Small subgraphs in the Achlioptas process
Reto Spöhel, ETH ZürichJoint work with Torsten Mütze and Henning Thomas
Introduction
• Goal: Avoid creating a copy of some fixed graph F
• Achlioptas process:
• start with the empty graph on n vertices
• in each step r edges are chosen uniformly at random (among all edges never seen before)
• select one of the r edges that is inserted into the graph, the remaining r – 1 edges are discarded
How long can we avoid F by this freedom of choice?
F = , r = 2
Introduction• N0=N0(F, r, n) is a threshold:
N0=N0(F, r, n) N = N(n) = # steps
There is a strategy that avoids
creating a copy of F withprobability 1-o(1)
N /N0
If F is a cycle, a clique or a complete bipartite graph with parts of equal
size, an explicit threshold function is known. (Krivelevich, Loh, Sudakov, 2007+)
Every strategy will be forced
to create a copy of F withprobability 1-o(1)
N [N0
n1.2
F = , r=2
n1.286…
r=3n1.333…
r=4
n1
r=1
Our Result
• Theorem:Let F be any fixed graph and let r R 2 be a fixed integer. Then the threshold for avoiding F in the Achlioptas process with parameter r is
where
• We solve the problem in full generality.
In the following we will try to convey an
intuition for the threshold formula.
r-matched graph r=2
Graph
r-matched Graphs
Random graph Random r-matched graph- generate Gn,m
- randomly partition the m edges into sets of size r
a family of disjoint r-sets of edges.
- have well-defined notion of subgraph inclusion
- Achlioptas process after N steps is distributed as
The Gluing Intuition
‚ gluingtogether’
E[ # copies of in ]
F
F 0 ³ E[ # copies of F in Gn,m ]
Threshold for „ contains “
= Threshold for „ Gn,m contains F “
F 0
³ nv(F) (m/n2)e(F)
Bollobás (1981): Threshold for the appearance of F in Gn,m is given by
m=m(n)
The Gluing Intuition
Example:
F = , r=2
Greedy strategy:
e/v = 5/4
As long asa.a.s. this subgraph does not appear, and hence we do not lose.
Our approach: view Achlioptas process as ‘static’ object and use (the r-matched version of) the above theorem.
1st Observation: Subgraph Sequences
F = , r=2
e/v = 11/8 = 1.375
Greedy strategy:
0 Maximization over a sequence of subgraphs of F
e/v = 15/10 = 1.5
Optimal strategy:
As long as a.a.s. this subgraph does not appear, and hence we do not lose.
2nd Observation: Edge OrderingsOrdered graph: pair
oldest edge
youngest edge2
3
7
4
6
1
5
0 Minimization over all possible edge orderings of F
1
2
2
e/v = 19/14 = 1.357...
2
F = , r=2
Edge ordering ¼1:
1
2
Optimal Strategy for ¼1:
12
2
12
e/v = 17/12 = 1.417...
Edge ordering ¼2:
12
12
2Optimal Strategy for ¼2:
F3-
F3-
F3-
F3-
F3-
F3-
F3-
F3-
4
4
4
4
F2-
F2-
F2-
F2-
3
4F1-3 F1-
r-1
4
Calculating the ThresholdMinimize over all possible edge orderings ¼ of FMaximize e(J)/v(J) over all
subgraphs J4F ¼
r2
34
F2
34
F 1
F ¼
Maximization over a sequence of subgraphs of F
5
J
H1
H2
H3
H1
H2H3
H3
H3
Calculating the Threshold (Example)
7
F6-…
F ¼
1
3 4
6
2
5
34
65
2
77
F = , r=2
Minimize over all possible edge orderings ¼ of FMaximize e(J)/v(J) over all
subgraphs J4F ¼5
3
7
4
6 5
3
7
4
65
7
7 7
7J
e(J)/v(J) = 19/14
Maximization over a sequence of subgraphs of F
1
3
7
4
6
2
5
F
(F, ¼) 23
7
4
6 5
F1-
3
7
4
6 5
F2-
Maximization over a sequence of subgraphs of F
Our Result ( explained)
• Theorem:Let F be any fixed graph and let r R 2 be a fixed integer. Then the threshold for avoiding F in the Achlioptas process with parameter r is
where
Minimization over all possible edge orderings of FMaximize e(J)/v(J) over all
subgraphs J4F ¼
A Canonical Strategy
• In order to avoid copies of F, we want to keep the number of copies of all (ordered!) subgraphs (H, ¼) of F as low as possible.
• For N = n2 – µ fixed:
• The problem has a recursive structure.
• We can compute a canonical strategy and values ¸µ(H, ¼) such that in N = n2 – µ many steps, at most copies of (H, ¼) are created (for all subgraphs (H, ¼) simultaneously!)
• In order to avoid copies of F we use this canonical strategy with
• Alternatively, the threshold can be defined directly via the recursion.
About the ¸-values
21
harm
les
sdangero
us
F = , r=2
¸µ(H, ¼) is the exponent of the expected number of copies of a ‚typical‘ history graph J of (H, ¼),
A Canonical Strategy
• For each edge calculate the level of danger it entails as the most dangerous (ordered) subgraph this edge would close
• Among all edges, pick the least dangerous one
f1 f2 fr…
is more dangerous than
5
Lower Bound ProofDeterministic Lemma: By our strategy, each black copy of some ordered graph is contained in a copy of a grey-black r-matched graph H’ with expectation at most . . .
1
2
H’“History graph”
• If , each possible history does a.a.s. not appear in .
• Constantly many histories H’of ending up with a copy of F
a.a.s. no copy of F is created.
Technical work!
F = r=2
21
12
harmless
dangerous
This might be the same edge
“Bastard”
• Goal: force a copy of F .
• We fix the most dangerous ordering ¼ and force a copy of (the r-matched version of) F
that contains a copy of F.
• This is enforced if we get r edges all closing the central copy of a copy of .
• We do this inductively in e(F) rounds, doing a small subgraphs type variance calculation in each round.
• works out if .
Upper Bound Proof
4
43
4
2
34
12
344
43
4
F ¼
F
Conclusion
• The problem can be solved completely using the methods of first and second moment only.
• The notion of r-matched graphs allows a combinatorial interpretation of the threshold formula.
• Open questions: (ongoing work with Michael Krivelevich)
•What if r = r(n) 1?
•What if we want to create a copy of F as quickly as possible?
Thank you! Questions?
Top Related