for linear matroids over a fixed field, one could try to derandomize the core construction or improve the running time to (kr)^{O(kr)} using more careful counting, which would be significant because the current exponent kr^4 is large even for moderate k and r.
The author leaves open whether the theorem extends to the weighted variant; if the exchange argument of Lemma 1 fails in the weighted setting, a plausible reason is that replacing an element outside X by an element inside X changes the weight, suggesting that weighted fault-tolerant bases may be genuinely harder than the unweighted case.
The NP-hardness for rank 3, based on planar point sets in general position, suggests that the rank-3 case is already as hard as a geometric packing problem, so the polynomial rank-2 result is likely the boundary of tractability for fixed ranks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main result is real: Fault-Tolerant Basis admits an FPT algorithm parameterized by k+r, with a matching dichotomy (NP-hard for k=1, para-NP-hard for r>=3, poly for r<=2). I think Theorem 1 survives. The h-uniform core lemma is a genuinely new device, and the reduction of the problem to a bounded core W is clean.

What the paper does well: it identifies a natural matroid formulation, relates it to bulk-robustness and k-edge-connected spanning subgraph, gives tight examples for the size bounds, and uses known hardness results correctly. The exchange argument in Lemma 1 is sound.

Soft spots. There is one real gap, exactly where the reader points: Lemma 2's Claim 1 asserts cl(X) = union over S of cl(Y_S), where Y_S is the recursive output on S. This requires the invariant that Y_S spans S. The paper never states or proves it. It does hold—the base case works, the big-Z case returns a superset of S, and the union case inherits the invariant from recursive calls—but it needs to be explicit. As written, a reader cannot verify the equality.

The Proposition 3 concern is less serious than the reviewer's note suggests. The proof never claims each X_i has size r; it only needs each X_i to be independent, so |B'| <= (k+1)r. The greedy deletion step is valid for that bound. The proof is terse and could use a sentence saying each |X_i| <= r and the recursion is well-defined, but that is a clarification, not a fix.

The running-time analysis is believable; the oracle model is explicit. Lower bounds are built on published reductions used as black boxes; no fitting or post hoc selection.

Who benefits: people working on parameterized algorithms for matroid problems, and anyone using fault-tolerant basis notions. It is not a practice-changing paper, but it settles a natural problem and introduces a reusable technique.

Recommendation: this deserves a serious referee. If I were handling it, I would ask for a revision that states and proves the spanning invariant in Lemma 2 and expands the few terse lines in Proposition 3. Conditional accept.

Referee Report

3 major / 3 minor

Summary. This paper studies k-fault-tolerant bases: minimum-cardinality subsets B of a matroid ground set such that rank(B \ F) = rank(M) for every F ⊆ B with |F| ≤ k. The main contribution is an FPT algorithm for Fault-Tolerant Basis parameterized by both k and the rank r, with running time (kr)^{O(kr^4)} · n^{O(1)} in the independence-oracle model. The algorithm constructs a bounded "important" set W via a recursive decomposition based on h-uniform sets, then enumerates candidate bases within W. The paper also proves the size bounds r + k ≤ |B| ≤ (k + 1)r, gives an O(n^2)-time algorithm for weighted partition matroids and for rank at most two, and presents lower bounds: W[1]-hardness for parameter k on linear matroids, NP-hardness for k = 1 on graphic matroids, and a claimed NP-hardness dichotomy for fixed rank r ≥ 3.

Significance. If the proof gaps are repaired, the FPT algorithm is a solid and nontrivial contribution: it gives a unifying treatment of fault-tolerant spanning structures in matroids and a tight dichotomy by k + r. The core derivations are from first principles; there is no circularity and no fitted-parameter tuning. The hardness parts use established external results as building blocks. The main reservation is that the fixed-rank lower-bound proof, as written, does not establish the stated dichotomy because the constructed matroid need not have the required rank.

major comments (3)

