Constraint satisfaction has proven to be an indispensable paradigm for solving complex problems in A.I. and industry. Indeed, many such problems can be expressed as a conjunction of constraints over variables, including logical constrains, mathematical relations and sophisticated global constraints such as automata.
However, for many applications, the true problem to be solved is not a satisfaction problem, though satisfaction is a critical component. For example, in many cases one is interested in finding a solution that minimizes an objective function, rather than any satisfying solution. Many other settings exist in which some satisfying solutions are more interesting or preferred than others, that is, some solutions dominate other solutions.
We introduce solution dominance as a way to express such dominance relations over the solutions of a CSP. A dominance relation here defines a preorder over the solutions. In line with Constraint Satisfaction Problems and Constraint Optimization Problems, we call the resulting problems Constraint Dominance Problems (CDPs). The goal is to find all non-dominating solutions to the CDP, that is, the Pareto optimal set. We discuss two variants, depending on whether equivalent solutions are allowed in the solution set or not.
Our work generalizes the work on preferences in SAT , where a strict partial order over literals is used. This captures MinOne, MaxSAT and Minimum Correction Subsets. Our work generalizes this and other works expressing preference as strict (irreflexive) partial orders, because: 1) (reflexive) preorders give us the freedom to reason both about solution sets that do or do not allow for equivalent solutions; 2) the goal is not to find all dominated (preferred) solutions, but rather all non-dominated solutions; hence 3) the investigated approach also capture multi-objective optimization and more. The set of non-dominated solutions is known as the Pareto frontier or the efficient set in multi-objective optimization  and our formalization is inspired by that, but can reason over arbitrary partial orders.
Preferences in SAT  are methodologically different from preferences expressed through Conditional Preference networks (CP-nets, ), because the latter requires expensive dominance checks. We investigate a novel dominance relation for CP-nets, and show that all these types of preferences can fit in the same framework and methodology. The framework can also express arbitrary solution dominance relations, including other forms of conditional preferences than CP-nets. This is motivated by recently studied data mining problems involving conditional dominance relations over CSPs . A further discussion of related work is provided in Section 7.
Inspired by declarative languages for modeling CSPs [21, 10, 28] we propose an extension to the MiniZinc modeling language that enables the formulation of CDPs. The idea is to specify a dominance nogood, a constraint pattern that can be used whenever a solution is found during search to exclude dominated solutions in the remaining search (analogous to the well-known branch-and-bound approach for optimization). Dominance nogoods can be derived from the solution dominance relation in a principled way. We present an intuitive implementation of a generic algorithm for dominance nogoods in the MiniSearch  meta-search language. We explore the possibilities of this approach on a number of problems and with a range of different solvers to show the potential of such a generic approach.
2 CSPs and solution dominance
Conceptually, solution dominance can be used to solve problems of the form: find where is the set of all solutions of a CSP.
A Constraint Satisfaction Problem (CSP) is a triple where is a set of variables, is a mapping from variables to a set of values, and is a set of constraints over (a subset of) . Constraints can represent arbitrary complex relations over the variables. A valuation is a mapping of variables to values: . A solution of CSP is a valuation that satisfies each constraint . We denote by the value of in solution .
A dominance relation over CSP solutions expresses when one solution dominates or is equivalent to another one. A simple example is that of a constrained optimization problem, where the optimization function defines a total (pre)order on the solutions. We follow the optimisation convention that smaller is better.
We define a dominance relation as a preorder over the set of solutions of a CSP . A preorder is a reflexive () and transitive () relation. It can be thought of as a partial order over equivalence classes. In other words, given two solutions either one dominates the other, or they are equivalent, or they are incomparable. The use of (reflexive) preorders instead of (irreflexive) partial orders allows us to discriminate between incomparable and equivalent solutions. This is also a key difference between the problem of finding all dominant/preferred solutions and all non-dominated solutions: neither of two equivalent solutions is strictly dominant, while both are non-dominant; incomparable solutions are also non-dominant.
