Permutation Constraint Research Articles

Comparing QUBO models for quantum annealing: integer encodings for permutation problems

AbstractQUBO (quadratic unconstrained binary optimization) has become the modeling language for quantum annealing and quantum‐inspired annealing solvers. We present different approaches in QUBO for the magic square problem and the quadratic assignment problem (QAP), which can be modeled by linear equations and a permutation constraint over integer variables. Different ways of encoding integers by Booleans in QUBO amount to models, the implementation of which could have very different performance. Experiments performed on the Fixstars Amplify Annealer Engine, a quantum‐inspired annealing solver, show that, compared to the classical one‐hot encoding, using unary encoding for integers performs slightly better for the QAP and much better for magic square.

Explicit K-symplectic-like algorithms for guiding center system

In this paper, for the guiding center system, we propose a type of explicit K-symplectic-like methods by extending the original guiding center phase space and constructing new augmented Hamiltonians. The original guiding center phase space is extended by making several copies in order to make the guiding center Hamiltonian separable to variables. In the extended phase space, the augmented guiding center Hamiltonian can be numerically solved by a K-symplectic method through the splitting technique and the composition of some simpler subsystems. Meanwhile, a midpoint permutation constraint is imposed on the extended phase space. Numerical experiments are carried out for guiding center motions in different magnetic fields using different numerical methods, including K-symplectic-like algorithms, canonical symplectic algorithms, and higher order implicit Runge-Kutta methods. Results show that energy errors of K-symplectic-like methods are bounded within small intervals over a long time, defeating higher order implicit Runge-Kutta methods. For comparison, explicit K-symplectic-like methods exhibit higher computational efficiency than existing canonicalized symplectic methods of the same order. We also verify that permutation constraints are important for the numerical properties of explicit K-symplectic methods. Among them, the method with the midpoint permutation constraint behaves better in long-term energy conservation and the elimination of secular drift errors than the same method without any permutation. The permutation that imposes a constraint on the Hamiltonian behaves best in energy preservation.

Back-to-Back Butterfly Network, an Adaptive Permutation Network for New Communication Standards

In this paper, we introduce a Back-to-Back Butterfly Network (B2BN) based on multiplexers (MUXs) in which any kind of permutation can be performed. However, for a given permutation, it is not an easy task to select the appropriate paths in B2BN without any conflict in terms of MUXs. In this paper, we propose a formal model to efficiently solve such conflicts. The proposed method relies on collecting the sets of potential paths that transfer an input to an output. Then, a path from each set is selected respecting a conflict free constraint. Once the appropriate paths are selected, the control signals of the MUXs are generated. This model has been experimented with 5G communication, showing how to process several frames in parallel with different permutation constraints.

Fully Parallel Circular-Shift Rotation Network for Communication Standards

This brief sheds light on a formal approach to design an innovative class of permutation network called Fully Parallel Circular-shift rotation Network. Thus, the contribution of such a design is the capability to process - in parallel - more than one set of elements with different lengths and permutation constraints. Yet, processing several sets simultaneously, arises from the increasing demand for higher throughput rate for applications such as the incoming 5G codes. Furthermore, the ability to handle several different sets of elements in parallel reduces the number of idled components in the designs (which improves the $performances/area$ ratio). Meanwhile, independent circular-shifter network blocks are generated in order to process the different sets simultaneously. Hence, the cost in terms of number of Multiplexers (MUXs) is low compared to the best existing works we know in the state-of-the-art, with the same number of stages.

Satisfying ternary permutation constraints by multiple linear orders or phylogenetic trees

A ternary permutation constraint satisfaction problem (CSP) is specified by a subset Π of the symmetric group S3. An instance of such a problem consists of a set of variables V and a set of constraints C, where each constraint is an ordered triple of distinct elements from V. The goal is to construct a linear ordering α on V such that, for each constraint (a,b,c)∈C, the ordering of a, b, c induced by α is in Π. Excluding symmetries and trivial cases there are eleven such problems, and their complexity is well known. Here we consider the variant of the problem, denoted 2-Π, where we are allowed to construct two linear orders α and β and each constraint needs to be satisfied by at least one of the two. We give a full complexity classification of all eleven 2-Π problems, observing that in the switch from one to two linear orders the complexity landscape changes quite abruptly and that hardness proofs become rather intricate. To establish the proofs, we use several computer-aided searches. Each such search returns a small instance with a unique solution for a given problem. We then focus on one of the eleven problems in particular, which is closely related to the 2-Caterpillar Compatibility problem in the phylogenetics literature. We show that this particular CSP remains hard on three linear orders, and also in the biologically relevant case when we swap three linear orders for three phylogenetic trees, yielding the 3-Tree Compatibility problem. Lastly, we give extremal results concerning the minimum number of trees required, in the worst case, to satisfy a set of rooted triplet constraints on n leaf labels.

