The Probabilistic Method

Transcription

1 The Probabilistic Method Ted, Cole, Reilly, Manny Combinatorics Math 372 November 23, 2015 Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

2 Overview 1 History created by Erds Paul Erds himself said, during his 80th birthday conference, that he believes the method will live long after him. 2 Basics describe the probability that an element of a set has a certain property use that to deduce the existence or nonexistence of elements with that property 3 Example: Planar Graphs 4 Example: Liar Game The Chip-Liar Game Expected Values Existence of a Winner Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

3 Planar Graphs Projecting simple graphs into the plane n vertices, m edges result from earlier: Planar graph G has at most 3n-6 elements Define cr(g) as minimum # of crossings in drawings of G cr(g)=0 iff G planar Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

4 basic rules no edge crosses itself edges with common end vertex cannot cross no two edges cross twice these all follow from redrawing Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

5 Non-probabilistic Result suppose G drawn with cr(g) crossings consider H where a new vertex is placed at each crossing H has n+cr(g) vertices and m+2cr(g) edges invoking bound we get m+2cr(g) 3(n+cr(G))-6 cr(g) m-3n+6 Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

6 Probabilistic Result What if there are many more vertices then edges? We can get a better bound using probabilistic method Assume m 4n Choose a subgraph G p including each vertex with probability p. Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

7 Probabilistic Result Let n p, m p, X p be random variables counting vertices, edges and crossings in G p Since cr(g)-m+3n 0 holds for all graphs, E(X p m p + 3n p ) 0 E(n p ) = pn, E(m p ) = p 2 m, E(X p ) = p 4 cr(g) By linearity of expectation, 0 p 4 cr(g) p 2 m + 3pn cr(g) m p 2 3n p 3 Choose p = 4n m cr(g) 1 m 3 64 n 2 Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

8 Example: The Liar Game Paul asks q questions in the form Is xɛs? for any subset S Determining xɛ {1,..., n} Carole doesn t necessarily need to select an x beforehand, but at least one x must be consistent with her answers. For which n numbers, q questions, and k lies are known by both Carole and Paul. Can Paul determine the number? Use Probabilistic method to prove the existence of a winning strategy for one player. Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

9 An Equivalent Game: Chip-Liar Chip for each x. Board is a method for keeping track of lies each x has remaining. Used to keep track of the possibility that each answer x S and x S could be a lie. Chips are removed from the board past the 0th slot, they are not possible x s. If there are more than one chips on the board at the end of the game then Paul cannot determine between them which is x and Carol wins. Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

10 Playing Chip-Liar: Examples Example 1 Imagine asking Is x {1, 2,..., n}?. Clearly it is. Carole could respond no, but would certainly use a lie and move the whole stack left one unit. Example 2 Imagine asking Is x {1, 2}? Carole could answer No, and has either lied or told the truth. Or she could answer Yes and has either lied or told the truth. At each round Carole can choose to either more S or S c one space to the left. Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

11 Does a Chip Survive: Expected Value Consider Carole playing randomly, moving S or S c. For each c {1,..., n} let I c is 1 iff c is on the board after q rounds If c is on position j, c will survive if chosen j times. Probability of Survival ( ) E [I c ] = 1 j q def 2 q i=0 = B(q, j) i B(q, j) to be the probability that a chip at j survives q rounds. Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

12 Expected Number of Chips Number chips on each space is given by {x 1,..., x k } The expected total number of chips is C is the number of chips in a position times the probability each will survive the game Expected total number of chips. E [C] = c E [I c] = k i=0 x ib(q, i) Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

13 Existence of a Winner Game Theory: Theorem In a game with perfect information, no hidden moves and no draws one player has a perfect winning strategy. E [C] weighted average of possibilities. If E [C] > 1, E [C] = C 1 µ (C 1 ) C p µ (C p ) > 1 implies C k > 1 with µ (C k ) > 0 If some C k > 1 with µ (C k ) > 0 then Carole is able to win. So no strategy allows Paul to always win. But by the Theorem someone has a perfect strategy that always wins, so it must be Carole. Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

14 Corollary E [C] = k i=0 x ib(q, i) gives k i=0 x ib(q, k) = {x 1,..., x k } = {0,..., n} When E [C] > 1 and Carole wins this gives k i=0 x ib(q, k) > 1 Rearranges to: k i=0 x i > 2q B(q,k) and k i=0 x i = n yield Corrolary - Carole wins the n number, q question, and k lie game n > k i=0 2 q q i (Probability of removing an x) * (the number of x s) > (total number of subsets of x s) Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

15 De-randomization Random play is expected to yield the same result, so the initial position will decide a better move. With l moves remaining Carole s dominant strategy is to maximize the expected number of chips on the board playing randomly after moving S or S c. Strategy E if E S [ C ] = ( ) x 1,..., x k and S c k x i B(l, i) and E i=0 [ C ] > E ( ) x 1,..., x k [ C ] = k i=0 x i B(l, i) [ C ] then move S, otherwise move S c Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

16 Homework Problems The Ramsey-number R(k, l) is the smallest number n such that in any two-coloring of the edges of a complete graph on n vertices K n by red and blue, either there is a red K k (i.e., a complete subgraph on k vertices all of whose edges are colored red) or there is a blue K l. 1 Draw the complete graph on 6 vertices, K 6 that is two-colored with a complete subgraph on 4 vertices with all edges colored the same. 2 Use the probabilistic method to prove this lower bound on diagonal Ramsey numbers, R(k, k): ( ) n 1 If < 1, then R(k, k) > n. k 2 (k 2) 1 (Hint: Consider the probability that a subgraph K k of the complete graph K n is monochromatic.) Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

17 References Noga Alon and Joel H. Spencer (2000) The Probabilistic Method (2nd Ed.) New York: Wiley-Interscience. ISBN Martin Aigner, Gnter M. Ziegler (2010) Proofs from THE BOOK (4th Ed.) Springer Publishing Company, Incorporated. ISBN: Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18

18 The End Ted, Cole, Reilly, Manny (Reed) The Probabilistic Method November 23, / 18