More formally, we define the equivalence relation ; and the negations and . Now, let be the set of all solutions of a CSP, and a dominance relation. We identify three possible properties for sets :
every solution in is dominated by or equivalent to a solution in : .
the solutions in are not dominated by any other in , except equivalent ones: . Equivalently, no is strictly dominated: .
no two solutions in are equivalent to each other: . In preference terms, they are indifferent to each other.
The set of complete and domination-free solutions is unique and defined as follows: . In the multi-objective optimization literature , this set is known as the Pareto frontier or the efficient set. In that context it is often written in the form .
In practice though, one is typically just interested in a complete and domination-free set that is also equivalence-free: with . This set is not unique (it contains one arbitrary solution per equivalence class).
3 Constraint Dominance Problems
We now show how a wide range of problems that are not captured by the classical CSP framework can be expressed with a dominance relation over a CSP. Generic solving methods are discussed in the next section.
A Constraint Optimization Problem (COP) is typically defined as a tuple where is a CSP as defined before and is a function over a valuation of the variables . A solution to a COP is a solution to the corresponding CSP that minimizes the function . One is typically interested in finding one such optimal solution, that is, one solution of for which .
In line with COPs we define a Constraint Dominance Problem (CDP) as a tuple where is a CSP and a dominance relation. Two types of queries exist for this problem (with or without equivalent solutions). We call the full solution to the CDP the set of complete and domination-free solutions of the CSP. A (non-unique) complete and domination-free set that is also equivalence free is simply called a solution to the CDP.
Given a COP , let be the total order corresponding to : . Then, finding a solution to the COP corresponds to finding a solution to the CDP .
- Lexicographic optimization.
Let be a CSP and a set of functions . The goal is to find a solution that lexicographically minimizes the functions. Given the preorder . A solution to the CDP is a lexicographically optimal solution.
- Multi-objective optimization.
The goal in multi-objective optimization is to find all Pareto optimal (non-dominated) solutions given a set of functions. Let be the following preorder: .
The full solution to the CDP corresponds to the set of all Pareto optimal solutions.
which is the classical definition of multi-objective optimization .
- -minimal models.
Let be Boolean (0/1) variables of a CSP . Given a solution , let be the variables of assigned to in solution . An -minimal model is a solution to the CSP such that there is no solution with . The set of all -minimal models is: .
We can define the preorder as: , that is, for each variable of an assignment to (domain value ) is preferred over (domain value ).
The full solution to the CDP corresponds to the set of -minimal models.
Using the same rewriting as for multi-objective optimization, we obtain that the set of non-dominated solutions is:
which equals .
The concept of minimal models can be extended to non-Boolean CSPs by defining a total order over the possible values of each variable , in line with .
- Weighted (partial) MaxCSP.
Given a CSP and a function that represents the weight of a constraint. Let be a valuation which need not be a solution to the CSP. The total weight of is the sum of the weights of the constraints that are satisfied by : . The goal is to find an assignment to that maximizes this weight. As our dominance relation is over solutions of a CSP, we define a new CSP as follows: a set of new Boolean variables is added to : , and each (soft) constraint is replaced by a reified version: . Let preorder be , with , then, the weighted MaxCSP problem given is equivalent to the CDP . Because is a total order, this is equivalent to a COP over .
- Valued CSPs.
A valued CSP with annotated constraints  has a valuation structure . Each constraint is mapped to a problem specific value in . The values are aggregated using operator , and defines a total over . Because of the total order, a similar encoding to that for weighted MaxCSP can be obtained.
- Maximally satisfiable subsets.
A maximally satisfiable subset of is such that is satisfiable and adding any other constraint leads to unsatisfiability: is unsatisfiable. Dually, we call a minimal correction subset.
Applying the same transformation of to as for MaxCSP problems, the goal is to find solutions with set of active constraints such that no additional constraints in can be added to it: . This corresponds to finding all -maximal models. In line with minimal models, we can define the preorder and corresponding CDP .
Figure 1: CP-net example over 3 variables.
A CP-net is an acyclic directed graph over a set of variables . Each node in a CP-net corresponds to a variable and has a conditional preference table . A CPT associates with each possible partial valuation of the parent variables in the graph, a strict total order over the values of (consistent with the rest of this paper, means is preferred over ). Figure 1 shows an example.
