Solvable Cases Research Articles

Complexity properties of complementary prisms

The complementary prism $$G\bar{G}$$GG? of a graph G arises from the disjoint union of the graph G and its complement $$\bar{G}$$G? by adding the edges of a perfect matching joining pairs of corresponding vertices of G and $$\bar{G}$$G?. Haynes, Henning, Slater, and van der Merwe introduced the complementary prism and as a variation of the well-known prism. We study algorithmic/complexity properties of complementary prisms with respect to cliques, independent sets, k-domination, and especially $$P_3$$P3-convexity. We establish hardness results and identify some efficiently solvable cases.

Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras

In previous work with Scullard, we have defined a graph polynomial PB(q, T) that gives access to the critical temperature Tc of the q-state Potts model defined on a general two-dimensional lattice It depends on a basis B, containing n × m unit cells of and the relevant root Tc(n, m) of PB(q, T) was observed to converge quickly to Tc in the limit Moreover, in exactly solvable cases there is no finite-size dependence at all. In this paper we show how to reformulate this method as an eigenvalue problem within the periodic Temperley–Lieb (TL) algebra. This corresponds to taking first, so that the bases B are semi-infinite cylinders of circumference n. The limit implies faster convergence in n, while maintaining the n-independence in exactly solvable cases. In this setup, Tc(n) is determined by equating the largest eigenvalues of two topologically distinct sectors of the transfer matrix. Crucially, these two sectors determine the same critical exponent in the continuum limit, and the observed fast convergence is thus corroborated by results of conformal field theory. We obtain similar results for the dense and dilute phases of the O(N) loop model, using now a transfer matrix within the dilute periodic TL algebra. Compared with our previous study, the eigenvalue formulation allows us to double the size n for which Tc(n) can be obtained, using the same computational effort. We study in details three significant cases: (i) bond percolation on the kagome lattice, up to nmax = 14; (ii) site percolation on the square lattice, to nmax = 21; and (iii) self-avoiding polygons on the square lattice, to nmax = 19. Convergence properties of Tc(n) and extrapolation schemes are studied in details for the first two cases. This leads to rather accurate values for the percolation thresholds: pc = 0.524 404 999 167 439(4) for bond percolation on the kagome lattice, and pc = 0.592 746 050 792 10(2) for site percolation on the square lattice.

Relative frequencies of constrained events in stochastic processes: An analytical approach.

The stochastic simulation algorithm (SSA) and the corresponding Monte Carlo (MC) method are among the most common approaches for studying stochastic processes. They relies on knowledge of interevent probability density functions (PDFs) and on information about dependencies between all possible events. Analytical representations of a PDF are difficult to specify in advance, in many real life applications. Knowing the shapes of PDFs, and using experimental data, different optimization schemes can be applied in order to evaluate probability density functions and, therefore, the properties of the studied system. Such methods, however, are computationally demanding, and often not feasible. We show that, in the case where experimentally accessed properties are directly related to the frequencies of events involved, it may be possible to replace the heavy Monte Carlo core of optimization schemes with an analytical solution. Such a replacement not only provides a more accurate estimation of the properties of the process, but also reduces the simulation time by a factor of order of the sample size (at least ≈10(4)). The proposed analytical approach is valid for any choice of PDF. The accuracy, computational efficiency, and advantages of the method over MC procedures are demonstrated in the exactly solvable case and in the evaluation of branching fractions in controlled radical polymerization (CRP) of acrylic monomers. This polymerization can be modeled by a constrained stochastic process. Constrained systems are quite common, and this makes the method useful for various applications.

Algorithms solving the Matching Cut problem

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete. This paper provides a first branching algorithm solving Matching Cut in time O⁎(2n/2)=O⁎(1.4143n) for an n-vertex input graph, and shows that Matching Cut parameterized by the vertex cover number τ(G) can be solved by a single-exponential algorithm in time 2τ(G)O(n2). Moreover, the paper also gives a polynomially solvable case for Matching Cut which covers previous known results on graphs of maximum degree three, line graphs, and claw-free graphs.

