some interesting research directions in satisfiability

Annals of Mathematics and Artificial Intelligence 28 (2000) 7–15 7

Perspectives

Some interesting research directions in satisfiability

John Franco

ECECS, University of Cincinnati, Cincinnati, OH 45221-0030, USAE-mail: [email protected]

The satisfiability problem and algorithms for solving it have received greatly increasedattention in the last few years. This interest comes from a variety of disciplines such asComputer Science, Operations Research, Graph Theory, and Physics, among others. Thispaper shows some of the current interesting directions in satisfiability research from thesedisciplines and presents some possible future directions.

1. CNF satisfiability

The question of determining whether a given Boolean formula in conjunctivenormal form is satisfiable (SAT) has been important for quite some time, but recentinterest in this problem has reached a level not previously observed. Part of thereason is that groups spanning several disciplines are now looking at SAT as an aidin furthering their main research interests. In the middle 1980s, SAT was considereda hard problem. Now, due to faster machinery, a better understanding of algorithmicmechanics, and the development of more effective algorithms, SAT is partly “easy”.Since many problems can easily be recast as SAT problems, it increasingly makessense to convert to SAT and solve it there.

SAT formulas have structural properties not unlike those of random graphs. ButSAT structures tend to be more complex and challenging than random graph struc-tures and have therefore received the attention of random graph theorists. In addition,random SAT structures exhibit properties analogous to those observed in matter; forexample, the transitory nature of magnetic spins during cooling. Because this analogysuggests a link to the “hardness” of random formulas, these transition phenomena havecaught the attention of Physicists (e.g., [49]), Artificial Intelligence Researchers, Com-puter Scientists, and Operations Researchers. The varied interest has opened severalvery active threads of research which did not exist in the 1980s. In this article wegive a brief description of some of the more interesting and productive threads of SATresearch, including some of the older ones as well as some which have developedrecently.

J.C. Baltzer AG, Science Publishers

8 J. Franco / Some interesting research directions in satisfiability

2. Significant recent threads of research

Among the primary goals of SAT research are to:

1. Determine properties of formulas that can either be exploited to produce solutionsefficiently or prove that no efficient solutions are possible.

2. Find and understand analogies between statistical properties of random formulasand statistical mechanics, particularly phase transitions.

3. Find algorithms for certifying unsatisfiability with good probabilistic performance.

4. Find algorithms for determining satisfiability with good probabilistic performance.

A number of research directions aimed at satisfying one or more research goalsstated above have developed or matured in the last decade. Here we mention some ofthe more significant ones.

2.1. Polynomial-time solvable subclasses of SAT

A considerable amount of effort, especially from the Operations Research com-munity, has revealed a multitude of classes of “easy” SAT formulas, many of whichare incomparable. The class of Horn formulas [27,44,56] has been known for a longtime to be solved in linear time by resolution on positive unit clauses. A solutionso obtained, if one exists, is uniquely minimum with respect to variables set to valuetrue. Similarly, 2-SAT formulas can be solved in linear time [3] simply by followingimplications, represented as edges of a graph, into strongly connected components ofthat graph where inconsistencies and equivalences can be revealed. Both these classesshow up frequently in real-world problems.

More recently, broader “easy” classes have been discovered. Several of thesehave come from studying properties of the polytopes of restricted formulations of CNFformulas as linear programs. For example, the extended Horn formulas of Chandruand Hooker [17] and the formulas corresponding to balanced matrices due to Confortiand Cornuejols [22], both of which rely on unit resolution for solutions. These havebeen generalized by the SLUR formulas [55]. Others are a consequence of fast ma-trix decompositions. For example, the q-Horn class, originally discovered by Boroset al. [8–10], was shown by Truemper [59,60] to be a special case of monotone de-composition applied to 0, +1,−1 matrix representations of CNF formulas. Monotonedecompositions, which may be computed in linear time, consist of a Horn quadrantabove a non-positive quadrant, and a 0 quadrant above an arbitrary 0, +1,−1 quad-rant. Because the unique minimum satisfying assignment with respect to true canbe easily found for Horn formulas, if it exists, a solution to a formula is determinedindependently by its Horn quadrant and its arbitrary quadrant. In the case of q-Hornformulas, the arbitrary quadrant is 2-SAT, so q-Horn formulas are solved in linear timeby solving a Horn subformula and a 2-SAT subformula.

