1. On the power of choices in random graph processes Reto Spöhel, PhD Defense February 17, 2010, ETH Zürich Examiners: Prof. Dr. Angelika Steger, ETH Zürich Prof. Dr. Michael Krivelevich, Tel Aviv University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A

2. The power of choices Assign n identical jobs to n servers independently u.a.r ) Maxload ¼ log n / log log n [Azar, Broder, Karlin, Upfal, STOC ‘94]: Draw two random servers for each job (sequentially), and assign job to server with lower load ) Maxload ¼ log log n / log 2 Applications in hashing, network routing, distributed computing, etc. We investigate similar phenomena in the context of random graph processes. ( ¼ 6 for n ¼ 10 7 ) ( ¼ 6 for n ¼ 10 27 )

3. G1G1 G2G2 G3G3 GNGN G4G4 G0G0 The random graph process Erdős-Rényi graph process: start with the empty graph on n vertices in each step a new edge appears uniformly at random and is inserted into the graph G N denotes the evolving graph after N steps. note: G N is distributed uniformly over all graphs with n vertices and N edges. N = 1N = 1N = 0N = 0N = 2N = 2N = 3N = 3N = 4N = 4

4. Average-case analysis Motivation for studying random graphs: average-case analysis of algorithms Usual worst-case complexity view may be too pessimistic; performance of an algorithm on ‘typical’ instances may be much better! Example: Testing whether a given graph G is 3-colorable is NP -hard. But [Wilf 84]: there is a simple deterministic algorithm that decides in constant expected time whether a graph drawn u.a.r. from all graphs on n vertices is 3-colorable. Reason: Typical instances contain many K 4 and are obviously not 3-colorable.

5. Average-case analysis Motivation for studying random graphs: average-case analysis of algorithms Usual worst-case complexity view may be too pessimistic; performance of an algorithm on ‘typical’ instances may be much better! Understanding random graphs can help explain why certain algorithms work well in practice.

6. The random graph process Erdős-Rényi graph process: start with the empty graph on n vertices in each step a new edge appears uniformly at random and is inserted into the graph GNGN

7. The random graph process Erdős-Rényi graph process: start with the empty graph on n vertices in each step a new edge appears uniformly at random and is inserted into the graph Question: How many steps N does it typically take (as n  1 ) until G N satisfies some monotone property P ? ) threshold N 0 ( P, n ) of the property P Examples [Erdős and Rényi 59-61]: The threshold for containing a linear-sized component is N 0 ( n ) = n / 2 Connectednessis N 0 ( n ) = n log n / 2 containing a triangle is ‘ N 0 ( n ) = £ ( n ) ’ [In the following we simply write N 0 ( n ) = n ]

8. Appearance of small subgraphs More generally: Bollobás (1981)

9. Power-of-choice random graph processes We study two random graph processes involving choices Achlioptas process [Bohman and Frieze 01]: in each step, select one out of r random edges Ramsey process [Friedgut et al. 03]: in each step, color a new random edge with one of r available colors Goal: delay or accelerate the occurrence of some monotone property P by careful edge selection/coloring strategies. What can be achieved by optimal strategies?

10. The Achlioptas process Achlioptas process: start with the empty graph on n vertices in each step r edges are drawn uniformly at random (from all edges never seen before) select one of these r edges for insertion into the evolving graph, and discard the remaining r – 1 edges note: r = 1 ) original Erdős-Rényi process r=2 r=2

11. The Achlioptas process Achlioptas process: start with the empty graph on n vertices in each step r edges are drawn uniformly at random (from all edges never seen before) select one of these r edges for insertion into the evolving graph, and discard the remaining r – 1 edges Goal: delay or accelerate the occurrence of some monotone property, e.g. containing a linear-sized (‘giant’) component [Bohman and Frieze 01], …, [Spencer and Wormald 07] containing a Hamilton cycle [Krivelevich, Lubetzky, Sudakov 10+] containing a copy of some fixed graph F [Krivelevich, Loh, Sudakov 09], [Mütze, S., Thomas 10+] Throughout the remainder of the talk, r ¸ 2 is a fixed integer, and the goal is to avoid copies of F for as long as possible. next few slides

