Unweighted Problem Research Articles

Optimal parameters related with continuity properties of the multilinear fractional integral operator between Lebesgue and Lipschitz spaces

We deal with the boundedness of the multilinear fractional integral operator $$I_{\gamma ,m}$$ from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We characterize the classes of weights for which the problem described above holds and show the optimal range of the parameters involved. The optimality is understood in the sense that the parameters defining the corresponding spaces belong to a certain region. We further exhibit examples of weights for the class which cover the mentioned area.

Computing valuations of the Dieudonné determinants

This paper addresses the problem of computing valuations of the Dieudonné determinants of matrices over discrete valuation skew fields (DVSFs). Under a reasonable computational model, we propose two algorithms for a class of DVSFs, called split. Our algorithms are extensions of the combinatorial relaxation of Murota (1995) and the matrix expansion by Moriyama and Murota (2013), both of which are based on combinatorial optimization. While our algorithms require an upper bound on the output, we give an estimation of the bound for skew polynomial matrices and show that the estimation is valid only for skew polynomial matrices.We consider two applications of this problem. The first one is the noncommutative weighted Edmonds' problem (nc-WEP), which is to compute the degree of the Dieudonné determinants of matrices having noncommutative symbols. We show that the presented algorithms reduce the nc-WEP to the unweighted problem in polynomial time. In particular, we show that the nc-WEP over the rational field is solvable in time polynomial in the input bit-length. We also present an application to analyses of degrees of freedom of linear time-varying systems by establishing formulas on the solution spaces of linear differential/difference equations.

Maximizing total early work in a distributed two‐machine flow‐shop

AbstractThe problem of maximizing total early work in a two‐machine flow‐shop, in which n jobs are to be scheduled subject to a common due date d, has been recently studied in the scheduling literature. An O(n2d4) time dynamic programming algorithm was presented first for the weighted case, and then for the unweighted case another O(n2d2) running time dynamic programming algorithm was proposed and converted into an time fully polynomial time approximation scheme (FPTAS). By establishing new problem properties, we present an O(nd2) time dynamic programming algorithm and an time FPTAS for the unweighted problem. We generalize the problem to a distributed setting of m parallel two‐machine flow‐shops, develop an O(nd3m) time dynamic programming algorithm, an time FPTAS, and three integer linear programming (ILP) formulations for it. Computational experiments are conducted to appraise the proposed ILP models.

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

A bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex uin U, the neighbors of u are consecutive in the ordering of V. An induced matchingH of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and mweighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex uin U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n+m) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of O(n^2) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. https://doi.org/10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.

Optimal algorithms for scheduling under time-of-use tariffs

We consider a natural generalization of classical scheduling problems to a setting in which using a time unit for processing a job causes some time-dependent cost, the time-of-use tariff, which must be paid in addition to the standard scheduling cost. We focus on preemptive single-machine scheduling and two classical scheduling cost functions, the sum of (weighted) completion times and the maximum completion time, that is, the makespan. While these problems are easy to solve in the classical scheduling setting, they are considerably more complex when time-of-use tariffs must be considered. We contribute optimal polynomial-time algorithms and best possible approximation algorithms. For the problem of minimizing the total (weighted) completion time on a single machine, we present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time slots to be used for preemptively scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for. For preemptive scheduling to minimize the makespan, we show that there is a comparably simple optimal algorithm with polynomial running time. This is true even in a certain generalized model with unrelated machines.