Min-Max Regret Version of the Linear Time–Cost Tradeoff Problem with Multiple Milestones and Completely Ordered Jobs

We consider a linear time–cost tradeoff problem with multiple milestones and uncertain processing times such that all jobs are completely ordered. The performance measure is expressed as the sum of total weighted number of tardy jobs and total crashing cost. The processing times uncertainty is described through two types of scenarios: discrete and interval scenarios. The objective is to minimize maximum deviation from optimality over all scenarios. For the discrete scenario case, we prove its NP-hardness, develop a pseudo-polynomial time approach, and present a polynomially solvable case. Finally, we show that the interval scenario case is also NP-hard.

Sextic potential for -rigid prolate nuclei

The equation of the Bohr–Mottelson Hamiltonian with a sextic oscillator potential is solved for -rigid prolate nuclei. The associated shape phase space is reduced to three variables which are exactly separated. The angular equation has the spherical harmonic functions as solutions, while the equation is converted to the quasi-exactly solvable case of the sextic oscillator potential with a centrifugal barrier. The energies and the corresponding wave functions are given in closed form and depend, up to a scaling factor, on a single parameter. The and states are exactly determined, having an important role in the assignment of some ambiguous states for the experimental bands. Due to the special properties of the sextic potential, the model can simulate, by varying the free parameter, a shape phase transition from a harmonic to an anharmonic prolate -soft rotor crossing through a critical point. Numerical applications are performed for 39 nuclei: Ru, Mo, Xe, Ce, Nd, Sm, Gd, Dy, 172Os, Pt, 190Hg and 222Ra. The best candidates for the critical point are found to be 104Ru and Xe, followed closely by 128Xe, 172Os, 196Pt and 148Nd.

A Pseudo-Polynomial Time Algorithm for a Special Multiobjective Order Picking Problem

In this study, we work on the order picking problem (OPP) in a specially designed warehouse with a single picker. Ratliff and Rosenthal [Operations Research31(3) (1983) 507–521] show that the special design of the warehouse and use of one picker lead to a polynomially solvable case. We address the multiobjective version of this special case and investigate the properties of the nondominated points. We develop an exact algorithm that finds any nondominated point and present an illustrative example. Finally we conduct a computational test and report the results.

On the complexity of role colouring planar graphs, trees and cographs

We prove several results about the complexity of the role colouring problem. A role colouring of a graph G is an assignment of colours to the vertices of G such that two vertices of the same colour have identical sets of colours in their neighbourhoods. We show that the problem of finding a role colouring with 1<k<n colours is NP-hard for planar graphs. We show that restricting the problem to trees yields a polynomially solvable case, as long as k is either constant or has a constant difference with n, the number of vertices in the tree. Finally, we prove that cographs are always k-role-colourable for 1<k≤n and construct such a colouring in polynomial time.

Complexity of the cluster deletion problem on subclasses of chordal graphs

We consider the following vertex-partition problem on graphs, known as the CLUSTER DELETION (CD) problem: given a graph with real nonnegative edge weights, partition the vertices into clusters (in this case, cliques) to minimize the total weight of edges outside the clusters. The decision version of this optimization problem is known to be NP-complete even for unweighted graphs and has been studied extensively. We investigate the complexity of the decision CD problem for the family of chordal graphs, showing that it is NP-complete for weighted split graphs, weighted interval graphs and unweighted chordal graphs. We also prove that the problem is NP-complete for weighted cographs. Some polynomial-time solvable cases of the optimization problem are also identified, in particular CD for unweighted split graphs, unweighted proper interval graphs and weighted block graphs.