A CP-net induces a set of preference rankings that are consistent with all CPTs, where a preference ranking is a total ordering over all complete valuations of . Traditionally, dominance in a CP-net is defined in terms of its preference rankings: dominates if in all of the preference rankings of the CP-net is ordered before . Even for binary-valued CP-nets, the complexity of a dominance check for an arbitrary network is NP-hard .
An easier to compute query is the ordering query : given two valuations and , is there a CPT where for all ancestor variables and have the same value and is preferred to according to the CPT: ? If so, there must exist a preference ordering where and hence this implies that does not dominate . It only implies non-dominance though (sufficient but not necessary), meaning that there may not exists such a CPT and yet does not dominate . This is because only CPTs are checked for which all ancestors have the same value in and ; hence there may be a CPT with that induces preference rankings in which but the CPT is not checked because of the ancestor condition.
We hence propose the following weaker form of CP-net dominance:
Given CP-net , a valuation locally dominates iff for all CPTs: when its parents have equal value in and but the CPT variable does not, then must be preferred to by the CPT;
Note that there is always at least one with corresponding CPT active, because a CP-net is acyclic and hence there is at least one node with an empty parent set.
Local dominance in a CP-net is a necessary but not a sufficient condition for traditional (preference ranking-based) dominance.
All preference rankings have to agree with all CPTs. Hence, if dominates then for all applicable (parent variables equal, current variable not) CPTs: , so local dominance must necessarily be true. However, if then by negation there must exists a node with applicable CPT for which and hence because of : . However, it may still be the case that dominates , because of an interaction between the grandparent variables and CPTs such that no preference ranking actually uses this entry of . This cannot be observed from the local CPTs directly, but must be verified through a (NP-hard) dominance check over the preference rankings.
Because each CPT in a CP-net encodes strict relations, two valuations can only be equivalent if they are identical: . We hence define .
Local dominance can be checked in time linear in the number of CPTs in the CP-net. We can hence realistically use it to enumerate all non-locally-dominated solutions of a CP-net by solving CDP . The resulting solution will be an over-approximation of the actual set of traditional non-dominated solutions. Should the application demand it, one could still filter the resulting set by post-processing with the NP-hard traditional dominance check.
- Domain-specific dominance relations.
We showcase the need for domain-specific dominance relations in data mining. Increasingly, constraint programming is used for data mining problems such as searching for patterns that appear frequently in a database . A pattern can be a set of items, a sequence or another structure such as a graph . A pattern is frequent if it is a subpattern of sufficiently many objects in the database. The problem is typically encoded as a set of constraints that define the pattern type, and that define when a pattern is frequent. One solution to this CSP is then one frequent pattern.
However, there are a number of pattern mining settings that do not fit the classical CSP framework, most notably closed and maximal patterns, relevant patterns and skyline patterns. Using the concept of dominance though, these problems can be modeled declaratively and combined with arbitrary constraints .
For example, a frequent pattern is maximal if there is no other frequent pattern that is a superpattern of this pattern. Let represent the subpattern relation, e.g. subset, subsequence or subgraph relation depending on the pattern type. The general dominance relation for maximal patterns is: . Let be the set of all frequent patterns, then the full solutions is , the set of all maximally large patterns such that no other frequent pattern is a superpattern of it. For patterns represented by a set , the subpattern relation is the subset relation over , hence; . Note that this is equivalent to an -maximal model with .
Closed frequent patterns have the weaker condition that there should not be a superpattern with the same frequency: dominance relation . The resulting full solution set is the set of all frequent patterns for which no superpattern exists that has the same frequency: , this is true for all pattern types. An algebra inspired by databases is used in  to show how a number of other pattern mining problems can be expressed as a combination of a constraint algebra and a dominance algebra. The framework we propose here focuses on one CSP and one dominance relation, which is sufficient for most settings.
4 Search and dominance nogoods
The main task that we consider is to find a solution to the constraint dominance problem , that is, a complete, domination-free and optionally equivalence-free set of solutions. Our methodology is to solve the problem through a chain of constraint satisfaction problems, similar in spirit to .
