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  "slug": "cover-time-robustness-separation",
  "title": "Cover-time robustness can stay positive while hitting-time robustness vanishes",
  "shortTitle": "Random walks: robust to cover, fragile to hit",
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  "oneLine": "On a huge clique with a private leaf on every vertex and one extra path of length L, an adversary nudging transition probabilities can be tolerated up to a fixed ε ≈ 0.293 before cover time doubles, but only ε ≈ 0.628/L before hitting time doubles, refuting Zuckerman's conjecture that the two tolerances always agree up to a constant.",
  "abstract": "A random walk explores a network by moving to a uniformly chosen neighbour at each step. The hitting time is the expected time to reach one given vertex in the worst case; the cover time is the expected time to visit every vertex. Zuckerman (Markov Chain Robustness, ITCS 2026) asked how much an adversary who may change each transition probability by a relative factor between 1−ε and 1+ε, with history-dependent choices, can slow these quantities down, and conjectured (Conjecture 1.17) that the largest tolerable ε for hitting time and for cover time always agree up to a constant factor. This candidate refutes that conjecture with sharp limiting tolerances. Take a complete graph on N = 2^(2^L) vertices, attach a private leaf to each vertex and attach one extra path of length L. For every fixed amplification budget C > 1, the hitting-time tolerance satisfies Lρ_hit → a_C, where (e^(2a_C)−1)/(2a_C) = C, because a small bias compounds along the path; the cover-time tolerance tends to min{1−C^(−1/2), 1/3} > 0, because collecting the leaves dominates and an adversary can inflate it only by (1−ε)^(−2). For C = 2 these are about 0.628/L and 0.293, and the ratio grows like log log |V|. The paper also proves a uniform cover-response formula, the exact critical multiplier 9/4 + 9/(8 log 2) ≈ 3.873 and tolerance expansions through order 1/L. The graphs have unbounded degree and are astronomically large: the theorem is asymptotic, and bounded-degree versions remain open. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-11-04",
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  "keywords": [
    "random walks",
    "Markov chain robustness",
    "cover time",
    "hitting time",
    "adversarial perturbation",
    "robust Markov decision processes",
    "counterexample",
    "Zuckerman conjecture"
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    "Hitting-time robustness vanishes (Theorem 1.1): on G_L, a clique on N = 2^(2^L) vertices with a private leaf on each vertex and one path of length L, L·ρ_hit(G_L;C) → a_C for every fixed budget C > 1, where (e^(2a_C)−1)/(2a_C) = C; for C = 2, a_2 ≈ 0.628.",
    "Cover-time robustness stays positive (Theorem 1.1): ρ_cov(G_L;C) → min{1−C^(−1/2), 1/3} > 0; for C = 2 this is 1 − 1/√2 ≈ 0.293. Below ε = 1/3 the worst cover-time multiplier tends to (1−ε)^(−2); above it, to infinity.",
    "Conjecture 1.17 fails (Theorem 1.1): ρ_cov/ρ_hit grows like a constant times L, that is Θ(log log |V|), for every fixed pair of hitting and cover budgets; this is consistent with Zuckerman's Theorem 1.15 because t_cov/(t_hit log|V|) → 0 here.",
    "Uniform response and critical window (Theorem 1.2, Section 9): t⁺_cov/(N² log N) = a_ε + b e^(−a_ε/b) + O(1/log N) uniformly in ε, with exact critical multiplier 9/4 + 9/(8 log 2) ≈ 3.873 at ε = 1/3, and cover-tolerance expansions through order 1/L for every budget.",
    "Scope: unbounded degree and astronomically large graphs; bounded-degree and regular versions, the converse comparison, the largest possible separation and the strong-adversary cover tolerance remain open."
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  "evidencePackage": "Theorems 1.1 and 1.2 have written analytic proofs: exact Bellman solutions for the hard endpoint and for leaf collection, a device deleting path excursions, and a martingale stopped at the successful endpoint trial; every upper bound allows arbitrary history-dependent adversaries. Supplementary checks: 63 exact rational Bellman and hitting-time checks and three small adaptive-cover solves (verify_markov.py), 60 endpoint-survival values at 80-128 digits in the critical window (verify_critical.py). All scripts were replayed with and without Python -O from fresh copies; a corrupted formula is rejected under -O; GitHub CI reruns them on Linux.",
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  "relatedWorks": [
    {
      "citation": "Zuckerman (2026). Markov Chain Robustness. ITCS 2026, LIPIcs 362. Defines relative-error robustness of hitting and cover times and states Conjecture 1.17, refuted here, and Theorem 1.15, the opposite comparison when Matthews' bound is tight.",
      "url": "https://doi.org/10.4230/LIPIcs.ITCS.2026.118",
      "doi": "10.4230/LIPIcs.ITCS.2026.118"
    },
    {
      "citation": "Anonymous (2026). Small diameter does not control hitting-time robustness. Evidence Press candidate, version 1.0.0-candidate. Sibling release answering Zuckerman's different Conjecture 1.16 (diameter versus hitting-time robustness) by a different mechanism; not independent confirmation.",
      "url": "https://doi.org/10.5281/zenodo.23575572",
      "doi": "10.5281/zenodo.23575572"
    },
    {
      "citation": "Matthews (1988). Covering Problems for Markov Chains. The Annals of Probability. The cover-time bound t_cov ≤ t_hit H_n behind Zuckerman's Theorem 1.15.",
      "url": "https://doi.org/10.1214/aop/1176991686",
      "doi": "10.1214/aop/1176991686"
    },
    {
      "citation": "Levin and Peres (2017). Markov Chains and Mixing Times, second edition. American Mathematical Society. Background on hitting times, commute times and cover times (Proposition 10.7, Chapter 11).",
      "url": "https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf",
      "doi": null
    },
    {
      "citation": "Azar, Broder, Karlin, Linial and Phillips (1996). Biased random walks. Combinatorica. Introduced the biased-walk model underlying Zuckerman's strong adversary.",
      "url": "https://doi.org/10.1007/BF01300124",
      "doi": "10.1007/BF01300124"
    },
    {
      "citation": "Iyengar (2005). Robust Dynamic Programming. Mathematics of Operations Research. The verification principle specialised in Lemma 2.1.",
      "url": "https://doi.org/10.1287/moor.1040.0129",
      "doi": "10.1287/moor.1040.0129"
    },
    {
      "citation": "Nilim and El Ghaoui (2005). Robust Control of Markov Decision Processes with Uncertain Transition Matrices. Operations Research. Robust dynamic programming with rectangular transition uncertainty.",
      "url": "https://doi.org/10.1287/opre.1050.0216",
      "doi": "10.1287/opre.1050.0216"
    },
    {
      "citation": "Haslegrave, Sauerwald and Sylvester (2022). Time Dependent Biased Random Walks. ACM Transactions on Algorithms. A controller model of biased walks; different constraints and objectives.",
      "url": "https://doi.org/10.1145/3498848",
      "doi": "10.1145/3498848"
    },
    {
      "citation": "Olesker-Taylor, Sauerwald and Sylvester (2026). Time-Biased Random Walks and Robustness of Expanders. SODA 2026. Perturbations of expanders in the biased-walk model; does not give this separation.",
      "url": "https://doi.org/10.1137/1.9781611978971.109",
      "doi": "10.1137/1.9781611978971.109"
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