Those who complain that there is never a policeman around when you need one will be mollified to know that street policing is one of the most difficult of all mathematical problems. Ideally there would be a member of the police force on every street corner, keeping watch along the various streets that intersect there. But no city police force can afford to patrol every crossroads. How, then, can every street be guarded?

Mathematically this is called the 'vertex cover (VC) problem'. A problem, Martin Weigt and Alexander Hartmann from the University of Göttingen now announce, that becomes first harder and then easier to solve as the street map gets bigger.

'Solving' the vertex problem means either finding a way to arrange the police officers so that no street is unobserved, or establishing that such an arrangement is impossible. It is clearly not necessary to have as many police as street intersections (vertices) to obtain total coverage (called simply 'cover'), because any particular street need be watched only from one end. But the more the police are outnumbered by vertices, the harder it is to provide cover.

The problem comprises the 'size of the city' (the number of vertices, assumed to be connected at random), the connectivity (number of streets meeting at each vertex) and the fraction of occupied vertices. In general, the only way to decide if cover exists for a given number of vertices and fractional occupancy is to look at all possible arrangements one by one. Even for a small town (say, with 40 vertices and a connectivity of just 2), this can take a long time.

The vertex cover problem belongs to a class called 'NP-complete' problems, which are the hardest to solve. It has emerged recently that some such problems display a transition: their solvability alters suddenly as the complexity (say, the number of possible configurations) increases. In the vertex cover problem, for instance, it is relatively easy to find a solution if there are almost as many police as intersections, and relatively easy to see that cover is impossible if the police are vastly outnumbered. But in between, deciding whether or not cover is possible is much harder.

This kind of abrupt change resembles the sudden freezing or boiling of a liquid as the temperature creeps up or down. Such changes are known as 'phase transitions'. The relationship between phase transitions in physical phenomena and those in NP-complete mathematical problems has recently become a focus of great interest in computer science and information processing.2

In Physical Review Letters 1, Weigt and Hartmann show that the vertex cover problem displays a phase transition that they have investigated in detail. For a connectivity of 2, the vertex cover problem is easy to solve when the fraction of occupied vertices is below about 20% or above about 40%, they say. But in the region from 30% to 40%, finding a solution consumes much more computer time.

They also calculated how this 'transition fraction' depends on the connectivity of the vertices. In a city like New York, where the grid-like street pattern gives a connectivity of 4 (four streets meet at each vertex), the transition occurs for an occupancy fraction of a little over 50%. In other words, it is hardest to figure out where to station the police officers when there is roughly one person for every two crossroads. (For a city as truly regular as Manhattan, however, this is not so hard.) Fewer police than this and it is clearly impossible to watch every street.

Of course, real policing doesn't work this way. Weight and Hartmann suggest that the problem is more akin to that of museum security, where stationary guards watch the exhibit spaces. So the message is: if a museum wants to employ the minimum number of guards to watch over its exhibits, it can do so only by encountering the greatest difficulty in working out where they should be stationed.