Let be an oracle that returns a satisfying solution to the CSP or fails if no such solution exists.
the following algorithm returns a complete solution to a CSP with dominance relation , using oracle :
Note how on line 4, a constraint is added over variables such that any (future) assignment to these variables may not be strictly dominated by the found solution . As the constraint set is monotonically increasing, the oracle can be incremental in that it can continue its search from where its last solution was found; the result is a branch-and-bound style algorithm where the bound is represented by .
Let be the sequence of solutions as found by the oracle in Algorithm 1.
The set returned by Algorithm 1 is a complete set: .
The set is complete: let be the CSP that yields solution , let be the set of all solutions of . is the original CSP and hence contains all solutions of the CSP and this set is complete. The subsequent only forbids solutions that are dominated and not equivalent to , hence is also complete as any solution is either dominated by or in . By induction, any set is complete. Hence, is complete as the stopping criterion on line 2 dictates that .
Complete and equivalence-free.
Algorithm 1 can easily be modified to return a complete and equivalence-free set. In this case, line 4 has to be changed to . This set is equivalence-free as now any solution to cannot be equivalent to . The set is still complete as only equivalent solutions are additionally removed by the modification, and hence for each solution in it will still be the case that .
Any solution found after cannot be strictly dominated by , as this is explicitly forbidden by the added constraint. That means that if the oracle enumerates the solutions from most to least preferred according to the preorder (e.g. first assign variables to 1, then to 0 for MaxCSP), then the complete and equivalence-free set is also domination-free. This is the approach used by .
However, if we assume no order on the solutions found by the oracle, it is possible to find, with , an that strictly dominates : . We can remove these by doing a backwards pass over the solutions in which we check for each that and if not we drop .
Note that we reuse oracle for this though it merely has to check whether a fixed assignment to the variables satisfies the constraints.
The same procedure can be applied to a complete and equivalence-free set to make it domination-free; this set is already equivalence-free so the process will only remove the strictly dominated solutions.
From dominance relation to dominance nogood.
We refer to the constraint added on line 4 of Algorithm 1 as the dominance nogood. It can be derived from the preorder in a principled way. We will demonstrate this for a number of earlier examples.
Since one is often interested in a complete, domination-free and equivalence-free set, we will demonstrate it for the equivalence-free dominance nogood . We denote the relation representing the dominance nogood as . It can often be obtained by negating the logical definition of the preorder . The dominance nogood with equivalence can be obtained following similar methods as well, perhaps more easily in the equivalent form . Recall that is the set of variables of the CSP and is a previously found solution.
For dominance relation the dominance nogood is . This guarantees that every new assignment to will have a smaller score than the previously found solution .
- Lexicographic optimization.
The dominance relation is . The negation is the dominance nogood: . Using the observation that we obtain . One could also use a
- Multi-objective optimization
. The dominance relation has the dominance nogood . Recall that as in the previous examples, this is the dominance nogood to obtain the equivalence-free set of complete and domination-free solutions.
- -minimal models
We have and dominance nogood .
- Minimal correction subsets
We noted earlier that minimal correction subsets is dual to finding the maximal satisfiable subsets, which corresponds to the problem of finding all maximal models of with . The corresponding dominance nogood is .
For CP-nets we have .
Each line in a CPT is mutually-exclusive, so the relation that is more preferred to according to , , can be formalized using a disjunction over all CPT entries. The preference in each entry can be modeled using implications. For example, for the first entry of in Figure 1 as follows: . Indeed, the parents have to take a specific (identical) value, and in that case if then is preferred only if takes value or ; if then must be to be preferred. Because of the mutual exclusivity and because is a solution of which we know the values, when posting the dominance nogood, at most one entry per CPT will be part of the logical formula.
Each time the oracle is called, it has to solve an NP-hard CSP in general. Furthermore, there can be an exponential number of non-dominated solutions and hence calls to the oracle: there exists a search order such that the algorithm has to enumerate all solutions to find all non-dominant ones. Brafman et al  show that the simpler problem of computing the next solution in a CSP or preference problem is in general also NP-hard, although there are some special cases (like tree CSPs) where it can be easier.
