Exploration of recolourable graphs
A single agent with little or no memory must visit every node of an anonymous graph whose nodes carry colours it can read and rewrite; how many colours are needed, and how much does it help to recolour, to see neighbours or to have port numbers?
Problem statement and measuresexpand
A single deterministic agent starts at a node \(s\) of a connected, simple, undirected, anonymous graph \(G=(V,E)\) with \(n\) nodes, \(m\) edges, maximum degree \(\Delta\) and diameter \(D\). Every node \(v\) holds a colour \(\psi(v)\) from a palette of \(c\) colours that persists between visits. In each step the agent at \(v\) observes \(\psi(v)\) (and, depending on the setting, \(\deg(v)\), port numbers, the entry port, and the multiset of its neighbours' colours), sets \(\psi(v)\) to a new colour (recolours \(v\)) if the setting allows it, and then moves to a neighbour or stops. In the main model Böckenhauer et al. 2023 there are no port numbers: the agent names a colour and the adversary moves it to any neighbour of that colour; the agent is oblivious (it keeps no memory between steps); all nodes start with the initial colour \(c_0\); and the agent must visit every node, return to \(s\) and stop there, from every start and against every adversary. Other settings fix the colours in advance (label-guided exploration), give port numbers, or ask only for perpetual exploration. Measures: the number of colours \(c\), counting \(c_0\) (the main measure); time, the number of moves (traversals of one edge) until termination (or until every node is visited, for perpetual exploration); and the agent's memory. All known results are for one agent.
Measures: Colours, number \(c\) of node colours, counting the initial colour \(c_0\) (for the rotor-router: per node); Time, moves until the agent stops (perpetual: until every node is visited); Memory, bits of agent memory; 0 for an oblivious agent.
Label-guided, oblivious agent
verifiedFixed colours cannot replace memory cheaply: an oblivious agent needs more than \(\lfloor\log\Delta\rfloor - 2\) designer colours.
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Graph dynamics: static graph
- \(G_t=G\) for all \(t\): the edge set never changes.
- Edge direction: undirected
- \(G=(V,E)\) with \(E\subseteq\binom V2\); every edge can be traversed in both directions.
- Port numbers: ports + entry port
- As for port numbers, and on arrival at \(v\) the agent also reads the port \(\mathrm{port}_v(e)\) of the edge \(e\) by which it entered. This is the standard convention of single-agent exploration.
- View: own node
- At node \(v\) the agent observes \(\psi(v)\), \(\deg(v)\) and nothing about its neighbours (and, where ports exist, what the port axis grants).
- Agent memory: oblivious
- The agent has no internal state: its action (new colour of \(v\), then a move, stay or stop) is a function of its current observation only. Called zero-memory, memoryless or a 1-state robot.
- Recolouring: fixed colours
- The colouring \(\psi : V \to \{1,\dots,c\}\) never changes; the agent reads \(\psi(v)\) (and, with neighbour view, its neighbours' colours) but cannot write.
- Initial colouring: designer's colours
- Before the agent starts, a designer who knows \(G\) (and the start node) chooses \(\psi_0\). Used with fixed colours this is label-guided exploration Cohen et al. 2008.
- Knowledge: nothing known
- The algorithm is the same for all graphs: it knows neither \(n\) nor \(\Delta\) nor any other parameter (a uniform algorithm in Böckenhauer et al. 2023).
- Termination requirement: perpetual
- Every node is visited at least once (for the rotor-router: infinitely often); the agent never needs to halt.
- Network: any graph
- A finite, simple, connected, undirected, anonymous graph \(G = (V,E)\) with \(n\) nodes.
What is known
- for every \(\Delta > 4\) and every one-state agent with at most \(\lfloor\log\Delta\rfloor - 2\) colours there is a simple graph of maximum degree \(\Delta\) it does not explore (Theorem 4.5)
- with self-loops allowed the bound rises to \(\Delta/2\) colours (Theorem 4.2)
- star-like graphs whose central node is entered and left by ports the agent's colour functions cannot cover; applies to visiting all nodes, so to every termination requirement
Results, in order of publication
No result recorded yet.
Open questions
How many designer colours does an oblivious agent need to explore all graphs of maximum degree \(\Delta\) Cohen et al. 2008?
Notes
With recolourable colours the same oblivious agent needs only a constant number of colours, even without port numbers (setting Recolourable nodes, oblivious agent, any graph): the power lies in writing, not in reading. Checked against the full text.
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