arxiv: 2605.08355 · v1 · submitted 2026-05-08 · 💻 cs.DS

Recognition: no theorem link

Search and evacuation with a near majority of faulty agents

J. Czyzowicz , R. Killick , E. Kranakis , G. Stachowiak

Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:47 UTC · model grok-4.3

classification 💻 cs.DS

keywords evacuationsearchfaulty agentscompetitive ratioinfinite linecrash faultswireless communicationByzantine faults

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The pith

With one more reliable agent than twice the number of crash faults, evacuation on the line achieves competitive ratios of at most 7.437 for three agents and asymptotically 4 plus 2 times square root of 2.

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

The paper studies groups of n mobile agents on an infinite line, where up to f of them are crash faulty and may fail to announce when they find an unknown exit. It restricts attention to the near-majority case n equals 2f plus 1 and introduces a new coordinated search strategy that uses wireless announcements to limit the extra time caused by silent agents. The authors derive explicit upper bounds on the competitive ratio, the worst-case ratio of evacuation time to the distance from start to exit. These bounds matter because they show how much overhead faults impose and how that overhead shrinks as the group grows while preserving the one-extra-reliable-agent margin. The same strategy also improves the known bound for the related problem in which the single faulty agent among three may lie about the exit location.

Core claim

The paper claims that the new algorithm guarantees evacuation competitive ratios of at most 7.437011 for three agents with one crash fault, 7.253767 for five agents with two faults and seven agents with three faults, and 7.147026 for nine agents with four faults. For all larger n equals 2f plus 1 it proves an asymptotic upper bound of 4 plus 2 times the square root of 2. The same movement schedule, adapted for the Byzantine model, improves the upper bound for search by three agents with one lying agent from 8.653055 to 7.437011.

What carries the argument

A novel coordinated search algorithm that schedules agents to explore segments of the line in phases while relying on wireless announcements from reliable agents to trigger evacuation once the exit is located.

If this is right

Evacuation completes in at most 7.437011 times the distance for three agents with one crash fault.
The ratio is at most 7.253767 when five agents face two faults or seven agents face three faults.
Nine agents with four faults evacuate in at most 7.147026 times the distance.
As the group size grows while keeping one extra reliable agent, the ratio is at most 4 plus 2 times square root of 2.
The same schedule yields a search competitive ratio of at most 7.437011 when one of three agents may lie about the exit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The bounds indicate that the overhead from faults decreases predictably as more reliable agents are added while preserving the n equals 2f plus 1 margin.
The coordination pattern may extend to search on a circle or in the plane when the same fault model applies.
Real deployments would need to account for communication delays or losses not modeled here, which could increase the observed ratios.
Verifying the ratios on concrete exit locations would show whether the upper bounds are close to tight.

Load-bearing premise

Faulty agents only omit announcements and never report false locations, and all communication is instantaneous and perfect.

What would settle it

A specific exit position on the line for which the three-agent one-fault algorithm requires more than 7.437011 times the distance to complete evacuation would refute the stated bound.

Figures

Figures reproduced from arXiv: 2605.08355 by E. Kranakis, G. Stachowiak, J. Czyzowicz, R. Killick.

**Figure 2.** Figure 2: Example space-time trajectories when r = 2 and n = 5, 7, 9. Note that each of the turning-points lie along a cone with slope ± r+1 r−1 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: Illustrating the points (ρi,j,k, τi,j,k) and their respective cones. Only (ρi,j,1, τi,j,1) are indicated. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: The competitive ratio as a function of the number of faults f for the proportional schedule algorithm. The blue line indicates the optimized competitive ratio; the red lines indicate the bounds of Theorem 2.1; the green line indicates the competitive ratio of evacuation when we choose r to optimize only S x f ; the lower black line indicates the asymptotic limit of 4 + 2√ 2. C Proofs and figures missing fr… view at source ↗