Dynamic Initial Weight Assignment for MaxSAT

The Maximum Satisfiability (Maximum Satisfiability (MaxSAT)) approach is the choice, and perhaps the only one, to deal with most real-world problems as most of them are unsatisfiable. Thus, the search for a complete and consistent solution to a real-world problem is impractical due to computational and time constraints. As a result, MaxSAT problems and solving techniques are of exceptional interest in the domain of Satisfiability (Satisfiability (SAT)). Our research experimentally investigated the performance gains of extending the most recently developed SAT dynamic Initial Weight assignment technique (InitWeight) to handle the MaxSAT problems. Specifically, we first investigated the performance gains of dynamically assigning the initial weights in the Divide and Distribute Fixed Weights solver (DDFW+Initial Weight for Maximum Satisfiability (DDFW+InitMaxSAT)) over Divide and Distribute Fixed Weights solver (DDFW) when applied to solve a wide range of well-known unweighted MaxSAT problems obtained from DIMACS. Secondly, we compared DDFW+InitMaxSAT’s performance against three known state-of-the-art SAT solving techniques: YalSAT, ProbSAT, and Sparrow. We showed that the assignment of dynamic initial weights increased the performance of DDFW+InitMaxSAT against DDFW by an order of magnitude on the majority of problems and performed similarly otherwise. Furthermore, we showed that the performance of DDFW+InitMaxSAT was superior to the other state-of-the-art algorithms. Eventually, we showed that the InitWeight technique could be extended to handling partial MaxSAT with minor modifications.

Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals

We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an ε-approximation for the d-variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on ε−1. The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on ε−1, we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential (s,t)-weak tractability with max(s,t)≥1. For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential (s,t)-weak tractability with max(s,t)<1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).

Recognition and Optimization Algorithms for P5-Free Graphs

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

Efficient computational method for the dynamic responses of a floating wind turbine

ABSTRACTThis paper investigates the computational methods for calculating the dynamic responses of a floating wind turbine. The mathematical procedures for a proposed new identification method for fitting a state space model to approximate the convolution term in the motion equation of the floating wind turbine have been specified, and these mathematical procedures have been executed in a stand-alone MATLAB programme in this study. This new identification method consists of finding the vector of parameters that give the best un-weighted least squares fitting to the frequency responses, and a damped Gauss–Newton algorithm is first proposed to solve this un-weighted least squares problem. The calculation results in this paper have been systematically analysed and compared, and the accuracy and efficiency of our proposed new method have been convincingly substantiated.

Approximating min-cost chain-constrained spanning trees: a reduction from weighted to unweighted problems

We study the min-cost chain-constrained spanning-tree (MCCST) problem: find a min-cost spanning tree in a graph subject to degree constraints on a nested family of node sets. We devise the first polytime algorithm that finds a spanning tree that (i) violates the degree constraints by at most a constant factor and (ii) whose cost is within a constant factor of the optimum. Previously, only an algorithm for unweighted CCST was known (Olver and Zenklusen in Proceedings of the 16th IPCO, pp 324–335, 2013), which satisfied (i) but did not yield any cost bounds. This also yields the first result that obtains an O(1)-factor for both the cost approximation and violation of degree constraints for any spanning-tree problem with general degree bounds on node sets, where an edge participates in a super-constant number of degree constraints. A notable feature of our algorithm is that we reduce MCCST to unweighted CCST (and then utilize Olver and Zenklusen in Proceedings of the 16th IPCO, pp 324–335, 2013) via a novel application of Lagrangian duality to simplify the cost structure of the underlying problem and obtain a decomposition into certain uniform-cost subproblems. We show that this Lagrangian-relaxation based idea is in fact applicable more generally and, for any cost-minimization problem with packing side-constraints, yields a reduction from the weighted to the unweighted problem. We believe that this reduction is of independent interest. As another application of our technique, we consider the k -budgeted matroid basis problem, where we build upon a recent rounding algorithm of Bansal and Nagarajan (Proceedings of IPCO 2016. arXiv:1512.02254 , 2015) to obtain an improved $$n^{O(k^{1.5}/\epsilon )}$$ -time algorithm that returns a solution that satisfies (any) one of the budget constraints exactly and incurs a $$(1+\epsilon )$$ -violation of the other budget constraints.

