I'm a professor of computer
science at the University of Picardie Jules Verne. My research is concerned
with the practical resolution of NP-hard problems, including SAT, CSP, MaxSAT, MinSAT, MaxClique, GCP. I am particularly interested in the intrinsical relationships between these problems. One of my
research directions is to find and exploit these relationships to solve them. A
recent example is the exploitation of the relationships between MaxSAT and MaxClique to solve MaxClique, see below.

**We
developed solvers for SAT, MaxSAT, MinSAT, MaxClique and Graph
Coloring. Their source codes are available below under the MIT Licence. Thanks for letting me know all your remarks or suggestions.**

**New: MaxCDCL1.2,**** a MaxSAT solver combining branch-and-bound and clause
learning, implemented by Chu-Min Li based on Maple_LCM
with the help of Jordi Coll. Its source code is available here.**

**Reference:
Chu-Min Li, Zhenxing Xu, Jordi Coll,
Felip Manyà, Djamal Habet, Kun He, Combining
Clause Learning and Branch and Bound for MaxSAT, in
CP-2021, to appear**

**Solvers
for SAT:**

Ø **Maple_CM**** :**** it won
the bronze medal of the main track ****of
the SAT competition 2017. ****Its
source code, as well as the detailed results, is available in the sat competition web page, or here.**

**Reference: **

Chu-Min Li, Fan
Xiao, Mao Luo, Felip Manyà, Zhipeng Lü, Yu Li, Clause vivification by unit
propagation in CDCL SAT solvers, Artificial
Intelligence, volume 279, February 2020, 103197, 23 pages,
https://doi.org/10.1016/j.artint.2019.103197

Ø **Maple_LCM_Dist**** and Maple_LCM
(Maple+ in the set): they won the main track of the SAT competition 2017 (see ****the sat competition web page**)**. **The source codes
of these solvers (for Mac OS and Linux, respectively) are available here.

**Reference: **

Mao Luo, Chu-Min Li, Fan Xiao,
Felip Many, Zhipeng Lu, An Effective Learnt
Clause Minimization Approach for CDCL SAT Solvers, In Proceedings of the
Twenty-Sixth International Joint Conference on Artificial Intelligence
(IJCAI-17), 2017, Melbourne, Pages 703-711.

**Solvers for ****the
Maximum Weight Clique Problem (weighted MaxClique):**

Ø **TSM-MWC: **it is an exact (BnB) solver highly effective for small, medium and massive
graphs, with or without very different vertex weight. It is the first exact
algorithm that reaches the best performance in both small/medium and massive
real-world graphs. Its source code is available here.

**Reference:**

Hua
Jiang, Chu-Min
Li, Yanli Liu, Felip Manya, A
Two-Stage MaxSAT Reasoning Approach for the Maximum
Weight Clique Problem, in proceedings of The Thirty-Second AAAI Conference on
Artificial Intelligence (AAAI-18), pages 1338-1346.

Ø
**WLMC: **it is
the first exact solver that finds better solutions than heuristic solvers in
shorter time in large graphs, even when including the time needed for proving
the optimality of its solutions, *refuting
**the prevailing
hypothesis that exact MaxClique algorithms, despite
proving optimality, are less adequate for large graphs than heuristic
algorithms. *A new release (Release
2017) of the solver (wlmc2) able to handle very large vertex weights is available here.

**Reference:**

Hua Jiang, Chu-Min Li, Felip Manya, An Exact
Algorithm for the Maximum Weight Clique Problem in large graphs. In
Proceedings of 31nd AAAI Conference on Artificial Intelligence (AAAI-17), AAAI
Press, Vol 2, pages 830-838, San Francisco, USA, 2017.

**Solvers for ****the
Maximum Clique Problem (non-Weighted MaxClique):**

Ø **LMC****: **A very efficient exact (BnB) solver for the
maximum clique problem in large graphs. It combines efficient
preprocessing to eliminate vertices that cannot belong to a maximum clique and
incremental MaxSAT reasoning to prune branches that
cannot lead to a maximum clique. The source code of
the algorithm is available here.

**Reference:**

Hua Jiang, Chu-Min Li, Felip
Manya, Combining
efficient preprocessing and incremental MaxSAT
reasoning for MaxClique in large graphs. In
Proceedings of 22nd European Conference On Artifical Intelligence (ECAI-2016), pages 939-947, 2016.