[§6, proof of Theorem 2] The reduction from General Position Subset Selection does not ensure rank(M) = 3. The text says "Note that rank(M) ≤ 3" and then applies Proposition 3 to conclude that any k-fault-tolerant basis has size at least p = k + 3; this uses rank(M) = 3, which is not guaranteed. If the input point set P is collinear, the lifted vectors have rank 2, and a rank-2 matroid on that ground set has a k-fault-tolerant basis of size k + 2 = p - 1, which is at most the budget p. Thus a no-instance with collinear points can map to a yes-instance. The same issue propagates to the extension r > 3. The authors need either to prove that General Position Subset Selection remains hard when the point set has rank 3, or to give a rank-forcing construction that preserves the answer; the current argument is not a valid reduction to the rank-r problem.
[Lemma 2, proof of Claim 1] The identity cl(X) = ∪_{S ⊆ Z, |S| = h-1} cl(Y_S) is asserted without proving that each recursive output Y_S spans its input S. The induction hypothesis stated in Claim 1 covers only properties (i) and (ii); it does not include cl(Y_S) = cl(S). The base case h = 1 works because any nonempty subset of cl({e}) spans {e}, but for h > 1 the spanning property must be proved by induction and stated explicitly. If it failed, the union of cl(Y_S) could be a proper subset of cl(X), so the bounded core W could fail to contain a k-fault-tolerant basis. This is load-bearing because Lemma 2 is the engine of Theorem 1.
[Proposition 3] The upper-bound proof says "as rank(B) = rank(M) and |B| > (k+1)r, such sets X_0, ..., X_k exist," where each X_i is an inclusion-maximal independent set in the remaining part of B. This is not justified: after removing previous blocks, the remaining set may have rank strictly smaller than r, so an inclusion-maximal independent set X_i can have size less than r. The proof should allow each X_i to have size at most r; then |B'| = Σ |X_i| ≤ (k+1)r < |B|, and the pigeonhole argument still works because B \ B' ⊆ cl(X_i) for the chosen i. The paper should state this correction explicitly.

minor comments (3)

[Lemma 2, Claim 1] In the proof of property (ii), the last displayed line ends with "= ∪_{i=1}^j Y_j"; the index should be i, i.e., "= ∪_{i=1}^j Y_i".
[§6, extension to r > 3] The reduction for r > 3 does not specify the budget b for the constructed instance. The proof should state b = (r - 3)(k + 1) + p and justify that any k-fault-tolerant basis meeting this budget must contain all newly added vectors and exactly p original vectors.
[Observation 1] The statement says "For every integer k ≥ 1, it is NP-hard..." but the cited hardness results are of different types: [13] handles k = 1 and [16] gives inapproximability for higher connectivity. The authors should clarify whether k is fixed or part of the input in this observation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the FPT algorithm and the lower-bound reductions are derived from first principles or rely on independent published results.

full rationale

The paper's central FPT construction (Lemma 2 and Theorem 1) computes a bounded important set W using matroid rank, closure, h-uniform sets, and the size bound from Proposition 3; it never defines the target k-fault-tolerant basis in terms of the algorithm's own output, nor does it fit any parameter to the answer. The only cited works with author overlap, [14] and [4], are not load-bearing in a circular sense: [14] supplies an external W[1]-hardness result for a different problem (Rank h-Reduction), and [4] is contextual related work. The NP-hardness reductions use external results of Fernandes [13], Gabow et al. [16], and Froese et al. [15] as standard building blocks. The proof gaps noted in the reader's take, namely the implicit spanning invariant in Claim 1 and the block-size justification in Proposition 3, are correctness concerns that are patchable, not circular identifications. No equation in the paper reduces by construction to its own input, and no fitted quantity is renamed as a prediction. Accordingly, the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The central claim rests on standard matroid axioms, the oracle model, and three external hardness results. The only internal gap is an unstated spanning invariant in Lemma 2, which we list under red flags.

assumptions (6)

standard math Matroid independence and closure axioms (I1)-(I3), (CL1)-(CL4)
Used throughout the paper as background; cited to Oxley [20].
domain assumption Independence oracle model: each oracle query answered in O(1) time
The FPT algorithm's input model; standard in matroid algorithms.
standard math Rank h-Reduction is W[1]-hard parameterized by k (Fomin et al. [14])
External published theorem used as black box for Proposition 1. Two current authors overlap with [14], but the result is peer-reviewed and independent of this paper.
standard math General Position Subset Selection is NP-hard for rational planar points (Froese et al. [15])
External NP-hardness result used for the r≥3 lower bound.
standard math An n-vertex graph has a 2-edge-connected spanning subgraph with at most n edges iff it is Hamiltonian (Fernandes [13]); higher connectivity hardness from Gabow et al. [16]
Basis of Observation 1 for the k=1 NP-hardness and its generalization.
standard math Linear matroid truncations can be represented in polynomial time (Lokshtanov et al. [19])
Needed for Proposition 1's reduction to produce a linear matroid.