5 Integration in a modeling language
Any constraint solver that supports incrementally adding constraints and retrieving the next solution can implement Algorithm 1. From the modeling perspective, we describe how MiniZinc , a modeling language for CSPs and COPs, can be extended to handle CDPs as new modeling primitive.
Instead of extending MiniZinc to express dominance relations, we instead add syntax for dominance nogoods, for two reasons: 1) users can explicitly specify either an equivalence-free dominance nogood, or a dominance nogood with equivalence; 2) we found it more intuitive to declare an invariant for the search (e.g. minimization as where is a previously found solution), rather then declaring when a previous solution dominates or is equivalent to a new one (e.g. ).
MiniZinc has keywords to define variables, constraints, predicates and a minimal search specification.
We add the keyword 24] function can be used to refer to the value of variable in a previously found solution. Hence, the dominance nogood for a minimization problem: can be expressed as:
with some array of variables of the accompanying CSP.
Here is an example for finding minimal correction subsets (note thein the output statement, where we convert the maximal satisfiable subset to a minimal correction subset).
. MiniSearch is an extension of MiniZinc with support for specifying meta-level search heuristics. The integration with MiniZinc allows any existing FlatZinc solver to be used by MiniSearch, and hence also for solving CDPs.
At the language level, a declaration 1 is then specified as follows in MiniSearch:is simply translated into a predicate declaration that posts the dominance nogood. Algorithm
Thefunction repeatedly queries a black-box solver for the next solution ( ), and if a solution is found, it remembers it ( ), prints it, and then posts the dominance nogood before continuing to search for the next solution. In MiniSearch, can either call an external solver process through a file based interface, restarting the search from scratch for each call; or use an incremental C++ API that permits adding constraints during the search.
We evaluate the viability of the framework and the opportunities of modeling constraint dominance problems in a declarative solver-independent language. We do not aim to beat the state-of-the-art on any one specific task, as they typically employ specialized bounds or additional inference mechanisms. The experiments below evaluate different categories of dominance nogoods and show that they can be handled through the generic framework presented in this paper, including the novel setting of optimizing over CP-nets using generic CSP solvers.
The experiments use MiniSearch with the following solvers: Gecode 4.4.0 (gecode), or-tools v2015-09 (ortools), Opturion CPX 1.0.2 (optcpx) and Chuffed b776ac2 (chuffed). Gecode and or-tools are classical depth-first search CP solvers, while Opturion CPX and Chuffed are lazy clause generation solvers . All solvers are called by MiniSearch as external processes using the standard file-based interface. Gecode additionally supports the direct, incremental C++ API (gecode-api).
We use benchmarks from the 2008 XCSP competition, MaxCSP with globals category. Table 1 shows a comparison of MiniSearch with different solvers on a selection of instances. For each solver, we compare a model where no variable order is given (’free’) with the specification of a most-to-least preferred strategy over the variables only (’ord’, e.g. assign to before ). The latter forces the solvers to first consider all constraints, then all but one (arbitrary) constraint, and so on.
We can see in Table 1 that providing the search order often leads to improved runtimes, but not always (quasigrp for gecode, cabinet for chuffed). gecode is slower than the incremental API approach gecode-api in case an order is given; when doing free search, the restarts of the file-based approach seem to improve runtime for q13 (and others, not shown). The remaining solvers seem to handle this task pretty well, especially optcpx. For a rough comparison, in the 2008 competition the quasigrp and latinSq instances were also solved within seconds, however runtimes of 600+ seconds were reported for cabinet and 40+ seconds for q13.
We consider traveling saleseman problems where two different costs are given between any two cities, e.g. duration and fuel cost. We report on the generation of all complete and equivalence-free solutions. The straightforward backward pass needed to make the set domination-free is omitted for reasons of simplicity. The instances are from the Oscar repository .
In Table 2 we compare the MiniSearch approach with different solvers and Oscar , which has an efficient dedicated propagator for multi-objective optimisation . The first three lines use the first-fail variable ordering used in Oscar, the last three use a max-regret ordering over the distance variables, as found in MiniZinc’s TSP models. The gecode-api results indicate that file-based restarts lead to much slower solving times. The number of intermediate solutions also has a big influence on runtime, as using a better variable order leads to both smaller solution sets and smaller runtimes.