Ø **MoMC****, SoMC, DoMC:
**three
efficient exact (BnB) solvers for the maximum clique
problem SoMC (Static ordering MaxClique
Solver) and DoMC (Dynamic ordering MaxClique Solver) correspond to SoMC2 and DoMC2 described
in the paper below. The paper presents also MoMC
(Mixed ordering MaxClique Solver), which switches
cleverly between the static and the dynamic vertex orderings and achieves
better performance than SoMC and DoMC
in general for solving MaxClique. The source code of the
algorithms is available here.

**Reference:**

Chu-Min Li, Hua Jiang, Felip
Manya, On
Minimization of the Number of Branches in Branch-and-Bound Algorithms for the
Maximum Clique Problem, Computers & Operations Research 84 (2017): 1-15

Compile for obtaining SoMC,
DoMC and MoMC,
respectively, from the same source:

gcc MoMC2016.c -O3 -DSOMC –o SoMC

gcc MoMC2016.c -O3 -DDOMC –o DoMC

gcc MoMC2016.c -O3 -DMOMC -o MoMC

for
finding a maximum clique, run ./MoMC
inputGraphInDimacsFormat

for
listing all maximum cliques

./MoMC instance -a n (where n is
the max number of cliques to be printed)

Ø **IncMC2:**** **it uses similar technology to SoMC and is
better than SoMC. When solving MaxClique
using a BnB procedure, a prevailing hypothesis is
that the larger the initial (incumbent) clique, the easier the search, because
it is believed that larger incumbent would allow to prune more search
space. *However, the performance of IncMC2 refutes this hypothesis*, as
showed in the paper below. Indeed, IncMC2 solves (i.e., finds and proves an
optimal solution of) certain instances more quickly from scratch than from an
optimal initial clique size, thanks to incremental upper bound.

The source of
IncMC2 is available here.

Use

./IncMC2 graphFile

to
solve the graph from scratch.

Use

./IncMC2 graphFile –i lb

to
find a clique larger than lb, where lb is a positive integer. When lb
is the optimal solution size, this command proves the optimality of lb.

**Reference:**

Chu-Min Li, Zhiwen Fang, Hua Jiang, Ke Xu, Incremental
Upper Bound for the Maximum Clique Problem, INFORMS Journal on
Computing, 2018, vol. 30, no. 1, pp. 137–153

**Other solvers:**

**Color6:
A Complete (Exact) Algorithm for Graph k-Coloring**

This algorithm solves efficiently the k-coloring problem by telling how
to color the vertices of a graph with k- colors if this is possible or proving
the graph cannot be colored with k colors. The source code of color6 is available here.

Compiling: gcc color6.c –O3 –o color6

Running: ./color6 inputGraphInDIMACSformat
–nbColors k

**Reference:**

Zhaoyang Zhou, Chu-Min
Li, Chong Huang, Ruchu Xu. An
exact algorithm with learning for the graph coloring problem. Computers
& Operations Research, 51(2014) pp 282-301.

**MaxCLQ**** for MaxClique**

**- Combining Incremental Upper Bound and MaxSAT
reasoning for MaxClique**

MaxSAT reasoning is powerful in computing upper bounds for the cardinality of
a maximum clique of a graph. However, existing upper bounds based on MaxSAT reasoning have two drawbacks: (1) at every node of
the search tree, MaxSAT reasoning has to be performed
from scratch to compute an upper bound and is time-consuming; (2) due to the
NP-hardness of the MaxSAT problem, MaxSAT reasoning generally cannot be complete at a node of
a search tree, and may not give an upper bound tight enough for pruning search
space. We thus propose an incremental upper bound and combine it with MaxSAT reasoning to remedy the two drawbacks. The new
approach is used to develop an efficient branch-and-bound algorithm for MaxClique, called **IncMaxCLQ**, which is very efficient and *closes an additional DIMACS instance **(**hamming10-4.clq)
in less than 7 days *on an INTEL
XEON E5-2680 (2.7Ghz, 20M Cache and 16GB memory) under
Linux*.*

An executable of the incremental MaxCLQ for Linux is available here.

The source code of the incremental MaxCLQ is available here.

__Reference :__

Chu Min LI, Zhiwen
Fang, Ke XU, Combining MaxSAT
Reasoning and Incremental Upper Bound for the Maximum Clique Problem, in proceedings of the 2013 IEEE 25th International Conference on Tools with
Artificial Intelligence (ICTAI2013), Pages 939-946.

**- Upper
Bound Based on MaxSAT Reasoning for MaxClique**

Branch-and-bound algorithms for the Maximum Clique
problem (MaxClique) usually partition a graph into
independent sets to compute an upper bound for the size of a maximum clique of
the graph, since every independent set can have at most one vertex in a clique.
However, this upper bound cannot be very tight for two reasons :
(1) partitioning a graph into independent sets is equivalent to coloring the
graph, which is even harder than MaxClique in
practice, so the partition cannot be optimal in every node of the search
tree ; (2) even if the partition is optimal, the number of independent
sets is strictly larger than the size of
a maximum clique for imperfect graphs (by definition of imperfect
graphs). Note that MaxClique and the graph coloring
problem are polynomial for perfect graphs.

