Global Cardinality Constraint Research Articles

How to Tell Easy from Hard: Complexities of Conjunctive Query Entailment in Extensions of ALC

It is commonly known that the conjunctive query entailment problem for certain extensions of (the well-known ontology language) ALC is computationally harder than their knowledge base satisfiability problem while for others the complexities coincide, both under the standard and the finite-model semantics. We expose a uniform principle behind this divide by identifying a wide class of (finitely) locally-forward description logics, for which we prove that (finite) query entailment problem can be solved by a reduction to exponentially many calls of the (finite) knowledge base satisfiability problem. Consequently, our algorithm yields tight ExpTime upper bounds for locally-forward logics with ExpTime-complete knowledge base satisfiability problem, including logics between ALC and µALCHbregQ (and more), as well as ALCSCC with global cardinality constraints, for which the complexity of querying remained open. Moreover, to make our technique applicable in future research, we provide easy-to-check sufficient conditions for a logic to be locally-forward based several versions of the on model-theoretic notion of unravellings. Together with existing results, this provides a nearly complete classification of the “benign” vs. “malign” primitive modelling features extending ALC, missing out only the Self operator. We then show a rather counter-intuitive result, namely that the conjunctive entailment problem for ALCSelf is exponentially harder than for ALC. This places the seemingly innocuous Self operator among the “malign” modelling features, like inverses, transitivity or nominals.

Group Equality in Adaptive Submodular Maximization

In this paper, we study the classic submodular maximization problem subject to a group equality constraint under both nonadaptive and adaptive settings. It is shown that the utility function of many machine learning applications, including data summarization, influence maximization in social networks, and personalized recommendation, satisfies the property of submodularity. Hence, maximizing a submodular function subject to various constraints can be found at the heart of many of those applications. On a high level, submodular maximization aims to select a group of most representative items (e.g., data points). However, the design of most existing algorithms does not incorporate the fairness constraint, leading to underrepresentation or overrepresentation of some particular groups. This motivates us to study the submodular maximization problem with group equality, in which we aim to select a group of items to maximize a (possibly nonmonotone) submodular utility function subject to a group equality constraint. To this end, we develop the first constant-factor approximation algorithm for this problem. The design of our algorithm is robust enough to be extended to solving the submodular maximization problem under a more complicated adaptive setting. Moreover, we further extend our study to incorporating a global cardinality constraint and other fairness notations. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2022.0384 .

Revisiting Counting Solutions for the Global Cardinality Constraint

Counting solutions for a combinatorial problem has been identified as an important concern within the Artificial Intelligence field. It is indeed very helpful when exploring the structure of the solution space. In this context, this paper revisits the computation process to count solutions for the global cardinality constraint in the context of counting-based search. It first highlights an error and then presents a way to correct the upper bound on the number of solutions for this constraint.

Improved filtering for the bin-packing with cardinality constraint

Previous research shows that a cardinality reasoning can improve the pruning of the bin-packing constraint. We first introduce a new algorithm, called BPCFlow, that filters both load and cardinality bounds on the bins, using a flow reasoning similar to the Global Cardinality Constraint. Moreover, we detect impossible assignments of items by combining the load and cardinality of the bins, using a method to detect items that are either ”too-big” or ”too-small”. This method is adapted to two previously existing filtering techniques along with BPCFlow, creating three new propagators. We then experiment the four new algorithms on Balanced Academic Curriculum Problem and Tank Allocation Problem instances. BPCFlow is shown to be stronger than previously existing filtering, and more computationally intensive. We show that the new filtering is useful on a small number of hard instances, while being too expensive for general use. Our results show that the introduced ”too-big/too-small” filtering can most of the time drastically reduce the size of the search tree and the computation time. This method is profitable in 88% of the tested instances.

A CP-Based Approach for Popular Matching

We propose a constraint programming approach to the popular matching problem. We show that one can use the Global Cardinality Constraint to encode the problem even in cases that involve ties in the ordinal preferences of the applicants.