12. Def: N 0 = N 0 ( F, r, n ) is the threshold for avoiding copies of F in the Achlioptas process with parameter r There is a strategy that avoids creating a copy of F with probability 1 - o ( 1 ) N ¿ N 0 Avoiding small subgraphs N 0 = N 0 ( F, r, n ) N = N ( n ) = # steps For F a cycle, a clique, or a complete bipartite graph with parts of equal size, this threshold was determined in [Krivelevich, Loh, Sudakov 09]. Every strategy will be forced to create a copy of F with probability 1 - o ( 1 ) N À N 0 n1.2n1.2 r=2 r=2 n 1. 286 … r=3r=3 n 1. 333 … r=4r=4 n2n2 F =

13. Avoiding small subgraphs Moreover, Krivelevich, Loh und Sudakov conjectured an explicit general threshold formula N 0 ( F, r, n ) for avoiding copies of F in the Achlioptas process with parameter r. In our first main result, we disprove this conjecture and give the true general threshold function of the problem. Mütze, S., Thomas (2010+)

14. Back to average-case analysis Our proofs show that for N ¿ N 0, a polynomial-time algorithm can compute selection decisions that avoid a copy of F with probability 1 - o ( 1 ) i.e., essentially best possible guarantees are achieved by a polynomial-time algorithm. This is in contrast with the fact that the underlying edge-selection problem is NP -hard.

15. Online vs. offline F = n1n1 r=1r=1 n 1. 286 … r=3r=3 n 1. 333 … r=4r=4 n1.2n1.2 r=2r=2 Compared to the classical case r = 1, there are two effects that influence the Achlioptas thresholds: the power of multiple choices the online nature of the process. next result [Erdős and Rényi 60] [Bollobás 81] [Krivelevich, Loh, Sudakov 09] [Mütze, S., Thomas 10+] n2n2 N=N(n)N=N(n)

16. The offline setting Achlioptas offline problem: We are given a ‘random r-matched graph’ sample a graph on n vertices and rN edges u.a.r. partition the rN edges into N sets of size r u.a.r. Achlioptas subgraph: select one edge from each r -set. Goal: Find an Achlioptas subgraph that does not contain a copy of F. a a e e b b f f c c g g d d F =F = r = 2

17. The offline setting Achlioptas offline problem: We are given a ‘random r-matched graph’ sample a graph on n vertices and rN edges u.a.r. partition the rN edges into N sets of size r u.a.r. Achlioptas subgraph: select one edge from each r -set. Goal: Find an Achlioptas subgraph that does not contain a copy of F. Note: easier than online setting! ) threshold of this offline problem is upper bound for threshold of Achlioptas process.

18. The offline setting The threshold does not depend on r (in order of magnitude) The threshold is actually ‘semi-sharp’ with constants depending on F and r. Krivelevich, S., Steger (2010)

19. Achlioptas: the full picture n1n1 r=1r=1 n 1. 286 … r=3r=3 n 1. 333 … r=4r=4 n1.2n1.2 r=2r=2 n 1. 499 … r = 1000 [Erdős and Rényi 60] [Bollobás 81] [Krivelevich, Loh, Sudakov 09] [Mütze, S., Thomas 10+] n2n2 N=N(n)N=N(n) n1n1 r=1r=1 n1.5n1.5 r¸2r¸2 [Krivelevich, S., Steger 10] online offline F =

20. The Ramsey process Ramsey process: start with the empty graph on n vertices in each step a new random edge appears color this edge with one of r available colors note: r = 1 ) original Erdős-Rényi process

21. The Ramsey process Ramsey process: start with the empty graph on n vertices in each step a new random edge appears color this edge with one of r available colors note: r = 1 ) original Erdős-Rényi process Friedgut, Kohayakawa, Rödl, Ruci ń ski, Tetali (2003)

22. We extend this result to a large class of graphs F including cliques and cycles of arbitrary size. For these graphs, a simple greedy strategy is optimal. The Ramsey process Marciniszyn, S., Steger (2009)

23. similar result for vertex-coloring analogue and arbitrary number of colors [Marciniszyn, S., SODA ‘07] both results apply to cliques and cycles of arbitrary size, but not e.g. to paths. The general case leads to difficult deterministic (!) combinatorial questions [Belfrage, Mütze, S. 10+], [Mütze, Rast, S. 10+] The Ramsey process Marciniszyn, S., Steger (2009)

24. Summary We investigated by how much the power of choice allows to delay the appearance of some fixed graph F in random graph processes. We showed that essentially best possible guarantees can be achieved by polynomial-time algorithms.