The upper bound based on independent sets can be
significantly improved using MaxSAT reasoning. In
every node of the search tree, we dynamically encode a MaxClique
instance into a partial MaxSAT problem and use MaxSAT reasoning to underestimate the number of independent
sets that cannot have a vertex in the maximum clique, obtaining MaxCLQ, a highly efficient branch-and-bound solver for MaxClique.

An executable MaxCLQ
for Linux is available here. This version is described in the ICATI2010
paper and improved from the version presented in the AAAI2010 paper.

The source code of MaxCLQ is here.

__References____ :__

Chu Min LI, Zhe Quan. 2010. Combining Graph Structure
Exploitation and Propositional Reasoning for the Maximum Clique Problem, in proceedings of the 22th IEEE International Conference
on Tools with Artificial Intelligence (ICTAI2010), Arras, France, October 2010,
Pages 344-351.

Li, C. M., and Quan, Z., 2010, An efﬁcient branch-and-bound algorithm based on
maxsat for the maximum clique problem. In *Proceedins** of AAAI-10*, Pages 128-133.

**MinSatz****: A branch and bound algorithm for
the minimum satisfiability problem.**

MinSAT is an optimization extension
of SAT and it is the dual problem of MaxSAT, aiming
at finding a truth assignment to minimize the number of satisfied clauses. It
is complementary with MaxSAT when solving
combinatorial optimization problems. Its code source is
available here.

References:

Chu Min Li, Zhu Zhu, Felip
Manya, Laurent Simon, Minimum satisfiability and its
applications, In : Proceedings of the 22nd International Joint Conference
on Artificial Intelligence (IJCAI-2011), pp 605-610, 2011.

Chu-Min Li, Zhu Zhu, Felip
Manya, Laurent Simon. Optimizing
with Minimum Satisfiability, In Artificial Intelligence 190 (2012) 32-44.

**Sattime****: An efficient local search solver
for SAT**

Sattime is a Stochastic
Local Search (SLS) solver for SAT, that exploits the satisfying history of
clauses, instead of falsifying history of clauses as in most other SLS solvers,
in selecting the next variable to flip. Sattime participated in the 2011
SAT competition and won a silver medal in the ran- dom category. Especially, Sattime
beat easily all the Conflict Driven Clause Learning (CDCL) solvers in the
crafted sat category (Sattime solved 109 instances
while the best CDCL only solved 93 instances), although SLS has been considered
less effective than CDCL for structured SAT problems for a long time. This is
the first time that a SLS solver enters the final phase of the SAT competition
in the crafted category and beats there all the CDCL algorithms on structured
SAT problems.

Sattime confirmed its performance
in the SAT challenge 2012, as the best SLS solver in the crafted (Hard Combinatorial SAT+UNSAT) category and the
second best single engine solver in the random SAT category.

The source codes of Sattime
are available here.

* Reference :*

Chu Min Li and Yu Li, Satisfying versus Falsifying in Local
Search for Satisfiability, in proceedings of SAT2012, Springer LNCS 7317, pp. 477–478, 2012.

**Maxsatz****: A branch and
bound solver for Max-SAT**

A branch and bound solver for Max-SAT that incorporates
into a Max-SAT solver some of the technology developed for **Satz**
(see below). At each node
of the proof tree it transforms the formula into an equivalent
formula that preserves the number of unsatisfied clauses by applying some
efficient refinements of unit resolution that the authors have defined for
Max-SAT (e.g., it replaces $x, y, \neg x \vee \neg
y$ with $\Box, x \vee y$, it replaces $x, \neg x \vee y, \neg x \vee z, \neg y \vee \neg z$ with $\Box, \neg x \vee y \vee
z, x \vee \neg y \vee \neg z$). MaxSatz implements a lower bound computation method that
consists of incrementing the lower bound by one for every disjoint inconsistent
subset that can be detected by unit propagation. Moreover, the lower bound
computation method is enhanced with failed literal detection. The variable
selection heuristics takes into account the number of positive and negative
occurrences in binary and ternary clauses Maxsatz and
its variants are the best performing solvers in the unweighted maxsat category in the 2006 maxsat solvers evaluation and the 2007 maxsat
solvers evaluation.

The source codes of maxsatz are available here.

The weighted version of maxsatz (wmaxsatz2009)
is available here.