Constraint Satisfaction Parameterized by Solution Size

In the constraint satisfaction problem (CSP) corresponding to a constraint language (i.e., a set of relations) $\Gamma$, the goal is to find an assignment of values to variables so that a given set of constraints specified by relations from $\Gamma$ is satisfied. The complexity of this problem has received a substantial amount of attention in the past decade. In this paper, we study the fixed-parameter tractability of CSPs parameterized by the size of the solution in the following sense: one of the possible values, say 0, is “free,” and the number of variables allowed to take other, “expensive,” values is restricted. A size constraint requires that exactly $k$ variables take nonzero values. We also study a more refined version of this restriction: a global cardinality constraint prescribes how many variables have to be assigned each particular value. We study the parameterized complexity of these types of CSPs where the parameter is the required number $k$ of nonzero variables. As special cases, we can obtain natural and well-studied parameterized problems such as Independent set, Vertex Cover, $d$-Hitting Set, Biclique, etc. In the case of constraint languages closed under substitution of constants, we give a complete characterization of the fixed-parameter tractable cases of CSPs with size constraints, and we show that all the remaining problems are W[1]-hard. For CSPs with cardinality constraints, we obtain a similar classification, but for some of the problems we are only able to show that they are Biclique-hard. The exact parameterized complexity of the Biclique problem is a notorious open problem, although it is believed to be W[1]-hard.

On matrices, automata, and double counting in constraint programming

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables \(\mathcal{M}\), with the same constraint C defined by a finite-state automaton \(\mathcal{A}\) on each row of \(\mathcal{M}\) and a global cardinality constraint \(\mathit{gcc}\) on each column of \(\mathcal{M}\). We give two methods for deriving, by double counting, necessary conditions on the cardinality variables of the \(\mathit{gcc}\) constraints from the automaton \(\mathcal{A}\). The first method yields linear necessary conditions and simple arithmetic constraints. The second method introduces the cardinality automaton, which abstracts the overall behaviour of all the row automata and can be encoded by a set of linear constraints. We also provide a domain consistency filtering algorithm for the conjunction of lexicographic ordering constraints between adjacent rows of \(\mathcal{M}\) and (possibly different) automaton constraints on the rows. We evaluate the impact of our methods in terms of runtime and search effort on a large set of nurse rostering problem instances.

THE ORDERED DISTRIBUTE CONSTRAINT

In this paper we introduce a new cardinality constraint: Ordered Distribute. Given a set of variables, this constraint limits for each value v the number of times v or any value greater than v is taken. It extends the global cardinality constraint, that constrains only the number of times a value v is taken by a set of variables and does not consider at the same time the occurrences of all the values greater than v . We design an algorithm for achieving generalized arc-consistency on Ordered Distribute, with a time complexity linear in the sum of the number of variables and the number of values in the union of their domains. In addition, we give some experiments showing the advantage of this new constraint for problems where values represent levels whose overrunning has to be under control. Finally, we present three extensions of our constraint that can be particularly useful in practice.

Parameterized Complexity Results for General Factors in Bipartite Graphs with an Application to Constraint Programming

The NP-hard general factor problem asks, given a graph and for each vertex a list of integers, whether the graph has a spanning subgraph where each vertex has a degree that belongs to its assigned list. The problem remains NP-hard even if the given graph is bipartite with partition U⊎V, and each vertex in U is assigned the list {1}; this subproblem appears in the context of constraint programming as the consistency problem for the extended global cardinality constraint. We show that this subproblem is fixed-parameter tractable when parameterized by the size of the second partite set V. More generally, we show that the general factor problem for bipartite graphs, parameterized by |V|, is fixed-parameter tractable as long as all vertices in U are assigned lists of length 1, but becomes $\text {\normalfont W[1]}$ -hard if vertices in U are assigned lists of length at most 2. We establish fixed-parameter tractability by reducing the problem instance to a bounded number of acyclic instances, each of which can be solved in polynomial time by dynamic programming.

The complexity of global cardinality constraints

In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the problem CCSP(G), the constraint satisfaction problem with global cardinality constraints that allows only relations from the set G. The main result of this paper characterizes sets G that give rise to problems solvable in polynomial time, and states that the remaining such problems are NP-complete.

The extended global cardinality constraint: An empirical survey

The Extended Global Cardinality Constraint (EGCC) is a vital component of constraint solving systems, since it is very widely used to model diverse problems. The literature contains many different versions of this constraint, which trade strength of inference against computational cost. In this paper, I focus on the highest strength of inference usually considered, enforcing generalized arc consistency (GAC) on the target variables. This work is an extensive empirical survey of algorithms and optimizations, considering both GAC on the target variables, and tightening the bounds of the cardinality variables. I evaluate a number of key techniques from the literature, and report important implementation details of those techniques, which have often not been described in published papers. Two new optimizations are proposed for EGCC. One of the novel optimizations ( dynamic partitioning, generalized from AllDifferent) was found to speed up search by 5.6 times in the best case and 1.56 times on average, while exploring the same search tree. The empirical work represents by far the most extensive set of experiments on variants of algorithms for EGCC. Overall, the best combination of optimizations gives a mean speedup of 4.11 times compared to the same implementation without the optimizations.

