robotic swarm engineers

Abstract: On ranked directed acyclic graphs coupled by vertex conflicts we study exact-arrival multi-agent path finding (departure and arrival pinned in place and time) posed as a distributed output task: every cell must emit its own bit of a valid joint path-support mask. The machines are cellular: one automaton per cell, state factored over cells, communication only along a fixed local fabric, synchronous rounds, a query-independent output round, no central decoder, and a per-topology transition code. "Necessary" here means: on an explicit hard topology, over all its feasible queries, no universally correct local-only architecture can renounce either temporal orientation; with global flags, impossibility becomes a bandwidth-round trade-off. The class is not vacuous: a flooding machine solves every instance in as many rounds as the diameter of the local fabric. The bound survives two concessions: we supply every agent's exact individual cavity, which alone solves the single-agent problem in zero rounds and makes the single-agent direction theorem vacuous; and we allow $k$ one-bit global aggregate flags. On a twelve-cell, two-agent, depth-three topology no forward-only machine with $k=0$ is correct on two feasible queries, for any number of rounds; the time-reversed topology does the same to backward-only machines. Placing $p$ copies side by side gives, for forward-only machines, $J_R+kR\ge p$ at constant depth, where $J_R$ measures how much of the query reaches the targets' cones and is $0$ here. A backward-only machine solves that family in constant rounds and a flag-streaming one in $\lceil p/k\rceil+O(1)$, so the bound is tight in order; on the composite family neither orientation can be renounced. The finite lemmas of the gadget and the pair-invariance contradictions are checked in Lean 4; the semantic bridge, the cone lemma and the counting are paper proofs. We report no benchmarks.

Abstract: We study the problem of treasure hunt by a group of $k \geq 1$ agents in vertex-permuted dynamic rings (VP). In this model, the $n$ vertices remain on a ring but are permuted at each time step. We first show that treasure hunt is impossible for any $k \leq n-3$ agents, if there are no restrictions on the sequence of permutations used in the dynamic ring. We then study the $VP(δ)$ setting, in which for every pair $i, j$ of vertices, the edge $(i, j)$ is guaranteed to appear within $δ$ steps. We show that the class $VP(δ)$ is feasible only for $δ\geq \left\lceil \frac{n-1}{2}\right\rceil$. For the one-agent case, we show a tight bound of $Θ(δn)$ on the worst-case search time as well as competitive ratio of any online algorithm for treasure hunt, provided $δ\geq 2n$. We then give an optimal algorithm for $k$ agents, thereby showing that $k$ agents can obtain a speedup of $k$ on the worst-case search time. Finally, in the R-VP setting, in which in every step, the vertices are arranged as a ring according to a random permutation, we show that treasure hunt takes expected $Θ(n)$ steps against an oblivious adversary and $Θ(n \log n)$ steps against an adaptive adversary.