An executable, called maxsatz2013f, that participated in the MaxSAT evaluation 2013 is available here. Maxsatz2013f is
significantly faster than the previous versions of maxsatz
and is one of the best solvers in the MaxSAT
evaluation 2013.

**
References: **

Chu Min LI, Felip Manya, Nouredine
Mohamedou, Jordi Planes, "Exploiting Cycle Structures in
Max-SAT". In proceedings of 12th international conference on the Theory
and Applications of Satisfiability Testing (SAT2009), Springer, LNCS 5584, pages 467-480, June-July
2009, Swansea, United Kindom.

Chu Min LI, Felip Manya,
Jordi Planes, "New Inference Rules for Max-SAT", in Journal of Artificial Intelligence
Research, October 2007, Volume 30, pages 321-359

Chu Min LI, Felip Manya,
Jordi Planes, "Detecting disjoint inconsistent subformulas for computing lower bounds for
Max-SAT". In Proceedings of the 21st National Conference on Artificial
Intel ligence (AAAI-06), Boston, USA, pp. 86–91. AAAI
Press.

**TNM** (Wanxia
Wei, Chu Min Li):

A local search procedure based
on **G2wsat** (see below) and using two different adaptive noise mechanisms:
Hoos mechanism and a new adaptive noise mechanism based on
the history of the most recent consecutive falsiﬁcations of a clause. TNM automatically switches
between these two mechanisms during search according to the variable weight
distribution. No parameter is needed to adjust when using TNM. TNM won a *GOLD
Medal *in the SAT2009 competition in satisfiable
random formula category. The version used in the competition
(using input and output format of the competition) as well as a
short presentation of the procedure can be obtained here.

* Reference:*

Chu Min Li, Wanxia
Wei, and Yu Li, Exploiting Historical Relationships of Clauses
and Variables in Local Search for Satisfiability, in proceedings of SAT2012,
Springer LNCS 7317, pp. 479–480, 2012.

**Adaptg2wsat2009++** (Chu Min Li, Wanxia Wei):

A local search procedure based
on **G2wsat** (see below) by integrating the adaptive noise mechanism of Hoos. No parameter is needed to adjust when using adaptg2wsat2009++.
Adaptg2wsat2009++ won a *Bronze Medal *in the SAT2009 competition in satisfiable
random formula category. The version used in the competition (using
input and output format of the competition) as well as a short presentation of
the procedure can be obtained here.

**Adaptg2wsat0** (Chu Min Li, Wanxia Wei and Harry Zhang):

A local search procedure based
on **G2wsat** (see below) by integrating the adaptive noise mechanism of Hoos. No parameter is needed to adjust when using
adaptg2wsat0. Adaptg2wsat0 won a *Silver Medal *in SAT2007 competition in satisfiable random formula category. The version used
in the competition (using input and output format of the competition) as well
as a short presentation of the procedure can be obtained here. The normal
version can be found here.

** Adaptg2wsat+** (Chu Min Li, Wanxia Wei and Harry Zhang):

A local search procedure based on G2wsat
by integrating the adaptive noise and random walk mechanisms of Hoos. No parameter is needed to adjust when using adaptg2wsat+.
Adaptg2wsat+ won a *Bronze Medal*
in SAT2007 competition in
satisfiable random formula
category. The version used in the competition (using input and output formats of the competition)
as well as a short presentation of the procedure can be obtained here.

* References for Adaptg2wsat0 and
Adaptg2wsat+:*

**
**Chu Min Li, Wanxia Wei,
Harry Zhang, "Combining Adaptive Noise and
Look-Ahead in Local Search for SAT".
In proceedings of 10th international conference on the Theory and Applications
of Satisfiability Testing (SAT2007), Lisbon, Portugal, May 2007.

**G2wsat **(Chu Min LI & Wenqi HUANG):

A local search procedure which
won a *Silver Medal* in SAT2005
competition in satisfiable random formula category.
The version used in the competition as well as a short presentation of the
procedure can be obtained here. The current
version (2005) is optimized after the competition and is
generally more than 50% faster. To get
the source code of current g2wsat (version 2005), click here.

**
Reference:**

Chu
Min LI & Wenqi HUANG, "Diversification and Determinism in
Local search

for Satisfiability", in proceedings of 8th international conference, SAT2005, LNCS
3569 Springer, St Andrews, UK, June 2005, Pages 158-172

** Satz215.2**:

Satz215 is Satz214 + Detection of implied literals
suggested by Daniel Le Berre. Satz215 solves more
structured instances than Satz214. Particularly, Satz215 solves 3bitadd_31 and
3bitadd_32 in a few seconds. Satz215.1 is Satz215 modified in which the
preprocessing only adds up to 10*INIT_NB_CLAUSE ternary resolvents
into a formule with initially INIT_NB_CLAUSE clauses.