On exact algorithms for the permutation CSP

In the Permutation Constraint Satisfaction Problem (Permutation CSP) we are given a set of variables V and a set of constraints C, in which the constraints are tuples of elements of V. The goal is to find a total ordering of the variables, π:V→[1,…,|V|], which satisfies as many constraints as possible. A constraint (v1,v2,…,vk) is satisfied by an ordering π when π(v1)<π(v2)<⋯<π(vk). An instance has arity k if all the constraints involve at most k elements.This problem expresses a variety of permutation problems including Feedback Arc Set and Betweenness problems. A naive algorithm, listing all the n! permutations, requires 2O(nlogn) time. Interestingly, Permutation CSP for arity 2 or 3 can be solved by Held–Karp-type algorithms in time O∗(2n), but no algorithm is known for arity at least 4 with running time significantly better than 2O(nlogn). In this paper we resolve the gap by showing that Arity 4 Permutation CSP cannot be solved in time 2o(nlogn) unless the exponential time hypothesis fails.

Every ternary permutation constraint satisfaction problem parameterized above average has a kernel with a quadratic number of variables

A ternary Permutation-CSP is specified by a subset Π of the symmetric group S 3 . An instance of such a problem consists of a set of variables V and a multiset of constraints, which are ordered triples of distinct variables of V. The objective is to find a linear ordering α of V that maximizes the number of triples whose rearrangement (under α) follows a permutation in Π. We prove that every ternary Permutation-CSP parameterized above average has a kernel with a quadratic number of variables.

MAX ONES Generalized to Larger Domains

We study a family of problems, called Maximum Solution, where the objective is to maximize a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints. When the domain is Boolean (i.e., restricted to $\{0,1\}$), the maximum solution problem is identical to the well-studied Max Ones problem, and the approximability is completely understood for all restrictions on the underlying constraints [S. Khanna, M. Sudan, L. Trevisan, and D. P. Williamson, SIAM J. Comput., 30 (2001), pp. 1863-1920]. We continue this line of research by considering domains containing more than two elements. We present two main results: a complete classification for the approximability of all maximal constraint languages over domains of cardinality at most $4$, and a complete classification of the approximability of the problem when the set of allowed constraints contains all permutation constraints. Under the assumption that a conjecture due to Szczepara [Minimal Clones Generated by Groupoids, Ph.D. thesis, Universite de Montreal, Montreal, QC, 1996] holds, we give a complete classification for all maximal constraint languages. These classes of languages are well studied in universal algebra and computer science; they have, for instance, been considered in connection with machine learning and constraint satisfaction. Our results are proved by using algebraic results from clone theory, and the results indicate that this approach is very powerful for classifying the approximability of certain optimization problems.

Generating Feasible Schedules for a Pick-up and Delivery Problem

Abstract In this research, we study a transportation problem that involves vehicle routing and driver scheduling for a bus station. The problem requires drivers to provide pick-up and delivery services to customers. Its solution involves planning itineraries for buses and establishing working schedule for drivers, subject to vehicle capacity limitation and time constraints. The objective is to efficiently schedule the fleet of vehicles for customer demand so as to reduce costs. This paper presents a complete constraint model and a solution method for solving the problem. For vehicle routing, a permutation constraint is used to impose a total order for visiting all customer locations regardless of different vehicle routes. This provides a global planning over all routes and plays an important role in the solution method. For driver scheduling, set partitioning constraints are used for assigning drivers and vehicles to requests. Based on this constraint model, efficient reactive planning and optimization algorithms are constructed to generate feasible schedules for the problem.

A note on scheduling jobs with missing operations in permutation flow shops

Flowshop scheduling problems are usually handled as permutation flowshops because of computational complexity although the technical framework in realworld problems often allows that jobs pass each other. Having a missing operation of a job on a machine, this job passing becomes a natural way to process jobs unless the underlying system is not an inflexible flow-line. However, when scheduling, missing operations are usually handled as zero processing time operations not allowing for any job passing. It is shown that even permitting job passing for missing operations while keeping the permutation constraint improves total completion time only under rather specific circumstances. We propose 'partial permutation flowshop sequencing' to overcome these specific formal requirements with respect to the real-world problem setting.

Total Ordering Problem

The problem of finding a total ordering of a finite set satisfying a given set of in-between restrictions is considered. It is shown that the problem is $NP$-complete.