Record statistics for random walk bridges

.We investigate the statistics of records in a random sequence of n time steps. The sequence ’s represents the position at step k of a random walk ‘bridge’ of n steps that starts and ends at the origin. At each step, the increment of the position is a random jump drawn from a specified symmetric distribution. We study the statistics of records and record ages for such a bridge sequence, for different jump distributions. In absence of the bridge condition, i.e. for a free random walk sequence, the statistics of the number and ages of records exhibits a ‘strong’ universality for all n, i.e. they are completely independent of the jump distribution as long as the distribution is continuous. We show that the presence of the bridge constraint destroys this strong ‘all n’ universality. Nevertheless a ‘weaker’ universality still remains for large n, where we show that the record statistics depends on the jump distributions only through a single parameter , known as the Lévy index of the walk, but are insensitive to the other details of the jump distribution. We derive the most general results (for arbitrary jump distributions) wherever possible and also present two exactly solvable cases. We present numerical simulations that verify our analytical results.

Selecting vertex disjoint paths in plane graphs

We study variants of the vertex disjoint paths problem in plane graphs where paths have to be selected from given sets of paths. We investigate the problem as a decision, maximization, and routing-in-rounds problem. Although all considered variants are NP-hard in planar graphs, restrictions on the locations of the terminals on the outer face of the given planar embedding of the graph lead to polynomially solvable cases for the decision and maximization versions of the problem. For the routing-in-rounds problem, we obtain a p-approximation algorithm, where p is the maximum number of alternative paths for a terminal pair, when restricting the locations of the terminals to the outer face such that they appear in a counterclockwise traversal of the boundary as a sequence s1,s2,',sk,tπ1,tπ2,',tπk for some permutation π. © 2015 Wiley Periodicals, Inc.NETWORKS, Vol. 662, 136-144 2015

On the two-stage hybrid flow shop with dedicated machines

In this paper we develop new elimination rules and discuss several polynomially solvable cases for the two-stage hybrid flow shop problem with dedicated machines. We also propose a worst case analysis for several heuristics. Furthermore, we point out and correct several errors in the paper of Yang [J. Yang, A two-stage hybrid flow shop with dedicated machines at the first stage. Comput. Oper. Res. 40 (2013) 2836−2843].

Two-machine flow shop scheduling with deteriorating jobs: minimizing the weighted sum of makespan and total completion time

This paper considers a two-machine flow shop scheduling problem with deteriorating jobs in which the processing times of jobs are dependent on their starting times in the sequence. The objective is to minimize the weighted sum of makespan and total completion time. To analyse the problem, we propose a mixed integer programming model, and discuss several polynomially solvable special cases. We also present a branch-and-bound algorithm with several dominance rules, an upper bound and a lower bound. Finally, we present results of computational experiments conducted to evaluate the performance of the proposed model and the exact algorithm.

The bipartite unconstrained 0–1 quadratic programming problem: Polynomially solvable cases

We consider the bipartite unconstrained 0–1 quadratic programming problem (BQP01) which is a generalization of the well studied unconstrained 0–1 quadratic programming problem (QP01). BQP01 has numerous applications and the problem is known to be MAX SNP hard. We show that if the rank of an associated m×n cost matrix Q=(qij) is fixed, then BQP01 can be solved in polynomial time. When Q is of rank one, we provide an O(nlogn) algorithm and this complexity reduces to O(n) with additional assumptions. Further, if qij=ai+bj for some ai and bj, then BQP01 is shown to be solvable in O(mnlogn) time. By restricting m=O(logn), we obtain yet another polynomially solvable case of BQP01 but the problem remains MAX SNP hard if m=O(nk) for a fixed k. Finally, if the minimum number of rows and columns to be deleted from Q to make the remaining matrix non-negative is O(logn), then we show that BQP01 is polynomially solvable but it is NP-hard if this number is O(nk) for any fixed k.