To get the source code of Satz215.2, click here.

** Satz214.2** (Li and Anbulagan) :

Satz214 is a very fast and a very simple Davis-Putnam
procedure to solve satisfiablility problems in DIMACS
format. Satz214 adds binary and ternary resolvents
into a formule when preprocessing it. Satz214.1 is
the last version in which only up to 10*INIT_NB_CLAUSE ternary resolvents
can be added into a formule with initially
INIT_NB_CLAUSE clauses To get the source
code of Satz214.2, click here. To compile it under Unix or
Linux system, the commande line looks like: gcc satz214.1.c -O3
-o satz

**
References: **

Chu Min LI, "A constrained-based approach to
narrow search trees for satisfiability",
Information processing letters 71(1999) page 75-80.

Chu Min LI & Anbulagan, "Look-ahead versus look-back for satisfiability
problems", in preceedings of third international conference on Principles
and Practice of Constraint
Programming--CP97, Springer-Verlag, LNCS 1330, Page
342-356, Autriche, 1997.

Chu Min LI & Anbulagan, "Heuristics based on unit propagation for
satisfiability problems", in
proceedings of 15th International Joint Conference on Artificial Interlligence (IJCAI'97), Morgan Kaufmann Publishers, ISBN
1-55860-480-4, Page 366-371, Japon, 1997.

** ****EqSatz** (Chu Min
LI) :

EqSatz is equivalency
reasoning enhanced satz to solve satisfiability
problems involving equivalency clauses (Xor or
modulo 2 arithmetics) such as 1<->2<->3
(equivalent to four CNF clauses: -1 or -2 or 3, -1 or 2 or -3, 1 or -2 or
-3, 1 or 2 or 3). EqSatz
is the first method to solve *all *the
ten DIMACS 32 bit parity par32-* instances (second
IJCAI-97 challenge on propositional reasoning and search) in reasonable
time. To get its source code click here. To compile it under Unix or Linux system, the
commande line looks like: gcc
eqsatz20.c -O3 -o eqsatz

**
References: **

Chu Min LI, "Integrating Equivalency reasoning
into Davis-Putnam procedure", in the
proceedings of AAAI-2000. Austin Texas, USA, July 2000, Page 291-296.

Chu Min LI, ``Equivalent literal propagation in
Davis-Putnam procedure'', in *Discrete Applied
Mathematics*, Vol 130/2 pp 251-276, 2003.

** Compactor **(Chu Min LI) **:**

The compactor is used to simplify an input formula in
DIMACS format. It is Satz without branching. To
get its source code, click here. Four operations are performed given an input formula:

(i)
the input formula is performed by adding resolvent of length < 4; e.g. if 1 2 3 and -1 2 4 are clauses,
add a clause 2 3 4 into the formula.

(ii)
elimination of pure literals;

(iii)
detection of failure literals ;

(iv)
rename remaining variables to be continuous;

The simplified formula is
written into the file named "out" in the
directory where the compactor is run. The old and new variables are
written into the file named "var_table" in
the directory where the compactor runs. This is a two-column file. The first column
is the old variables
in the original input formula which are replaced in the simplified formula
by the variables in the second column.

Please don't hesitate to
contact the author for any question and any suggestion.

** ****ParaSatz**** **(Bernard Jurkowiak, Chu Min LI, Gil Utard) : ** **

ParaSatz is the
distributed/parallel version of Satz which can run on
any machines in a local network. For example, it can run on any Linux or Unix
machines in a local network. Based on sequential Satz,
ParaSatz is developed by Bernard Jurkowiak,
Chu Min Li and Gil Utard. Particular thanks to
Dominique Lazure for material help.

ParaSatz uses the simple master/slave model for communication and dynamically
balances workload among slaves, i.e. a slave cannot be idle whenever the
resolution is not finished, since the most loaded working slave is capable of
giving a part of its work to the slave which has finished its own work.

Moreover,
ParaSatz
supports fault-tolerant computing, i.e. the work of any died slave will be continued by other slaves. ParaSatz also supports partial computing, i.e. the entire
resolution can be stopped at any moment and continued later, which is
important for the so-called "global computing" using idle machines in
the world.

To get the source code of ParaSatz, click here.

**
Reference: **

Bernard Jurkowiak,
Chu Min LI, and Gil Utard, A parallelization scheme based on
work stealing for a class of SAT solvers, in
Journal of Automated Reasoning (2005) 34:73-101.