The Probabilistic Method Techniques Union bound Argument from - PowerPoint PPT Presentation

The Probabilistic Method Techniques Union bound Argument from expectation Alterations The second moment method The (Lovasz) Local Lemma And much more Alon and Spencer, “The Probabilistic Method” Bolobas, “Random Graphs” � Hung Q. Ngo (SUNY at Buffalo) c CSE 694 – A Fun Course 1 / 9

Lovasz Local Lemma: Main Idea Recall the union bound technique: want to prove Prob [ A ] > 0 ¯ A ⇒ (or ⇔ ) some bad events B 1 ∪ · · · ∪ B n done if Prob [ B 1 ∪ · · · ∪ B n ] < 1 Could also have tried to show Prob [ ¯ B 1 ∩ · · · ∩ ¯ B n ] > 0 Would be much simpler if the B i were mutually independent, because n Prob [ ¯ B 1 ∩ · · · ∩ ¯ � Prob [ ¯ B n ] = B i ] > 0 i =1 Main Idea Lovasz Local Lemma is a sort of generalization of this idea when the “bad” events are not mutually independent � Hung Q. Ngo (SUNY at Buffalo) c CSE 694 – A Fun Course 2 / 9

PTCF: Mutual Independence Definition (Recall) A set B 1 , . . . , B n of events are said to be mutually independent (or simply independent) if and only if, for any subset S ⊆ [ n ] , �� � � Prob B i = Prob [ B i ] i ∈ S i ∈ S Definition (New) An event B is mutually independent of events B 1 , · · · , B k if, for any subset S ⊆ [ k ] , � � � Prob B | B i = Prob [ B ] i ∈ S Question: can you find B, B 1 , B 2 , B 3 such that B is mutually independent of B 1 and B 2 but not from all three? � Hung Q. Ngo (SUNY at Buffalo) c CSE 694 – A Fun Course 3 / 9

PTCF: Dependency Graph Definition Given a set of events B 1 , · · · , B n , a directed graph D = ([ n ] , E ) is called a dependency digraph for the events if every event B i is independent of all events B j for which ( i, j ) / ∈ E . What’s a dependency digraph of a set of mutually independence events? Dependency digraph is not unique ! � Hung Q. Ngo (SUNY at Buffalo) c CSE 694 – A Fun Course 4 / 9

The Local Lemma Lemma (General Case) Let B 1 , · · · , B n be events in some probability space. Suppose D = ([ n ] , E ) is a dependency digraph of these events, and suppose there are real numbers x 1 , · · · , x n such that 0 ≤ x i < 1 � Prob [ B i ] ≤ x i (1 − x j ) for all i ∈ [ n ] ( i,j ) ∈ E Then, � n � n � ¯ � Prob B i ≥ (1 − x i ) i =1 i =1 � Hung Q. Ngo (SUNY at Buffalo) c CSE 694 – A Fun Course 5 / 9

The Local Lemma Lemma (Symmetric Case) Let B 1 , · · · , B n be events in some probability space. Suppose D = ([ n ] , E ) is a dependency digraph of these events with maximum out-degree at most ∆ . If, for all i , 1 Prob [ B i ] ≤ p ≤ e (∆ + 1) then � n � � ¯ Prob B i > 0 . i =1 The conclusion also holds if 1 Prob [ B i ] ≤ p ≤ 4∆ � Hung Q. Ngo (SUNY at Buffalo) c CSE 694 – A Fun Course 6 / 9

Example 1: Hypergraph Coloring G = ( V, E ) a hypergraph, each edge has ≥ k vertices Each edge f intersects at most ∆ other edges Color each vertex randomly with red or blue B f : event that f is monochromatic 2 1 2 | f | ≤ Prob [ B f ] = 2 k − 1 There’s a dependency digraph for the B f with max out-degree ≤ ∆ Theorem G is 2 -colorable if 1 1 2 k − 1 ≤ e (∆ + 1) � Hung Q. Ngo (SUNY at Buffalo) c CSE 694 – A Fun Course 7 / 9

Example 2: k -SAT Theorem In a k -CNF formula ϕ , if no variable appears in more than 2 k /e clauses, then ϕ is satisfiable. Recently Moser (and Moser-Tardos) showed how to find such a truth assignment. � Hung Q. Ngo (SUNY at Buffalo) c CSE 694 – A Fun Course 8 / 9

Example 3: Edge-Disjoint Paths N a directed graph with n inputs and n outputs From input a i to output b i there is a set P i of m paths In switching networks, we often want to find (or want to know if there exists) a set of edge-disjoint ( a i → b i ) -paths Theorem Suppose 8 nk ≤ m and each path in P i shares an edge with at most k paths in any P j , j � = i . Then, there exists a set of edge-disjoint ( a i → b i ) -paths. � Hung Q. Ngo (SUNY at Buffalo) c CSE 694 – A Fun Course 9 / 9