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Cite this review

Pith. "Pith review of Fault-Tolerant Matroid Bases." pith.science (2026). https://pith.science/paper/CT4VEK5Y

@misc{pith2026250622010,
  author       = {Pith},
  title        = {Pith review of: Fault-Tolerant Matroid Bases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CT4VEK5Y}},
  note         = {Machine review of arXiv:2506.22010}
}

read the original abstract

We investigate the problem of constructing fault-tolerant bases in matroids. Given a matroid M and a redundancy parameter k, a k-fault-tolerant basis is a minimum-size set of elements such that, even after the removal of any k elements, the remaining subset still spans the entire ground set. Since matroids generalize linear independence across structures such as vector spaces, graphs, and set systems, this problem unifies and extends several fault-tolerant concepts appearing in prior research. Our main contribution is a fixed-parameter tractable (FPT) algorithm for the k-fault-tolerant basis problem, parameterized by both k and the rank r of the matroid. This two-variable parameterization by k + r is shown to be tight in the following sense. On the one hand, the problem is already NP-hard for k=1. On the other hand, it is Para-NP-hard for r \geq 3 and polynomial-time solvable for r \leq 2.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

Adjiashvili, F

D. Adjiashvili, F. Hommelsheim, M. M ¨uhlenthaler, and O. Schaudt , Fault- tolerant edge-disjoint s-t paths – beyond uniform faults , in Proceedings of the 18th Scan- dinavian Symposium and Workshops on Algorithm Theory (SW AT), Schloss Dagstuhl - Leibniz-Zentrum f¨ ur Informatik, 2022, pp. 5:1–5:19

work page 2022
[2]

Adjiashvili, S

D. Adjiashvili, S. Stiller, and R. Zenklusen , Bulk-robust combinatorial optimiza- tion, Mathematical Programming, 149 (2015), pp. 361–390. 1Given a partition matroid defined by a partition ( P1, . . . , Pd) of the ground set and a d-tuple of capaci- ties ( c1, . . . , cd), to solve Weighted F ault-Tolerant Basis, we have to choose ci + k elements of mini...

work page 2015
[3]

Bansal, J

I. Bansal, J. Cheriyan, L. Grout, and S. Ibrahimpur , Improved approximation al- gorithms by generalizing the primal-dual method beyond uncrossable functions , in Proceed- ings of the 50th International Colloquium on Automata, Languages, and Programming (ICALP), Schloss Dagstuhl–Leibniz-Zentrum f¨ ur Informatik, 2023, pp. 15:1–15:19

work page 2023
[4]

Bentert, J

M. Bentert, J. Schestag, and F. Sommer, On the complexity of finding a sparse con- nected spanning subgraph in a non-uniform failure model , in Proceedings of the 18th Inter- national Symposium on Parameterized and Exact Computation (IPEC), Schloss Dagstuhl - Leibniz-Zentrum f¨ ur Informatik, 2023, pp. 4:1–4:12

work page 2023
[5]

Bertsimas, D

D. Bertsimas, D. B. Brown, and C. Caramanis , Theory and applications of robust optimization, SIAM Review, 53 (2011), pp. 464–501

work page 2011
[6]

S. C. Boyd, J. Cheriyan, A. Haddadan, and S. Ibrahimpur , Approximation algo- rithms for flexible graph connectivity , in Proceedings of the 41st IARCS Annual Confer- ence on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), Schloss Dagstuhl–Leibniz-Zentrum f¨ ur Informatik, 2021, pp. 9:1–9:14

work page 2021
[7]

Chalermsook, C

P. Chalermsook, C. Huang, D. Nanongkai, T. Saranurak, P. Sukprasert, and S. Yingchareonthawornchai, Approximating k-edge-connected spanning subgraphs via a near-linear time LP solver , in Proceedings of the 49th International Colloquium on Automata, Languages, and Programming (ICALP), Schloss Dagstuhl - Leibniz-Zentrum f¨ ur Informatik, 2022, pp. 37:1–37:20

work page 2022
[8]

Chekuri and R

C. Chekuri and R. Jain , Approximation algorithms for network design in non-uniform fault models , in Proceedings of the 50th International Colloquium on Automata, Lan- guages, and Programming (ICALP), Schloss Dagstuhl–Leibniz-Zentrum f¨ ur Informatik, 2023, pp. 36:1–36:20

work page 2023

Show all 20 references

[9]

Cygan, F

M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh , Parameterized Algorithms, Springer, 2015

2015
[10]

A. G. Dimakis, P. B. Godfrey, Y. Wu, M. J. W ainwright, and K. Ramchandran, Network coding for distributed storage systems , IEEE Transactions on Information Theory, 56 (2010), pp. 4539–4551

2010
[11]

M. C. Dourado, D. Meierling, L. D. Penso, D. Rautenbach, F. Protti, and A. R. de Almeida , Robust recoverable perfect matchings, Networks, 66 (2015), pp. 210– 213

2015
[12]

Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing, Springer, 2010

M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing, Springer, 2010

2010
[13]

C. G. Fernandes , A better approximation ratio for the minimum size k-edge-connected spanning subgraph problem, Journal of Algorithms, 28 (1998), pp. 105–124

1998
[14]

