the expanding search ratio of a graph spyros angelopoulos* christoph dürr* thomas lidbetter**...

The expanding search ratioof a graph

Spyros Angelopoulos*Christoph Dürr*

Thomas Lidbetter**

*Sorbonne Universités, UPMC Univ Paris 06, CNRS, LIP6, Paris, France**Department of Mathematics, London School of Economics, UK

Background

• Searching a fixed graph (Koutsoupias, Papadimitriou, Yannakakis, 1996)

• Mining coal or finding terrorists: The expanding search paradigm (Alpern, L., 2013)

Expanding search

3

21

2

31

𝑂

An expanding search of a (weighted, connected) graph with root is a sequence of edges each one of which is incident to a previously searched vertex.

Search time

For a search , and vertex , the search time is the time is first discovered.Eg. The normalized search time, is .Eg.

𝑖

3+2+2¿7

3

21

2

31

𝑂

Search ratio

The search ratio of is .

The search ratio of a graph is .

If minimises the search ratio we say is optimal.

Eg. 3

21

2

31

𝑂

Proposition

For trees or graphs with unit edge weights, it is optimal to search the vertices in order of their distance from O.

3

21

2

31

𝑂

Counterexample for weighted graphs

5 10

76

𝑂

Counterexample for weighted graphs

5 10

76

𝑂

Theorem

It is NP-complete to decide whether .

Proof: Reduction from 3-SAT.

TheoremThere is a polynomial time algorithm that approximates the search ratio within a factor of

Proof sketch:

𝐺

𝑂Min. cost tree containing all vertices at distance from .

Min. cost tree con-taining all vertices at distance from .

Min. cost tree con-taining all vertices at distance from .

Randomized search ratio

For a randomized search and a vertex , the expected search time and expecte normalized search time are denoted by and

The randomized search ratio of a random search is .

The randomized search ratio of a graph is .

Game theoretic interpretationFinding the optimal randomized search is equivalent to finding the optimal strategy in a zero-sum search game between a Searcher and Hider.

21

𝑂

Hider/Searcher 1,2 2,11 1 32 3/2 1

Optimal randomized search: start with short edge with probability and long edge with probability .

Randomized search ratio, .

2-approximate strategyProposition: For trees or graphs with unit length edges, the optimal deterministic strategy is a 2-approximation for the optimal randomized strategy.

Example

11

𝑂

1

𝑛

and

Randomization can be very bad

𝐿≫1

1𝑂

but searching in a random order has search ratio

Randomized star search

𝑑1

𝑂

𝑑2𝑑3

𝑑𝑛

1=𝑑1≤ 𝑑2≤…≤𝑑𝑛

𝑑2 𝑑3𝑑1 𝑑𝑛

Idea: randomize in “stages”

Randomize between all edges with length satisfying for

Unfortunately, it doesn’t work…

1𝑂

𝑛

2− 2

This has search ratio .

But .2−2

−

Better idea: randomize in “random stages”

1 2 4 8 2𝑘− 2 2𝑘−12𝑘

𝑥1 𝑥2 𝑥3 𝑥𝑘− 1 𝑥𝑘

𝑂

Better idea: randomize in “random stages”

1 2 4 8 2𝑘− 2 2𝑘−12𝑘

𝑥1 𝑥2 𝑥3 𝑥𝑘− 1 𝑥𝑘

𝑂

Better idea: randomize in “random stages”

1 2 4 8 2𝑘− 2 2𝑘−12𝑘

𝑥1 𝑥2 𝑥3 𝑥𝑘− 1 𝑥𝑘

𝑂

Better idea: randomize in “random stages”

1 2 4 8 2𝑘− 2 2𝑘−12𝑘

𝑥1 𝑥2 𝑥3 𝑥𝑘− 1 𝑥𝑘

𝑂

Better idea: randomize in “random stages”

1 2 4 8 2𝑘− 2 2𝑘−12𝑘

𝑥1 𝑥2 𝑥3 𝑥𝑘− 1 𝑥𝑘

𝑂

Theorem: This has an approximation ratio of 5/4.

Idea of proof

Bound from below using a collection of mixed Hider strategies:

Lemma: If the Hider chooses from the edges with probability proportional to the square of the length of the edge, the expected search ratio is at least

.

An exactly optimal randomized search

Theorem: If the lengths of the edges “don’t increase too fast” then the optimal randomized search can be found inductively and

.

Theorem: The graph with edges that has maximum randomized search ratio is the one with equal length edges, i.e. .

Further directions

• Computational complexity of finding randomized search ratio?

• Continuous version…