SCIENTIA SINICA Informationis, Volume 47, Issue 12: 1646-1661(2017) https://doi.org/10.1360/N112016-00248

## Sufficient conditions for convergence of the survey propagation algorithm

• AcceptedDec 12, 2016
• PublishedSep 5, 2017
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### Abstract

Message propagation algorithms are very effective at finding satisfying assignments for SAT instances, and hard regions of a random SAT have become narrower. However, message propagation algorithms do not always converge for some random SAT instances. Unfortunately, a rigorous theoretical proof of this phenomenon is still lacking. The survey propagation (SP) algorithm is very effective at solving SAT instances, and a theoretical analysis of SP is very important in designing other message passing algorithms. Through this study, we derived the sufficient conditions for convergence of SP with extending message [0,~1] to message $(-\infty,\infty)$. Finally, the experiment results show that the conditions for the convergence of SP are very effective in random 3-SAT instances.

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• Figure 1

Factor graph

• Figure 2

Part of factor graph

• Figure 3

Properties change of random 3-SAT instance with parameter $\alpha~=\frac{m}{n}$. (a) The phase transition of satisfaction; (b) the hardness of solving random 3-SAT instance

• Figure 4

(Color online) Properties change of convergence (a) and time (b) of SP with parameter $\alpha~=\frac{m}{n}$.

• Figure 5

(Color online) The change of spectral radius $\rho$ with parameter $\alpha~=\frac{m}{n}$

•

Algorithm 1 SP algorithm

Require:the factor graph of formula $F$; a maximal number of iterations $t_{\rm~max}$; a requested precision $\varepsilon$.

Output:unconvergence if SP has not converged after $t_{\rm~max}$ sweeps. If it has converged: the messages $\eta^*~_{a~\to~i}$.

At time $t=0$: For every edge $a~\to~i$ of the factor graph, randomly initialize the messages $\eta~_{a~\to~i}(t=0)\in~[0,1]$;

For $t=0$ to $t=t_{\rm~max}$:2.1. Sweep the set of edges in a random order, and update sequentially the warning on all the edges of the graph, generating the values $\eta~_{a~\to~i}(t)$, using subroutine SP-UPDATE;2.2. If $|\eta~_{a~\to~i}(t)-\eta~_{a~\to~i}(t-1)|<\varepsilon$ on all the edges, the iteration has converged and generated $\eta^*~_{a~\to~i}=\eta~_{a~\to~i}(t)$: go to 3;

If $t==t_{\rm~max}$ return UN-CONVERGENCE. If $t<t_{\rm~max}$ return the set of fixed point survey $\eta^*~~_{a~\to~i}=\eta~_{a~\to~i}(t)$.

•

Algorithm 2 Subroutine SP-UPDATE ($\eta~_{a~\to~i}$)

Require:set of all messages arriving onto each variable node $j~\in~V(a)\backslash~i$.

Output:new value for the message $\eta~_{a~\to~i}$.

For every $j\in~V(a)\verb|\|i$, compute the three numbers $\Pi^u_{j~\to~a}$, $\Pi^s_{j~\to~a}$, $\Pi^0_{j~\to~a}$; If a set like $V^{s}_{a}(j)$ is empty, the corresponding product takes value 1 by definition;

Using these numbers, compute and return the new survey $\eta~_{a~\to~i}$.

•

Algorithm 3 SID algorithm

Require:the factor graph of formula $F$; a maximal number of iterations $t_{\rm~max}$; a requested precision $\varepsilon$.

Output:one assignment which satisfies all clauses, or SP UN-CONVERGENCE, or probably UNSAT.

Random initial condition for the surveys;

Run SP. If SP does not converge, return SP UN-CONVERGENCE and stop. If SP converges, use the fixed-point surveys $\eta^*_{a~\to~i}$ in order to:

Decimate: 3.1. If non-trivial surveys $(\eta~\ne~0)$ are found, then(a) Evaluate, for each variable node $i$, the three `biases $\{W^+_{i},W^-_{i},W^0_{i}\}$;(b) Fix the variable with the largest $|W^+_{i}-W^-_{i}|$ to the value $x_{i}=1$ if $|W^+_{i}>W^-_{i}|$, to the value $x_{i}=0$ if$|W^+_{i}<W^-_{i}|$. Clean the graph;3.2. If all surveys are trivial $(\eta~=~0)$, then output the simplified sub-formula and run on it a local search process (e.g., WALKSAT);

If the problem is solved completely by unit clause propagation, then output SAT and stop. If no contradiction is found then continue the decimation process on the smaller problem, go to 1, else stop.

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