F. V. Fomin, P. A. Golovach, D. Lokshtanov, and S. Saurabh , Covering vectors by spaces: Regular matroids, SIAM Journal on Discrete Mathematics, 32 (2018), pp. 2512– 2565

2018
[15]

Froese, I

V. Froese, I. A. Kanj, A. Nichterlein, and R. Niedermeier , Finding points in general position, International Journal of Computational Geometry and Applications, 27 (2017), pp. 277–296

2017
[16]

H. N. Gabow, M. X. Goemans, ´E. Tardos, and D. P. Williamson , Approximating the smallest k-edge connected spanning subgraph by LP-rounding , Networks, 53 (2009), pp. 345–357. 14

2009
[17]

Hommelsheim, M

F. Hommelsheim, M. M ¨uhlenthaler, and O. Schaudt , How to secure matchings against edge failures , SIAM Journal on Discrete Mathematics, 35 (2021), pp. 2265–2292

2021
[18]

Khuller and U

S. Khuller and U. Vishkin , Biconnectivity approximations and graph carvings , in Pro- ceedings of the 24th Annual ACM Symposium on Theory of Computing (STOC), ACM, 1992, pp. 759–770

1992
[19]

Lokshtanov, P

D. Lokshtanov, P. Misra, F. Panolan, and S. Saurabh , Deterministic truncation of linear matroids , ACM Transactions on Algorithms, 14 (2018), pp. 14:1–14:20

2018
[20]

J. G. Oxley, Matroid theory, Oxford University Press, 1992. 15

1992

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Adjiashvili, F

D. Adjiashvili, F. Hommelsheim, M. M ¨uhlenthaler, and O. Schaudt , Fault- tolerant edge-disjoint s-t paths – beyond uniform faults , in Proceedings of the 18th Scan- dinavian Symposium and Workshops on Algorithm Theory (SW AT), Schloss Dagstuhl - Leibniz-Zentrum f¨ ur Informatik, 2022, pp. 5:1–5:19

work page 2022

[2] [2]

Adjiashvili, S

D. Adjiashvili, S. Stiller, and R. Zenklusen , Bulk-robust combinatorial optimiza- tion, Mathematical Programming, 149 (2015), pp. 361–390. 1Given a partition matroid defined by a partition ( P1, . . . , Pd) of the ground set and a d-tuple of capaci- ties ( c1, . . . , cd), to solve Weighted F ault-Tolerant Basis, we have to choose ci + k elements of mini...

work page 2015

[3] [3]

Bansal, J

I. Bansal, J. Cheriyan, L. Grout, and S. Ibrahimpur , Improved approximation al- gorithms by generalizing the primal-dual method beyond uncrossable functions , in Proceed- ings of the 50th International Colloquium on Automata, Languages, and Programming (ICALP), Schloss Dagstuhl–Leibniz-Zentrum f¨ ur Informatik, 2023, pp. 15:1–15:19

work page 2023

[4] [4]

Bentert, J

M. Bentert, J. Schestag, and F. Sommer, On the complexity of finding a sparse con- nected spanning subgraph in a non-uniform failure model , in Proceedings of the 18th Inter- national Symposium on Parameterized and Exact Computation (IPEC), Schloss Dagstuhl - Leibniz-Zentrum f¨ ur Informatik, 2023, pp. 4:1–4:12

work page 2023

[5] [5]

Bertsimas, D

D. Bertsimas, D. B. Brown, and C. Caramanis , Theory and applications of robust optimization, SIAM Review, 53 (2011), pp. 464–501

work page 2011

[6] [6]

S. C. Boyd, J. Cheriyan, A. Haddadan, and S. Ibrahimpur , Approximation algo- rithms for flexible graph connectivity , in Proceedings of the 41st IARCS Annual Confer- ence on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), Schloss Dagstuhl–Leibniz-Zentrum f¨ ur Informatik, 2021, pp. 9:1–9:14

work page 2021

[7] [7]

Chalermsook, C

P. Chalermsook, C. Huang, D. Nanongkai, T. Saranurak, P. Sukprasert, and S. Yingchareonthawornchai, Approximating k-edge-connected spanning subgraphs via a near-linear time LP solver , in Proceedings of the 49th International Colloquium on Automata, Languages, and Programming (ICALP), Schloss Dagstuhl - Leibniz-Zentrum f¨ ur Informatik, 2022, pp. 37:1–37:20

work page 2022

[8] [8]

Chekuri and R

C. Chekuri and R. Jain , Approximation algorithms for network design in non-uniform fault models , in Proceedings of the 50th International Colloquium on Automata, Lan- guages, and Programming (ICALP), Schloss Dagstuhl–Leibniz-Zentrum f¨ ur Informatik, 2023, pp. 36:1–36:20

work page 2023