Evaluation of optimality in the fuzzy single machine scheduling problem including discounted costs

The single machine scheduling problem has been often regarded as a simplified representation that contains many polynomial solvable cases. However, in real-world applications, the imprecision of data at the level of each job can be critical for the implementation of scheduling strategies. Therefore, the single machine scheduling problem with the weighted discounted sum of completion times is treated in this paper, where we assume that the processing times, weighting coefficients and discount factor are all described using trapezoidal fuzzy numbers. Our aim in this study is to elaborate adequate measures in the context of possibility theory for the assessment of the optimality of a fixed schedule. Two optimization approaches namely genetic algorithm and pattern search are proposed as computational tools for the validation of the obtained properties and results. The proposed approaches are experimented on the benchmark problem instances and a sensitivity analysis with respect to some configuration parameters is conducted. Modeling and resolution frameworks considered in this research offer promise to deal with optimality in the wide class of fuzzy scheduling problems, which is recognized to be a difficult task by both researchers and practitioners.

Strongly solvable spherical subgroups and their combinatorial invariants

A subgroup H of an algebraic group G is said to be strongly solvable if H is contained in a Borel subgroup of G. This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in reductive complex algebraic groups: Luna's general classification of arbitrary spherical subgroups restricted to the strongly solvable case, Luna's 1993 classification of strongly solvable wonderful subgroups, and the author's 2011 classification of strongly solvable spherical subgroups. We give a detailed presentation of all the three classifications and exhibit interrelations between the corresponding combinatorial invariants, which enables one to pass from one of these classifications to any other.

Quantum phase transitions and string orders in the spin-1/2 Heisenberg–Ising alternating chain with Dzyaloshinskii–Moriya interaction

Quantum phase transitions (QPTs) and the ground-state phase diagram of the spin-1/2 Heisenberg–Ising alternating chain (HIAC) with uniform Dzyaloshinskii–Moriya (DM) interaction are investigated by a matrix-product-state (MPS) method. By calculating the odd- and even-string order parameters, we recognize two kinds of Haldane phases, i.e. the odd- and even-Haldane phases. Furthermore, doubly degenerate entanglement spectra on odd and even bonds are observed in odd- and even-Haldane phases, respectively. A rich phase diagram including four different phases, i.e. an antiferromagnetic (AF), AF stripe, odd- and even-Haldane phases, is obtained. These phases are found to be separated by continuous QPTs: the topological QPT between the odd- and even-Haldane phases is verified to be continuous and corresponds to conformal field theory with central charge c = 1; while the rest of the phase transitions in the phase diagram are found to be c = 1/2. We also revisit, with our MPS method, the exactly solvable case of HIAC model with DM interactions only on odd bonds and find that the even-Haldane phase disappears, but the other three phases, i.e. the AF, AF stripe and odd-Haldane phases, still remain in the phase diagram. We exhibit the evolution of the even-Haldane phase by tuning the DM interactions on the even bonds gradually.

The bottleneck transportation problem with auxiliary resources

In this paper, we introduce a new extension of the bottleneck transportation problem where additionally auxiliary resources are needed to support the transports. A single commodity has to be sent from supply to demand nodes such that the total demand is satisfied and the time at which all units of the commodity have arrived at the demand nodes is minimized. We show that already the problem with a single demand node and a single auxiliary resource is NP-hard and consider some polynomially solvable special cases.

Bias of group generators in the solvable case

Babai and Pak demonstrated a weakness in the product replacement algorithm, a widely used heuristic algorithm intended to rapidly generate nearly uniformly distributed random elements in a finite group G. It was an open question whether the same weakness can be exhibited if one considers only finite solvable groups. We give an affirmative solution to this problem. We consider the distribution of the first component in a k-tuple chosen uniformly in the set of all the k-tuples generating G and construct an infinite sequence of finite solvable groups G for which this distribution turns out to be very far from uniform.

Band structure analysis of an analytically solvable Hill equation with continuous potential

This paper concerns analytically solvable cases of Hill’s equation containing a continuously differentiable periodic potential. We outline a procedure for constructing the Floquet–Bloch fundamental system, and analyze the band structure of the system. The similarities to, and differences from, the cases of a piecewise constant periodic potential and the Mathieu potential, are illuminated.