SCIENCE CHINA Information Sciences, Volume 62 , Issue 11 : 212104(2019) https://doi.org/10.1007/s11432-019-9881-0

## Accelerating MUS enumeration by inconsistency graph partitioning

• ReceivedJan 11, 2019
• AcceptedApr 19, 2019
• PublishedOct 9, 2019
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### Abstract

The problem of finding minimal unsatisfiable subsets (MUSes) has been studied frequently because of itstheoretical importance and wide range of applications in domains such as electronic design automation,software, and integrated circuit verification. In this paper, a method for accelerating theenumeration of MUSes based on inconsistency graph partitioning is proposed. First, an inconsistency graphof a set of clauses is constructed by extracting the inconsistencyrelations between literals of different clauses. In this paper, we show that by partitioning the inconsistency graphinto small connected components through a vertex cut, the enumeration of MUSes in different componentsbecomes independent and it is possible to compute them separately. Moreover, the MUSes of the original clause setcan be constructed by merging the unit clauses in the MUSes of these connected components back into the clauses inthe vertex cut. Experiments show that by integrating the acceleration method into the MARCO MUSes enumerator,there is a 2–3 times improvement in the average runtime of solved instances for randomly generatedbenchmarks. By integrating the acceleration method into itself as an MUS enumerator, there isanother 3–4 times improvement when compared with the accelerated MARCO.

### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61690202, 61502022) and State Key Laboratory of Software Development Environment (Grant No. SKLSDE-2017ZX-17).

### References

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• Figure 1

(Color online) Modularity of inconsistency graph partitioning.

• Figure 2

(Color online) Comparison of MARCO, GPMUS+MARCO, and GPMUS+GPMUS. (a) $N_{\text{TO}}$; (b) $T^-_{\text{Ave}}$;protect łinebreak (c) $T_{\text{Ave}}$.

•

Algorithm 1 $\text{GPMUS}(\Gamma_S,~\{\Gamma_1,~\ldots,~\Gamma_k\})$

Require:$\Gamma$ is a clause set whose inconsistency graph is $G$; $\Gamma_S~=~\{A_1,~\ldots,~A_m\}$ is a subset of $\Gamma$ whose corresponding vertex set is a cut of $G$; $\{\Gamma_1,~\ldots,~\Gamma_k\}$ is the set of corresponding clause sets of connected components after a cut.

Output:The set of all MUSes of $\Gamma$.

for $i~=~1$ TO $k$

Compute the set $\mathcal{M}_i$ of all MUSes of $\Gamma_i~\cup~\mathcal{C}(\Gamma_i,~\Gamma_S)$;

end for

Compute the set $\mathcal{M}_S$ of all MUSes of $\mathcal{L}(\Gamma_S)$;

$M_0~:=~(\bigcup_{i=1}^{k}~\mathcal{M}_i)~\cup~\mathcal{M}_S$;

for $i~=~1$ TO $m$

if $A_i$ is not a unit clause then

$M_i~:=~\text{Merge}(M_{i-1},~A_i)$;

else

$M_i~:=~M_{i-1}$;

end if

end for

return $M_m$.

• Table 1   Comparison of MARCO with GPMUS+MARCO
 Classes MARCO GPMUS+MARCO $N_{\text{TO}}$ $T^-_{\text{Ave}}$ $T_{\text{Ave}}$ $N_{\text{TO}}$ $T^-_{\text{Ave}}$ $T_{\text{Ave}}$ mus100 12 3.71 21.48 6 1.85 10.79 mus200 76 16.93 124.50 44 4.79 69.74 mus400 182 71.01 279.39 113 20.92 178.60 mus600 200 – 300 161 22.03 245.80 mus800 200 – 300 186 39.76 281.78 mus1000 200 – 300 194 16.02 291.48
• Table 2   Comparison of GPMUS+MARCO with GPMUS+GPMUS
 Classes GPMUS+MARCO GPMUS+GPMUS $N_{\text{TO}}$ $T^-_{\text{Ave}}$ $T_{\text{Ave}}$ $N_{\text{TO}}$ $T^-_{\text{Ave}}$ $T_{\text{Ave}}$ mus100 6 1.85 10.79 4 0.48 6.47 mus200 44 4.79 69.74 17 1.58 26.95 mus400 113 20.92 178.60 35 2.21 54.32 mus600 161 22.03 245.80 54 4.98 84.63 mus800 186 39.76 281.78 63 1.55 95.56 mus1000 194 16.02 291.48 69 2.04 104.83
•

Algorithm 2 $\text{Merge}(M_{i-1},~A_i)$

Require:$M_{i-1}$ is the set of all MUSes of $\Sigma_{i-1}$; $A_i~=~L^i_1~\lor~\cdots~\lor~L^i_{n_i}$ is a clause.

Output:The set of all MUSes of $\Sigma_{i}$.

$M&apos;_{i}~:=~\emptyset$;

$N_{i}:=~\{\Phi~\mid~\Phi~\in~M_{i-1}~\text{~and~}~\Phi~\cap~\{L^i_1,~\ldots,~L^i_{n_i}\}=\emptyset\}$;

$S_{i}~:=~\{(\Phi_1^i,~\ldots,~\Phi_{n_i}^i)~\mid~\Phi^i_j~\in~M_{i-1},~L^i_j~\in~\Phi^i_j,~1~\leq~j~\leq~n_i\}$;

for all $(\Phi_1^i,~\ldots,~\Phi_{n_i}^i)~\in~S_i$

$\Phi&apos;~:=~\bigcup_{j~=1}^{n_i}~(\Phi^i_j~-~\{L^i_j\})$;

if literals in $\Phi&apos;$ are all from different clauses then

$M&apos;_{i}~:=~M&apos;_{i}~\cup~\{\{A_i\}~\cup~\Phi&apos;\}$;

end if

end for

$M_i~:=~\MS(N_{i}~\cup~M&apos;_{i})$;

return $M_i$.

•

Algorithm 3 $\text{Partition}(G)$

Require:$G$ is the inconsistency graph of $\Gamma$;

Output:A vertex cut of $G$.

$\mathcal{S}~:=~\emptyset$;

Filter out all clauses in $\Gamma$ that contain pure literals;

Group vertices of $G$ such that each group is a smallest balanced set;

Partition $G$ using the Louvain algorithm and obtain connected components $K_1,~\ldots,~K_s$;

while there exist $K_i$ and $K_j$ that are connected in $G$ do

Compute the number $N$ of components that intersect with the corresponding group of each edge vertex;

Select all groups that have the maximal number $N$ as $\mathcal{C}$;

if $|\mathcal{C}|~=~1$ then

Let the unique group in $\mathcal{C}$ be $\Phi$;

else

Compute the number of edges (degree) connecting vertices in each group of $\mathcal{C}$ to other components;

Select a group $\Phi$ with the maximal degree;

end if

Delete all edges in $G$ connect to vertices in $\Phi$;

$\mathcal{S}~:=~\mathcal{S}~\cup~\Phi$;

end while

return $\mathcal{S}$.

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