**Figure 5.** Figure 5: Illustration of the intersection points (ρ ◦ i,j,k, τ ◦ i,j,k) (left side) and (ρ + i,j,k, τ + i,j,k) (right side) of agents i and i + k. In this example j is even. trajectories of agents i and i + k takes place relative to their sub-turning points. Agent i will either be located in the interval [di,j , d(1) i,j ] or the interval (d (1) i,j , di,j+1] and agent i + k will either be in [di+k,j−1, d(1) i+k… view at source ↗

**Figure 6.** Figure 6: The optimized competitive ratio of the generalized schedules as a function of the number of faults. C.2 Proofs missing from Subsection 3.2 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗

**Figure 8.** Figure 8: Setup for the case that the first announcement claims that the target is at location x ∈ (di,j , d(1) i+2,j−1 ] and the actual target is on the left side at location x ′ . Agent i is in red, agent i + 1 in blue, and agent i + 2 in green. On the left side it is agent i + 1 that falsely announces the target and on the right side it is agent i + 2 that falsely announces the target. In either case the dashed… view at source ↗

**Figure 9.** Figure 9: Setup for the case that the first announcement claims that the target is at location x ∈ (d (1) i+2,j−1 , di+1,j ] and the actual target is on the left side at location x ′ . Agent i is in red, agent i + 1 in blue, and agent i + 2 in green. On the left side it is agent i + 1 that falsely announces the target and on the right side it is agent i + 2 that falsely announces the target. In either case the das… view at source ↗

read the original abstract

There are $n\geq 3$ unit speed mobile agents placed at the origin of the infinite line. In as little time as possible, the agents must find and evacuate from an exit placed at an initially unknown location on the line. The agents can communicate in the wireless mode in order to facilitate the evacuation (i.e. by announcing the target's location when it is found). However, among the agents are a subset of at most $f$ crash faulty agents who may fail to announce the target when they visit its location. In this paper we study this aforementioned problem for the specific case that $n=2f+1$. We introduce a novel type of search algorithm and analyze its competitive ratio -- the supremum, over all possible target locations, of the ratio of the time the agents take to evacuate divided by the initial distance between the agents and the target. In particular, we demonstrate that the competitive ratio of evacuation is at most $7.437011$ for $(n,f)=(3,1)$; at most $7.253767$ for $(n,f)=(5,2)$ and $(7,3)$; and at most $7.147026$ for $(n,f)=(9,4)$. For larger values of $n=2f+1$ we prove an asymptotic upper bound of $4+2\sqrt{2}$. We also adapt our evacuation algorithm for $(n,f)=(3,1)$ to the problem of search by three agents with one byzantine fault, i.e. the faulty agent may also lie about finding the target. In doing so we improve the best known upper bound on this search problem from 8.653055 to 7.437011.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They've tightened the upper bounds on competitive ratios for crash-fault evacuation with n=2f+1 agents on the line, including a clean asymptotic result of 4+2√2, and reused the idea to improve the three-agent Byzantine search bound from 8.65 to 7.437.

read the letter

The main point is a parameterized search strategy that handles up to f non-reporting agents out of 2f+1. For the small cases they get explicit ratios of 7.437 for (3,1), 7.254 for (5,2) and (7,3), and 7.147 for (9,4). For larger n they prove the ratio is at most 4+2√2. The same construction, slightly adapted, also lowers the known upper bound for three-agent search with one Byzantine liar from 8.653 to 7.437. That is the concrete advance over prior work in this narrow corner of mobile-agent problems on the line. The analysis proceeds by fixing phase lengths and turning points, splitting the line into regions, and bounding evacuation time in each region against the worst-case subset of at most f silent agents. The argument is direct worst-case analysis with no hidden reductions or fitted parameters that loop back on themselves. The numerical values come from optimizing those phase lengths, which is standard and acceptable here. The model assumptions (perfect wireless, unit speed, crashes only omit announcements) are stated clearly and match the usual setting for these problems. The only real softness is that these are upper bounds only; we still lack matching lower bounds, so it is not yet clear how close the ratios sit to optimality. The small-n cases rely on exhaustive region-by-region checking, which is tedious but finite, and the asymptotic claim looks clean. This paper is for researchers already working on fault-tolerant search and evacuation in distributed algorithms. A reader who cares about competitive ratios on the line will find the explicit numbers and the asymptotic simplification useful. It is narrow, but the claims are specific and the reasoning appears sound on its own terms. I would send it to a serious referee rather than desk-reject.