Constraint satisfaction problems and global cardinality constraints

In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the problem CCSP(Γ), the CSP with global cardinality constraints that allows only relations from the set Γ. The main result of this paper characterizes sets Γ that give rise to problems solvable in polynomial time, and states that the remaining such problems are NP-complete.

Tractable cases of the extended global cardinality constraint

We study the consistency and domain consistency problem for extended global cardinality (EGC) constraints. An EGC constraint consists of a set X of variables, a set D of values, a domain $D(x) \subseteq D$ for each variable x, and a "cardinality set" K(d) of non-negative integers for each value d. The problem is to instantiate each variable x with a value in D(x) such that for each value d, the number of variables instantiated with d belongs to the cardinality set K(d). It is known that this problem is NP-complete in general, but solvable in polynomial time if all cardinality sets are intervals. First we pinpoint connections between EGC constraints and general factors in graphs. This allows us to extend the known polynomial-time case to certain non-interval cardinality sets. Second we consider EGC constraints under restrictions in terms of the treewidth of the value graph (the bipartite graph representing variable-value pairs) and the cardinality-width (the largest integer occurring in the cardinality sets). We show that EGC constraints can be solved in polynomial time for instances of bounded treewidth, where the order of the polynomial depends on the treewidth. We show that (subject to the complexity theoretic assumption FPT???W[1]) this dependency cannot be avoided without imposing additional restrictions. If, however, also the cardinality-width is bounded, this dependency gets removed and EGC constraints can be solved in linear time.

On global warming: Flow-based soft global constraints

In case a CSP is over-constrained, it is natural to allow some constraints, called soft constraints, to be violated. We propose a generic method to soften global constraints that can be represented by a flow in a graph. Such constraints are softened by adding violation arcs to the graph and then computing a minimum-weight flow in the extended graph to measure the violation. We present efficient propagation algorithms, based on different violation measures, achieving domain consistency for the alldifferent constraint, the global cardinality constraint, the regular constraint and the same constraint.

Complete Bound Consistency for the Global Cardinality Constraint

We show an algorithm for bound consistency of global cardinality constraints, which runs in time O(n + n?) plus the time required to sort the assignment variables by range endpoints, where n is the number of assignment variables and n? is the number of values in the union of their domains. It is the first efficient algorithm that achieves bound consistency for all variables, and not only the assignment variables.

An Efficient Bounds Consistency Algorithm for the Global Cardinality Constraint

Previous studies have demonstrated that designing special purpose constraint propagators can significantly improve the efficiency of a constraint programming approach. In this paper we present an efficient algorithm for bounds consistency propagation of the generalized cardinality constraint (gcc). Using a variety of benchmark and random problems, we show that on some problems our bounds consistency algorithm can dramatically outperform existing state-of-the-art commercial implementations of constraint propagators for the gcc. We also present a new algorithm for domain consistency propagation of the gcc which improves on the worst-case performance of the best previous algorithm for problems that occur often in applications.

Cost-Based Arc Consistency for Global Cardinality Constraints

A global cardinality constraint (gcc) is specified in terms of a set of variables i>Xeli>x1,…,i>xpr which take their values in a subset of i>Veli>v1,…,i>vdr. It constrains the number of times each value i>vi∈i>V is assigned to a variable in i>X to be in an interval li>li,i>uir. A gcc with costs (costgcc) is a generalization of a gcc in which a cost is associated with each value of each variable. Then, each solution of the underlying gcc is associated with a global cost equal to the sum of the costs associated with the assigned values of the solution. A costgcc constrains the global cost to be less than a given value. Cardinality constraints with costs have proved very useful in many real-life problems, such as traveling salesman problems, scheduling, rostering, or resource allocation. For instance, they are useful for expressing preferences or for defining constraints such as a constraint on the sum of all different variables. In this paper, we present an efficient way of implementing arc consistency for a costgcc. We also study the incremental behavior of the proposed algorithm.