User–Base-Station Association in HetSNets: Complexity and Efficient Algorithms

This work considers the problem of user association to small-cell base stations (SBSs) in a heterogeneous and small-cell network (HetSNet). Two optimization problems are investigated, which are maximizing the set of associated users to the SBSs (the unweighted problem) and maximizing the set of weighted associated users to the SBSs (the weighted problem), under signal-to-interference-plus-noise ratio (SINR) constraints. Both problems are formulated as linear integer programs. The weighted problem is known to be NP-hard and, in this paper, the unweighted problem is proved to be NP-hard as well. Therefore, this paper develops two heuristic polynomial-time algorithms to solve both problems. The computational complexity of the proposed algorithms is evaluated and is shown to be far more efficient than the complexity of the optimal brute-force (BF) algorithm. Moreover, the paper benchmarks the performance of the proposed algorithms against the BF algorithm, the branch-and-bound (B\&B) algorithm and standard algorithms, through numerical simulations. The results demonstrate the close-to-optimal performance of the proposed algorithms. They also show that the weighted problem can be solved to provide solutions that are fair between users or to balance the load among SBSs.

Application of genetic algorithm for the set-covering problem solution

The weighed and unweighted minimal set-cover problem, its applicability for the solution of the major optimization practical tasks, such as arrangement of service points, assignment of crews in transport, as well as the integrated-circuit and conveyer lines designing is considered. The paper objective is to describe methods of improving efficiency of this task solution. The principle of operation of the genetic algorithm and the applicability of its modification as a method of the solution to the set-cover problem are defined. Greedy strategy of Chvatal for the set-cover problem solution is considered. An exhaustive algorithm as an exact algorithm for the solution of small-size tasks is developed. The modified genetic algorithm developed by Nguyen M. H. is described. A software tool for comparing the performance of these algorithms is created. It is concluded that the solution of the set-cover problem by our genetic algorithm modification is more effective than by the genetic algorithm of Nguyen M. H. or the greedy strategy. And the results obtained in small-size tasks are noted for insignificant error.

Heuristic and Exact Algorithms for the Two-Machine Just in Time Job Shop Scheduling Problem

The problem addressed in this paper is the two-machine job shop scheduling problem when the objective is to minimize the total earliness and tardiness from a common due date (CDD) for a set of jobs when their weights equal 1 (unweighted problem). This objective became very significant after the introduction of the Just in Time manufacturing approach. A procedure to determine whether the CDD is restricted or unrestricted is developed and a semirestricted CDD is defined. Algorithms are introduced to find the optimal solution when the CDD is unrestricted and semirestricted. When the CDD is restricted, which is a much harder problem, a heuristic algorithm is proposed to find approximate solutions. Through computational experiments, the heuristic algorithms’ performance is evaluated with problems up to 500 jobs.

On weak tractability of the Clenshaw–Curtis Smolyak algorithm

We consider the problem of integration of d-variate analytic functions defined on the unit cube with directional derivatives of all orders bounded by 1. We prove that the Clenshaw–Curtis Smolyak algorithm leads to weak tractability of the problem. This seems to be the first positive tractability result for the Smolyak algorithm for a normalized and unweighted problem. The space of integrands is not a tensor product space and therefore we have to develop a different proof technique. We use the polynomial exactness of the algorithm as well as an explicit bound on the operator norm of the algorithm.

Scheduling with deteriorating jobs and setup times on a single machine

This paper addresses several single-machine scheduling problems with deteriorating jobs and setup times to minimise the maximum lateness, the weighted number of tardy jobs and the total weighted completion time. The actual job processing times and setup times are both proportional functions of their starting times. For the maximum lateness problem, we first show that the general problem is -hard, which also resolves an open question on its complexity raised in the literature, and then we show that the problem with a fixed number of families is polynomially solvable. For the weighted number of tardy jobs problem, we first present a pseudo-polynomial-time algorithm when the number of families is fixed, and then we provide an O(n2)-time algorithm for the unweighted problem with uniform family due dates. For the total weighted completion time problem, we present a polynomial-time algorithm when the number of families is fixed.