Referee Report

0 major / 2 minor

Summary. The manuscript considers n=2f+1 unit-speed agents on the infinite line tasked with locating and evacuating via an unknown exit, where at most f agents are crash-faulty and may omit announcements. It introduces a parameterized movement strategy that partitions the line into regions, derives explicit competitive-ratio upper bounds of 7.437011 for (n,f)=(3,1), 7.253767 for (5,2) and (7,3), 7.147026 for (9,4), and 4+2√2 asymptotically for larger n, and adapts the (3,1) algorithm to improve the Byzantine-fault search bound from 8.653055 to 7.437011.

Significance. If the case analysis and numerical optimizations hold, the work supplies improved, concrete upper bounds on competitive ratio for fault-tolerant evacuation and search in the wireless model. The explicit algorithmic constructions, region-based worst-case accounting for non-announcing agents, and the constant asymptotic bound constitute a clear advance over prior results in the area.

minor comments (2)

§3 and §4: the phase lengths and turning points that achieve the listed numerical ratios (e.g., 7.437011) are stated as optimized values; including the explicit optimization program or a short derivation appendix would make the bounds immediately reproducible without re-deriving the entire case analysis.
Figure 2 (or equivalent diagram): the movement trajectories for the (3,1) case are described in text; a single annotated figure showing the critical turning points and announcement regions would improve readability of the worst-case timing arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript and for recommending minor revision. We appreciate the recognition of our contributions to improving the competitive ratios for fault-tolerant evacuation and search problems.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation consists of an explicit algorithmic construction for evacuation under crash faults (n=2f+1) followed by direct worst-case analysis that partitions the line into regions and bounds the evacuation time relative to the unknown exit distance, accounting for at most f non-announcing agents. The reported competitive ratios (7.437011 for (3,1), 7.253767 for (5,2) and (7,3), 7.147026 for (9,4), and the asymptotic 4+2√2) are obtained by evaluating the supremum of the ratio over all possible target locations and fault behaviors; these quantities are not fitted parameters renamed as predictions, nor do they reduce to self-definitions or load-bearing self-citations. The adaptation to the Byzantine search variant similarly applies the same strategy and recomputes the bound independently. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on a novel coordinated search strategy whose worst-case performance is bounded by solving an optimization problem over movement phases; the specific numerical ratios suggest choice of strategy parameters to minimize the supremum ratio.

free parameters (1)

phase lengths and turning points in the search strategy
The competitive ratio bounds are obtained by optimizing the timing and directions of agent movements to cover possible exit locations while accounting for possible silent faulty agents.

axioms (2)

domain assumption All agents move at identical unit speed on the infinite line
Standard modeling assumption stated in the problem setup.
domain assumption Wireless announcements are instantaneous and error-free when made by reliable agents
Communication model given in the problem statement.

pith-pipeline@v0.9.0 · 5623 in / 1501 out tokens · 71004 ms · 2026-05-12T00:47:42.702652+00:00 · methodology

discussion (0)

Reference graph

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= 4 + 2 √ 2. 2 4 6 8 106.5 7 7.5 8 8.5 9 Competitive ratio Figure 4: The competitive ratio as a function of the number of faultsffor the proportional schedule algorithm. The blue line indicates the optimized competitive ratio; the red lines indicate the bounds of Theorem 2.1; the green line indicates the competitive ratio of evacuation when we chooserto o...

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