At least one family of classes is tied to the structure of Davis–Putnam tree refu-tations. A CNF formula is minimally unsatisfiable if it is unsatisfiable but removal

J. Franco / Some interesting research directions in satisfiability 9

of any one of its clauses makes it satisfiable. A minimally unsatisfiable formula musthave more clauses than variables. A minimally unsatisfiable formula such that dif-ference k between the number of clauses and the number of variables is 1 can becertified unsatisfiable by an irreducible tree refutation branching on every variableexactly once (both true and false values), and every such refutation applies to someminimally unsatisfiable formula with k = 1 [2,14]. Minimally unsatisfiable formulaswith k growing up to logarithmically in the number of variables are solved in poly-nomial time. Recent advances in this area have been due to and inspired by Kleine,Buening and Kullman [46].

Finally, we wish to mention an interesting line of research involving hierachiesof formulas and fixed parameter tractability [28]. Classical recursively defined hierar-chies of incrementally harder classes have been defined and studied in [14,23,38,47,51]among other places. However, the complexity of such hierarchies is O(nf (k)) wherek is the level of the hierarchy and f is usually linear in k. More recently, Heusch [43]and several others [33] found a O(kkn2) hierarchy involving pure implicational formu-las. Because k is not in the exponent of n, this is a fixed parameter tractable hierachy.

These are only some of the new classes which have been discovered recently. Itis interesting but unfortunate that few of these originated from an actual application.Despite this, discovery and study of polynomial time solvable classes may be useful forseveral reasons but primarily because a SAT solver may be improved by a preprocessingstep which attempts to either recognize and solve a given formula as a member of one ormore special subclasses or decompose a given formula into an easily solved subformulaand a relatively small, orthogonal general subformula. The reader is encouraged tolook at Truemper’s book Effective Logic Computation [60] for some idea how this canbe done.

2.2. SAT algorithms which perform well almost always

If there exists a SAT algorithm which can run efficiently on all but a few patholog-ical cases, then one may consider SAT to be effectively an efficiently solved problem.On the other hand, if an application generates mostly pathological formulas then suchan algorithm is useless. Thus, if we are to measure how useful a particular SATalgorithm is, some knowledge of the input frequencies is needed. Unfortunately, inmost cases, this is not readily obtainable. One possibility is to rely on random inputs.Random inputs are typically not biased to a particular application and this is good ifour algorithm is to be shown generally useful. But, currently, random inputs need tocontain statistically independent components to support analysis and such structurescannot be expected to appear frequently in practice.

Nevertheless, numerous results using random formulas from several differentgenerators have appeared, often on Davis–Putnam–Loveland–Logemann (DPLL) stylealgorithms [24,25]. These results have a form similar to the following: with probabil-ity tending to 1 in the limit, algorithm X finds, in polynomial time, a truth assignmentwhich satisfies a given satisfiable random formula. Although such results are of-


ten presented in a manner suggesting they are derived for their own sake, in fact,they are more seriously used to illuminate the reasons why an algorithm is empiri-cally successful. A notable example is the family of probabilistic results on variantsof single-branch DPLL algorithms where partial truth assignments are extended bychoosing a variable from a clause containing the smallest number of non-falsified lit-erals. Due to arguments which keep track of the average accumulation of clauses ofi-literal clauses, for all possible values of i, and the average flow of clauses betweenaccumulations, we have a fairly clear understanding of why such heuristics are effec-tive and how they work. It can be argued that this has led to better SAT solvers: forexample, the winners of recent SAT solver competitions (e.g., [16]) use such heuristics,but it is hard to find a discussion of these heuristics which predates the probabilisticresults. Notable contributions in this area have come from Achlioptas [1], Broderet al. [11], Purdom, Brown, Haven, Bugrara and others [12,15,41,52], Chao, Francoand Paul [18,19,31,32,35], Chvatal and Reed [20], Frieze and Suen [37], among others.

2.3. Difficulty of resolution for unsatisfiability: probabilistic lower bounds

It has been known for some time that general resolution has weaknesses relatedto the “sparseness” of a formula [42,61]. Chvatal and Szemeredi [21] showed thatnearly all random k-SAT formulas are sparse when the ratio of clauses to variablesis held fixed as formula size is increased. Thus, they showed resolution refutationsare usually exponentially large for fixed ratios, when most formulas are unsatisfiable.Others, particularly Pitassi, Beame, Karp, Saks [4,5], Widgerson, and Ben-Sasoon [6]have extended these results, steadily refining their arguments, until we now have afairly clear and relatively simple picture of the nature of the difficulty experienced byresolution, even for ratios that are not fixed. The useful outcome of this research isthe conclusion that the size of a linear refutation is exponential in the maximum widthof a clause derived in that refutation.

2.4. Threshold results

Probably the thread currently being investigated most intensely has to do withthreshold results. Consider random k-SAT formulas with m clauses taken uniformlyand with replacement from the set of all non-tautological k-literal clauses over n vari-ables. In general, it is difficult to produce an expression for the probability that aformula is satisfiable as a function of m, n, and k. Early on, it was found that k-SATformulas, k > 2, are unsatisfiable with probability tending to 1 ifm/n > c1·2k [35], forsome constant c1, and satisfiable with probability tending to 1 if m/n < c2 ·2k/k [20],for some constant c2. More recently the lower bound has been improved somewhat(see [29]). For 3-SAT, these two bounds have been steadily tightened by numerousideas and results [30] to something close to m/n > 4.54 for unsatisfiability with highprobability and m/n < 3.145 [1] for satisfiability with high probability. Recently,it became known [36] that for any k > 2 there is a sequence rk(n) such that theprobability of satisfiability tends to 1 if m/n < rk(n)− ε and of unsatisfiability tends


to 1 if m/n > rk(n) + ε, where ε is an arbitrarily small constant. This is known as asharp threshold.

Unfortunately, we do not understand the sequence: it could actually oscillate or,as most believe, converge to some asymptotic value. We do not know the convergencerate, called the scaling window (assuming convergence) for k-SAT, k > 2. But recentlyBollobas et al. [7] have computed the scaling window for 2-SAT. It was shown in 1992by three groups that r2(n) converges to 1 [20,26,40].

There is, perhaps, a slight misunderstanding in the interpretation of some thresh-old related results. Many people have observed that a variety of algorithms for SATrequire increasing amounts of time on random k-SAT formulas as the ratio m/n isincreased to about 4.25 (roughly, the threshold) and then decreasing amounts as theratio is increased beyond that point. Thus, the expression “easy–hard–easy” is oftenused to suggest that the hard random k-SAT problems are concentrated around thethreshold.

However, we do not know whether SAT becomes easy soon after the threshold:all we really know is that resolution is inefficient during and after the threshold, evenif m/n grows, unless m/n > nk−1 (approximately). After that, resolution is certainlyefficient since random formulas then almost always contain minimally unsatisfiablecomponents with a constant variable-clause difference.

2.5. Upper bounds

The worst-case complexity for all known SAT algorithms is exponential in the firstpower of the input size. The naive algorithm that tries every variable setting requirestime 2n for n variable formulas. For 3-SAT, the best known upper bound on worst-casecomplexity has been worked down from 1.618n [50] to around 1.5n [48,53,54]. Otherwork on the topic is given in [39].

3. Future research

The following are an interesting set of problems which hopefully will be attackedin the near future in addition to the current threads mentioned above. They are listedin no particular order.

3.1. What constitutes a good probabilistic model for SAT?

Probabilistic results have been obtained for a variety of input models, but littlehas been done to justify these models. It has been observed that several, if not most,SAT solvers find random k-SAT formulas more difficult than formulas arising frompractical applications. Does this help or hinder our understanding of SAT algorithmsthrough probabilistic results on random k-SAT formulas?


3.2. Easy classes which frequently appear

Recently discovered polynomial-time solvable classes of CNF formulas usuallyhave no clear relationship to interesting sets of actually encountered problems. Furtherinvestigations of such classes should take this into account. Relevant fixed parametertractable classes will hopefully be revealed.

3.3. Problems which can benefit from a lot of preprocessing

Consider the following problem. We are given m arbitrary boolean functions,each arbitrarily taking k out of n inputs. For a particular output vector, find an inputvector that will produce it. For such a problem, preprocessing can be amortized over allthe requests for input vectors. For this and other such problems, how can preprocessingbest be taken advantage of?

3.4. The why 2k problem

It was mentioned that random k-SAT formulas are satisfiable with probabilitytending to 1 if m/n < c2 · 2k/k and unsatisfiable with probability tending to 1 ifm/n > c1 · 2k, for some constants c1 and c2. It is believed by many that the secondinequality is a tight bound. However, no compelling reason for this conjecture hasbeen offered. Thus, this has been called the “why 2k” problem. The reason forthe current gap is that two very different techniques have been applied to each setof bounds: the lower bounds are due to clause-flow analyses of shortest-clause-firstalgorithms whereas the upper bounds are computed by finding expectations of subsetsof satisfying assignments. Clearly, there is a need to improve or replace at least oneof these techniques. Doing so would likely lead to a much improved understanding ofSAT formulas around the threshold.

3.5. Alternative methods

It is well known that CNF satisfiability can be treated as a Linear Integer Program.Then, for example, linear cutting plane methods can be applied. Recently, non-linearcuts have shown some promise [62,64]. We look forward to further development ofthis area. We also mention the work of Wah [63] using Lagrangian techniques aspromising.

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some interesting research directions in satisfiability

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