Theoretical and computational advances for network diversion

The network‐diversion problem (ND) is defined on a directedor undirected graph G = (V,E) having non‐negative edge weights, a source vertex s, a sink vertex t, and a “diversion edge” . This problem, with intelligence‐gathering and war‐fighting applications, seeks a minimum‐weight, minimal s‐t cut in G such that . We present (a) a new NP‐completeness proof for ND on directed graphs, (b) the first polynomial‐time solution algorithm for a special graph topology, (c) an improved mixed‐integer programming formulation (MIP), and (d) useful valid inequalities for that MIP. The proof strengthens known results by showing, for instance, that ND is strongly NP‐complete on a directed graph even when is incident from s or into t, but not both, and even when G is acyclic; a corollary shows the NP‐completeness of a vertex‐deletion version of ND on undirected graphs. The polynomial‐time algorithm solves ND on s‐t planar graphs. Compared to a MIP from the literature, the new MIP, coupled with valid inequalities, reduces the average duality gap by 10–50% on certain classes of test problems. It can also reduce solution times by an order of magnitude. We successfully solve unweighted problems with roughly 90,000 vertices and 360,000 edges and weighted problems with roughly 10,000 vertices and 40,000 edges. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 000(00), 000–000 2013

APPROXIMATE WEIGHTED FARTHEST NEIGHBORS AND MINIMUM DILATION STARS

We provide an efficient reduction from the problem of querying approximate multiplicatively weighted farthest neighbors in a metric space to the unweighted problem. Combining our techniques with core-sets for approximate unweighted farthest neighbors, we show how to find approximate farthest neighbors that are farther than a factor (1 - ∊) of optimal in time O( log n) per query in D-dimensional Euclidean space for any constants D and ∊. As an application, we find an O(n log n) expected time algorithm for choosing the center of a star topology network connecting a given set of points, so as to approximately minimize the maximum dilation between any pair of points.

Quasi-polynomial tractability

Tractability of multivariate problems has become a popular research subject. Polynomial tractability means that the solution of a d -variate problem can be solved to within ε with polynomial cost in ε − 1 and d . Unfortunately, many multivariate problems are not polynomially tractable. This holds for all non-trivial unweighted linear tensor product problems. By an unweighted problem we mean the case when all variables and groups of variables play the same role. It seems natural to ask what is the “smallest” non-exponential function T : [ 1 , ∞ ) × [ 1 , ∞ ) → [ 1 , ∞ ) for which we have T -tractability of unweighted linear tensor product problems; that is, when the cost of a multivariate problem can be bounded by a multiple of a power of T ( ε − 1 , d ) . Under natural assumptions, it turns out that this function is T qpol ( x , y ) = exp ( ( 1 + ln x ) ( 1 + ln y ) ) for all x , y ∈ [ 1 , ∞ ) . The function T qpol goes to infinity faster than any polynomial although not “much” faster, and that is why we refer to T qpol -tractability as quasi-polynomial tractability. The main purpose of this paper is to promote quasi-polynomial tractability, especially for the study of unweighted multivariate problems. We do this for the worst case and randomized settings and for algorithms using arbitrary linear functionals or only function values. We prove relations between quasi-polynomial tractability in these two settings and for the two classes of algorithms.

Infinite split scheduling: a new lower bound of total weighted completion time on parallel machines with job release dates and unavailability periods

This paper addresses an identical parallel machine scheduling problem with job release dates and unavailability periods to minimize total weighted completion time. This problem is known to be NP-hard in the strong sense. We propose a new lower bound that can be computed in polynomial time. The test on more than 8 400 randomly generated instances shows a very significant improvement with respect to existing results for previously studied special cases: without unavailability constraints, unweighted version, or identical job release dates. For instance, the average improvement for the unweighted problem is as much as 20.43% for 2 machines, 53.03% for 7 machines and 66.70% for 15 machines. For some instances, the improvement can be even as much as 93%.

On Dynamic Shortest Paths Problems

We obtain the following results related to dynamic versions of the shortest-paths problem: Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems. A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time of $\tilde {O}(m\sqrt{n})$ (we use $\tilde {O}$ to hide small poly-logarithmic factors) and a worst case query time is O(n3/4). A deterministic O(n2log n) time algorithm for constructing an O(log n)-spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance.