Approximating Bounded Degree Boolean #CSP . Jingcheng Liu - - PowerPoint PPT Presentation

. .

Approximating Bounded Degree Boolean #CSP

Jingcheng Liu

SJTU-ACM Class 2010

October 13, 2013

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 1 / 16

Outline

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1

Introduction Constraint Satisfaction Problem A Combinatorial Detour . .

2

Approximate Counting Algorithms . .

3

Hardness and Phase Transitions Hardness Phase Transitions Open Problems

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 2 / 16

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

• nly outputs ˆ

N to the desired precision with high probability.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 3 / 16

Constraint language

.

Definition

. . A constraint language is a set of boolean relations, usually denoted as Γ. . . . . . .

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 4 / 16

Constraint language

.

Definition

. . A constraint language is a set of boolean relations, usually denoted as Γ. .

Example

. . OR2(x, y) is true if x or y is true, and false otherwise. NAND2(x, y) is false if x and y is true, and true otherwise. Pinning, Pin0(x) is true if and only if x is false, likewise Pin1(x) := x. Equality, EQ(x, y) is true iff x = y. . . .

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 4 / 16

Constraint language

.

Definition

. . A constraint language is a set of boolean relations, usually denoted as Γ. .

Example

. . OR2(x, y) is true if x or y is true, and false otherwise. NAND2(x, y) is false if x and y is true, and true otherwise. Pinning, Pin0(x) is true if and only if x is false, likewise Pin1(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 ORs, 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) Approximating #CSP October 13, 2013 4 / 16

CSP instance

.

Definition

. . A degree d CSP instance with constraint language Γ, denoted as CSPd(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 #CSPd(Γ).

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 5 / 16

Covering Problem vs. Monotone CNF

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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) Approximating #CSP October 13, 2013 6 / 16

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) Approximating #CSP October 13, 2013 6 / 16

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) Approximating #CSP October 13, 2013 6 / 16

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 CSPk({OR2}), or CSPk({NAND2}).

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 6 / 16

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) Approximating #CSP October 13, 2013 7 / 16

Devising subproblems.

.

Definition

. . A dangling edge e = (u, ) of a graph is such singleton edge with exactly

• ne 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

• ne 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) Approximating #CSP October 13, 2013 8 / 16

Dangling instances

For example, given a degree-3 vertex u with dangling edge e shown in Figure 1a , the result of e1 − u is shown in Figure 1b and the result of G − e − u ≜ (G − e) − u is shown in Figure 1c.

⑥

• ⑥

❅ ❅ ❅ ⑥

u e e1 e2

(a) G ⑥ ⑥ ❅ ❅ ❅ ⑥

u e e1 e2

(b) e1 − u ⑥ ⑥

e1 e2

(c) G − e − u Figure : Dangling edges examples.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 9 / 16

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 e1, e2, . . . , em be an enumeration of the edges E where ei = (ui, vi). Define G1 ≜ G, Gi ≜ Gi−1 − ei−1 − ui−1 − vi−1, 1 < i ≤ m. Then Z(G) = 1 ∏m

i=1(1 − P(Gi, ei))

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 10 / 16

Computation Tree Recursion

Denote G′ ≜ G − e, G′′ ≜ G − e − u − v, G1

i ≜ G′′ − ∑i−1 k=1 ek,

G2

i ≜ G′′ − ∑d1 k=1 ek − ∑i−1 k=1 fk, G3 i ≜ G′′ − ∑i−1 k=1 fk, and

G1 ≜ G − e − u, and ∀i ≥ 2, Gi ≜ Gi−1 − ei−1. Let X = ∏d1

i=1 P(G1 i , ei, L), Y = ∏d2 i=1 P(G2 i , fi, L), Z = ∏d2 i=1 P(G3 i , fi, L),

L′ ≜ L − ⌈log6 (d + 1)⌉ P(G, e, L) =       

1 2, e is free or L ≤ 0. 1−∏d

i=1 P(Gi,ei,L′)

2−∏d

i=1 P(Gi,ei,L′), e is dangling.

1 −

1 2+X·Y −X−Z , otherwise.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 11 / 16

Computation Tree Recursion

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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) Approximating #CSP October 13, 2013 12 / 16

Hardness

(More than) Ten years of story to tell. .

Remark

. . It has been long conjectured that d = 6 for approximately counting independent sets in graphs with maximum degree d is hard, for which a weighted version coincides exactly with a phase transition in the hardcore model in statistical physics.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 13 / 16

Hardness

(More than) Ten years of story to tell. .

Remark

. . It has been long conjectured that d = 6 for approximately counting independent sets in graphs with maximum degree d is hard, for which a weighted version coincides exactly with a phase transition in the hardcore model in statistical physics. This correspondence is first rigorously established first by Allan Sly (FOCS 2010) on a small region around this uniqueness threshold and then together with Nike Sun, they established the whole correspondence in a FOCS 2012 paper.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 13 / 16

Hardness

(More than) Ten years of story to tell. .

Remark

. . It has been long conjectured that d = 6 for approximately counting independent sets in graphs with maximum degree d is hard, for which a weighted version coincides exactly with a phase transition in the hardcore model in statistical physics. This correspondence is first rigorously established first by Allan Sly (FOCS 2010) on a small region around this uniqueness threshold and then together with Nike Sun, they established the whole correspondence in a FOCS 2012 paper. Their proof is utilized a symmetry breaking phenomenon on regular bipartite graph, and construct gadget to encode MAX-CUT.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 13 / 16

Hardness

(More than) Ten years of story to tell. .

Remark

. . It has been long conjectured that d = 6 for approximately counting independent sets in graphs with maximum degree d is hard, for which a weighted version coincides exactly with a phase transition in the hardcore model in statistical physics. This correspondence is first rigorously established first by Allan Sly (FOCS 2010) on a small region around this uniqueness threshold and then together with Nike Sun, they established the whole correspondence in a FOCS 2012 paper. Their proof is utilized a symmetry breaking phenomenon on regular bipartite graph, and construct gadget to encode MAX-CUT. Their results enabled a easy classification for CSPd(Γ) with d ≥ 6 (STACS 2010 by Martin E. Dyer et.al.).

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 13 / 16

Phase Transitions

Many of such non-rigorous results for predicting the computational hardness transition arise in the context of statistical physics. .

Remark

. . Hardness of approximate counting is closely related to the uniqueness

• f the translation-invariant Gibbs measure on the infinite d-regular

computation tree, i.e. uniqueness usually implies short-range correlation (thus an FPTAS is likely) and non-uniqueness coincides with long-range correlation (thus a hardness is likely).

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 14 / 16

Phase Transitions

Many of such non-rigorous results for predicting the computational hardness transition arise in the context of statistical physics. .

Remark

. . Hardness of approximate counting is closely related to the uniqueness

• f the translation-invariant Gibbs measure on the infinite d-regular

computation tree, i.e. uniqueness usually implies short-range correlation (thus an FPTAS is likely) and non-uniqueness coincides with long-range correlation (thus a hardness is likely). Although a rigorous structural result like this would be of great interest, even for a particular problem class it’s not yet thoroughly

• understood. By far the short-range correlation in counting matching

has only enabled an FPTAS for constant-degree bounded graph.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 14 / 16

Phase Transitions

Many of such non-rigorous results for predicting the computational hardness transition arise in the context of statistical physics. .

Remark

. . Hardness of approximate counting is closely related to the uniqueness

• f the translation-invariant Gibbs measure on the infinite d-regular

computation tree, i.e. uniqueness usually implies short-range correlation (thus an FPTAS is likely) and non-uniqueness coincides with long-range correlation (thus a hardness is likely). Although a rigorous structural result like this would be of great interest, even for a particular problem class it’s not yet thoroughly

• understood. By far the short-range correlation in counting matching

has only enabled an FPTAS for constant-degree bounded graph. In view of this our work provides evidence in support of such a structural conjecture, and hopefully the technique used here can be generalized to other settings.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 14 / 16

Open Problems: Lower degrees?

Recall that with 3-simulating equality, one has simple classification of constraint language Γ in #CSP3(Γ): either Γ is in OR-conj or NAND-conj, or #CSP3(Γ) instance is as hard as general #SAT. And our results already showed an FPTAS for Read-5-Monotone CNF, bridging up the OR cases up to degree 5.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 15 / 16

Open Problems: Lower degrees?

Recall that with 3-simulating equality, one has simple classification of constraint language Γ in #CSP3(Γ): either Γ is in OR-conj or NAND-conj, or #CSP3(Γ) instance is as hard as general #SAT. And our results already showed an FPTAS for Read-5-Monotone CNF, bridging up the OR cases up to degree 5. However this is still non-trivial to complete the classification of #CSP, as in the language of #CSP, in particular, #CSP3(∆), #CSP3(K4) has FPTAS, which actually corresponds to instances in Read-6-Monotone CNF and Read-9-Monotone CNF.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 15 / 16

Open Problems: Lower degrees?

Recall that with 3-simulating equality, one has simple classification of constraint language Γ in #CSP3(Γ): either Γ is in OR-conj or NAND-conj, or #CSP3(Γ) instance is as hard as general #SAT. And our results already showed an FPTAS for Read-5-Monotone CNF, bridging up the OR cases up to degree 5. However this is still non-trivial to complete the classification of #CSP, as in the language of #CSP, in particular, #CSP3(∆), #CSP3(K4) has FPTAS, which actually corresponds to instances in Read-6-Monotone CNF and Read-9-Monotone CNF. For degree 2, we have FPTAS for #CSP2(Kc+1) for any constant c, this turns out to be the same as counting matchings.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 15 / 16

Open Problems: Lower degrees?

Recall that with 3-simulating equality, one has simple classification of constraint language Γ in #CSP3(Γ): either Γ is in OR-conj or NAND-conj, or #CSP3(Γ) instance is as hard as general #SAT. And our results already showed an FPTAS for Read-5-Monotone CNF, bridging up the OR cases up to degree 5. However this is still non-trivial to complete the classification of #CSP, as in the language of #CSP, in particular, #CSP3(∆), #CSP3(K4) has FPTAS, which actually corresponds to instances in Read-6-Monotone CNF and Read-9-Monotone CNF. For degree 2, we have FPTAS for #CSP2(Kc+1) for any constant c, this turns out to be the same as counting matchings. We also showed that CSP5(S2) is hard, and conjectured that both CSP4(S2), CSP4(∆) is hard.

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 15 / 16

Thank you!

Jingcheng Liu (SJTU-ACM Class 2010) Approximating #CSP October 13, 2013 16 / 16