. Approximating Bounded Degree Boolean #CSP . Jingcheng Liu SJTU-ACM Class 2010 October 13, 2013 Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 1 / 16 Approximating #CSP
Outline . . Introduction 1 Constraint Satisfaction Problem A Combinatorial Detour . . Approximate Counting Algorithms 2 . . Hardness and Phase Transitions 3 Hardness Phase Transitions Open Problems Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 2 / 16 Approximating #CSP
Approximation Schemes We are interested primarily in two type of polynomial time approximation scheme: . Definition . (Informally) For given parameter ε > 0 and an instance of a particular problem class, if the algorithm outputs a number ˆ N such that (1 − ε ) N ≤ ˆ N ≤ (1 + ε ) N , where N is the accurate answer of the problem instance, and the running time is bounded by poly ( n, 1 /ε ) with n being the size of instance, this is called the FPTAS (fully polynomial time approximation scheme) . A randomized relaxation of FPTAS is known as FPRAS (fully polynomial time randomized approximation scheme) , which uses random bits and only outputs ˆ N to the desired precision with high probability. . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 3 / 16 Approximating #CSP
. . . . . . Constraint language . Definition . A constraint language is a set of boolean relations, usually denoted as Γ . . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 4 / 16 Approximating #CSP
. . . Constraint language . Definition . A constraint language is a set of boolean relations, usually denoted as Γ . . . Example . OR 2 ( x, y ) is true if x or y is true, and false otherwise. NAND 2 ( x, y ) is false if x and y is true, and true otherwise. Pinning, Pin 0 ( x ) is true if and only if x is false, likewise Pin 1 ( x ) := x . Equality, EQ ( x, y ) is true iff x = y . . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 4 / 16 Approximating #CSP
Constraint language . Definition . A constraint language is a set of boolean relations, usually denoted as Γ . . . Example . OR 2 ( x, y ) is true if x or y is true, and false otherwise. NAND 2 ( x, y ) is false if x and y is true, and true otherwise. Pinning, Pin 0 ( x ) is true if and only if x is false, likewise Pin 1 ( x ) := x . Equality, EQ ( x, y ) is true iff x = y . . . Remark . A boolean relation is in OR-conj if it can be expressed as a conjunction of OR s, similarly for NAND-conj. It’s well-known that given pinning, every boolean relations is either in OR-conj or NAND-conj, or it can simulate equality with only one extra occurrence (a.k.a 3-simulates equality). . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 4 / 16 Approximating #CSP
CSP instance . Definition . A degree d CSP instance with constraint language Γ , denoted as CSP d (Γ) , consists of a set of variables and set of constraints (expressed using relations in Γ ), where each variable appears in at most d constraints. . We are interested in the number of satisfying assignment to such a bounded degree CSP instance, and denote such problem as # CSP d (Γ) . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 5 / 16 Approximating #CSP
. . . Covering Problem vs. Monotone CNF . Definition . An edge cover of a graph is a set of edges such that every vertex has at least one adjacent edge in it. . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 6 / 16 Approximating #CSP
Covering Problem vs. Monotone CNF . Definition . An edge cover of a graph is a set of edges such that every vertex has at least one adjacent edge in it. . . Remark . An alternative view of edge cover is Rtw-Mon-CNF, where each edge is viewed as a Boolean variable and it is connected with two vertices (read twice); and the constraint on each vertex is a monotone OR, requring that at least one of the edge be chosen (assigned True). . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 6 / 16 Approximating #CSP
Covering Problem vs. Monotone CNF . Definition . An edge cover of a graph is a set of edges such that every vertex has at least one adjacent edge in it. . . Remark . An alternative view of edge cover is Rtw-Mon-CNF, where each edge is viewed as a Boolean variable and it is connected with two vertices (read twice); and the constraint on each vertex is a monotone OR, requring that at least one of the edge be chosen (assigned True). Following the same spirit we have Read- k -Monotone CNF is just edge covers in hyper-graph with edge size at most k , or vertex covers (or independent sets) in hyper-graphs with maximum degree k . . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 6 / 16 Approximating #CSP
Covering Problem vs. Monotone CNF . Definition . An edge cover of a graph is a set of edges such that every vertex has at least one adjacent edge in it. . . Remark . An alternative view of edge cover is Rtw-Mon-CNF, where each edge is viewed as a Boolean variable and it is connected with two vertices (read twice); and the constraint on each vertex is a monotone OR, requring that at least one of the edge be chosen (assigned True). Following the same spirit we have Read- k -Monotone CNF is just edge covers in hyper-graph with edge size at most k , or vertex covers (or independent sets) in hyper-graphs with maximum degree k . From the point of view of CSP, Read- k -Monotone CNF is simply CSP k ( { OR 2 } ) , or CSP k ( { NAND 2 } ) . . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 6 / 16 Approximating #CSP
General Paradigm. Let’s take the example of applying correlation decay technique to counting edge covers, in which we first devise subproblem for computation tree recursion, and show its correlation decay property, and eventually from the probability distribution we approximate the count. This result is to appear in SODA’14, and is a joint work with Chengyu Lin and Pinyan Lu. This approach is very similar to the Markov Chain Monte Carlo method, where one instead of designing computation tree recursion, design a Markov Chain, and instead of showing the correlation decay property of the tree recursion, one shows the rapid mixing of the markov chain. Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 7 / 16 Approximating #CSP
Devising subproblems. . Definition . A dangling edge e = ( u, ) of a graph is such singleton edge with exactly one end-point vertex u , as shown in the Figure 1a. A free edge e = ( , ) of a graph is such edge with no end-point vertex. . Note that an alternative view to these combinatorial definitions is from Rtw-Mon-CNF. A dangling edge is simply a variable which only appears at one clause, and a free edge is a variable that does not appear at all, whereas normal edge just corresponds to variables appearing twice. Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 8 / 16 Approximating #CSP
Dangling instances For example, given a degree-3 vertex u with dangling edge e shown in Figure 1a , the result of e 1 − u is shown in Figure 1b and the result of G − e − u ≜ ( G − e ) − u is shown in Figure 1c. e e e 1 e 2 u u ⑥ ⑥ � ❅ ❅ e 1 e 1 � ❅ e 2 ❅ e 2 ⑥ ⑥ � ❅ ❅ ⑥ ⑥ ⑥ ⑥ (c) (a) G (b) e 1 − u G − e − u Figure : Dangling edges examples. Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 9 / 16 Approximating #CSP
Counting and probability distributions Let EC ( G ) denote the set of edge covers for graph G , P ( G, e ) be the probability that an edge is not chosen for a randomly chosen edge cover in EC ( G ) . This P ( G, e ) is the probability distribution in concern. . Proposition . Let Z ( G ) ≜ | EC ( G ) | ̸ = 0 and e 1 , e 2 , . . . , e m be an enumeration of the edges E where e i = ( u i , v i ) . Define G 1 ≜ G, G i ≜ G i − 1 − e i − 1 − u i − 1 − v i − 1 , 1 < i ≤ m . Then 1 Z ( G ) = ∏ m i =1 (1 − P ( G i , e i )) . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 10 / 16 Approximating #CSP
Computation Tree Recursion Denote G ′ ≜ G − e, G ′′ ≜ G − e − u − v , G 1 i ≜ G ′′ − ∑ i − 1 k =1 e k , i ≜ G ′′ − ∑ d 1 i ≜ G ′′ − ∑ i − 1 k =1 e k − ∑ i − 1 G 2 k =1 f k , G 3 k =1 f k , and G 1 ≜ G − e − u , and ∀ i ≥ 2 , G i ≜ G i − 1 − e i − 1 . Let X = ∏ d 1 i =1 P ( G 1 i , e i , L ) , Y = ∏ d 2 i =1 P ( G 2 i , f i , L ) , Z = ∏ d 2 i =1 P ( G 3 i , f i , L ) , L ′ ≜ L − ⌈ log 6 ( d + 1) ⌉ 1 2 , e is free or L ≤ 0. 1 − ∏ d i =1 P ( G i ,e i ,L ′ ) P ( G, e, L ) = i =1 P ( G i ,e i ,L ′ ) , e is dangling. 2 − ∏ d 1 1 − 2+ X · Y − X − Z , otherwise . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 11 / 16 Approximating #CSP
Computation Tree Recursion . Remark . . 1 The major technique for deriving such tree recursion is called self-avoiding walk. . . 2 We truncate the normal computation tree up to recursion depth L . As a side note on this, the recursion depth is modified in such a way that, while maintaining the correlation decay, it overcomes the usual degree bound. . . 3 Our later work improve this up to Read- 5 -Monotone CNF. As we shall see, Read- 6 -Monotone CNF does not admit FPTAS unless RP = NP . . Jingcheng Liu ( SJTU-ACM Class 2010 ) October 13, 2013 12 / 16 Approximating #CSP
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