The following experiments consider a variant of the Photo problem, where the goal is to find an ordering of friends such that the number of preferences regarding whom to stand next to for a group photo is maximized. We here consider the case that preferences are supplied as a CP-net: each person indicates a number of people (parents in the CP-net) and their preferences considering the locations of these people. Such a CP-net can be partial, i.e., it can contain disconnected components, which our method can handle without any modification.
We randomly generated CP-nets with people () and for each person between 0 and parents (sampled uniformly per person, ). The induced order in the CPTs corresponds to preferring smaller average distance to the parent(s). We again report only the forward pass of computing all complete and equivalence-free solutions. Larger and larger lead to larger CP-nets and runtimes, though that depends very much on the actual CP-net generated.
Table 3 shows results on 5 different networks generated with and . No variable/value ordering strategy was imposed. The number of solutions found clearly has an influence on runtime, where the number of solutions is not only specific to the problem at hand, but also to the search order chosen by the solver. We expect a method that uses the expensive (traditional) dominance checks to perform worse.
7 Related Work and discussion
As discussed in the introduction, most related is the work on preferences in SAT [25, 7], where a preference can be defined over individual literals. They identify preference formulas for these tasks, which correspond to dominance nogoods, and use incremental SAT solving. Our framework generalizes this to a wider range of tasks and different solving technology.
As with any generic method, one cannot expect to obtain the most efficient method for each of the covered tasks. Indeed, specialized methods have been developed for MaxSAT , minimum correction subsets  and X-minimal models  that can be more efficient than a SAT with preferences approach, but only for their specific task. For multi-objective optimization in CSPs, specialized propagation algorithms exist that filter the search space more effectively [11, 15]. Similarly for other forms of preference .
Nevertheless, recent applications of data mining using SAT  and CSP  demonstrate the need for generic methods for handling novel solution dominance settings, for example involving conditional dominance relations.
Many works in CP-nets focus on consistency and dominance testing [8, 1, 2]. A branch-and-bound style algorithm for finding all non-dominated solutions given additional constraints has been studied ; it uses expensive (PSPACE complete ) dominance checks in case the ordering query returns false. Furthermore the search (variable order) in their method is driven by the CP-net’s structure. In contrast, our method proposes a novel dominance relation that is cheap to evaluate and provides an over-approximation; it can be used with any existing solver and for any variable order.
A recent extension of Answer Set Programming  covers some of the tasks in this paper too (no CP-nets or domain-specific relations), but within the stable model semantics. They provide a language extension for expressing preference relations with a preference type (e.g. less, subset, pareto) and preference elements (the variables). Our language extension is closer to the original dominance relation which can make it easier to specify domain-specific dominance nogoods.
The concept of dominance is also used in different contexts in the constraint programming community. Dominance breaking  for COPs creates constraints that, given a mapping , prevent the finding of solutions such that is a better solution of the COP. They can drastically improve solving performance.  rely on a notion of dominance relation that applies to all valuations and not just solutions, so that sub-trees can be pruned during the search. In that sense they are complementary to the solution dominance we consider in this paper. Indeed an interesting direction for further work is to extend dominance breaking to arbitrary solution dominance problems.
We introduced the concept of solution dominance, where the dominance relation is a preorder over the solutions of a CSP. We call the resulting problems Constraint Dominance Problems, and this captures single/lexicographic/multi-objective optimization, X-minimal models, weighted MaxCSP, minimum correction subsets as well as a novel dominance relation for reasoning over CP-nets, as well as other dominance relations. We provide a natural and declarative extension to MiniZinc for specifying Constraint Dominance Problems, based on MiniSearch.
Preferences and (solution) dominance have a history in CP research, as discussed above. Two directions for future work hence emerge: 1) which other preference  or dynamic solving settings  fit the solution dominance framework; and 2) what specialised solving methods that have been investigated for one category (e.g. MaxCSP or Multi-objective) can also be applied to other settings? From a modeling perspective one can also ask the question whether it can be automatically detected from a dominance nogood specification, that a more specialised algorithm could be used (e.g. for lexicographic or bi-objective optimisation).
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