Fast Polynomial Time Algorithms Research Articles

Integrated Robotics Networks with Co-Optimization of Drone Placement and AirGround Communications

Terrestrial robots, i.e., unmanned ground vehicles (UGVs), and aerial robots, i.e., unmanned aerial vehicles (UAVs), operate in separate spaces. To exploit their complementary features (e.g., fields of views, communication links, computing capabilities), a promising paradigm termed integrated robotics network therefore emerges, which provides communications for cooperative UAVs-UGVs applications. However, how to efficiently deploy UAVs and schedule the UAVs-UGVs connections according to different UGV tasks become challenging. In this paper, we consider the sum-rate maximization problem, where UGVs plan their trajectories autonomously and are dynamically associated with UAVs according to their planned trajectories. Although this problem is a NP-hard mixed integer program, a fast polynomial time algorithm using alternating gradient descent and penalty-based binary relaxation, is devised. Simulation results demonstrate the effectiveness of the proposed algorithm.

Directed negative-weight percolation.

We consider the negative-weight percolation model on directed graphs. In particular, we study the model on a two-dimensional, weighted, periodic, centered square lattice. Bond weights are taken from a distribution that allows for both positive and negative values. For a given realization of the disorder, a minimally weighted directed loop and path configuration is determined by performing a nontrivial transformation of the original lattice into a minimum-weight perfect matching problem. For this problem, fast polynomial-time algorithms are available, thus we could study large systems with high accuracy. Depending on the fraction of negatively and positively weighted bonds on the lattice, a continuous phase transition is identified and its characterizing critical exponents are estimated by means of a finite-size scaling analysis. We observe a strong change of the universality class with respect to standard directed percolation as well as with respect to undirected negative-weight percolation. Furthermore, the relation to directed polymers in random media is illustrated.

Scheduling Mutual Exclusion Accesses in Equal-Length Jobs

A fundamental problem in parallel and distributed processing is the partial serialization that is imposed due to the need for mutually exclusive access to common resources. In this article, we investigate the problem of optimally scheduling (in terms of makespan) a set of jobs, where each job consists of the same number L of unit-duration tasks, and each task either accesses exclusively one resource from a given set of resources or accesses a fully shareable resource. We develop and establish the optimality of a fast polynomial-time algorithm to find a schedule with the shortest makespan for any number of jobs and for any number of resources for the case of L = 2. In the notation commonly used for job-shop scheduling problems, this result means that the problem J | d ij =1, n j =2| C max is polynomially solvable, adding to the polynomial solutions known for the problems J 2 | n j ≤ 2 | C max and J 2 | d ij = 1 | C max (whereas other closely related versions such as J 2 | n j ≤ 3 | C max , J 2 | d ij ∈ { 1,2} | C max , J 3 | n j ≤ 2 | C max , J 3 | d ij =1 | C max , and J | d ij =1, n j ≤ 3| C max are all known to be NP-complete). For the general case L > 2 (i.e., for the job-shop problem J | d ij =1, n j = L > 2| C max ), we present a competitive heuristic and provide experimental comparisons with other heuristic versions and, when possible, with the ideal integer linear programming formulation.

Fast solution of single-machine scheduling problem with embedded jobs

We present a fast polynomial-time algorithm for single-machine scheduling problem with release times (rj), processing times (pj) and due dates (dj) with the objective to minimize the maximum job lateness. The general setting is strongly NP-hard. We expose a particular embedded structure that typically possess a subset of jobs in an instance of the problem. We establish useful properties of the problem instances that tolerate this structure, i.e., in such an instance no subset of embedded jobs overlaps in time with the jobs that do not belong to that subset. Relying on these properties, our algorithm finds an optimal solution for the corresponding class of problem instances. Whether a given problem instance is tolerant of the revealed embedded structure becomes known during the execution of the algorithm. Nevertheless, some sufficient conditions in terms of job parameters can a priori be given. One such a condition is that for any pair of jobs j and i with ri>rj and di<dj, dj−rj−pj≤di−ri−pi, and if ri+pi≥rj+pj, then di≥dj. As we show, our setting remains powerful enough to model important real-life applications.

A Simple NOVCA: Near Optimal Vertex Cover Algorithm

This paper describes an extremely fast polynomial time algorithm, the Near Optimal Vertex Cover Algorithm (NOVCA) that produces an optimal or near optimal vertex cover for any known undirected graph G (V, E). NOVCA constructs the vertex cover by repeatedly adding, at each step, all vertices adjacent to the vertex of minimal degree; in the case of a tie, it selects the one having the maximum sum of degrees of its neighbors. The results identifying bounds on the size of the minimum vertex cover as well as polynomial complexity of algorithm are given with experimental verification. Future research efforts will be directed at tuning the algorithm and providing proof for better approximation ratio with NOVCA compared to any other available vertex cover algorithms.

Transitive orientations in bull-reducible Berge graphs

A bull is a graph with five vertices r , y , x , z , s and five edges r y , y x , y z , x z , z s . A graph G is bull-reducible if every vertex of G lies in at most one bull of G . We prove that every bull-reducible Berge graph G that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color the vertices of such a graph exactly.

A fast polynomial time algorithm for logistics network flows

For a multi-tiered logistics network containing the manufacturers, distributors and consumers, in this paper we present a fast polynomial time algorithm to find product flow with the minimum congestion by shipping products from manufacturers to distributors and from distributors to consumers. Global convergence is delivered for the proposed algorithm. The values of congestion and the potential function can be successfully decreased by conducting the proposed algorithm in logistics networks. Numerical computations and comparisons with well-known linear programming package CPLEX on large-scale logistics networks are performed where significant savings in computations are gained from the proposed algorithm.

Hyperreconfigurable architectures and the partition into hypercontexts problem

Dynamically reconfigurable architectures or systems are able to reconfigure their function and/or structure to suit the changing needs of a computation during run time. The increasing flexibility of modern dynamically reconfigurable systems improves their adaptability to computational needs but also makes fast reconfiguration difficult because of the large amount of reconfiguration information which has to be transferred. However, even when a computation uses this flexibility it will not use it all the time. Therefore, we propose to make the potential for reconfiguration itself reconfigurable. Such architectures are called hyperreconfigurable. Different models of hyperreconfigurable architectures are proposed in this paper. We also study a fundamental problem that emerges on such architectures, namely, to determine for a given computation when and how the potential for reconfiguration should be changed during run time so that the reconfiguration overhead is minimal. It is shown that the general problem is NP-hard but fast polynomial time algorithms are given to solve this problem for special types of hyperreconfigurable architectures. We define two example hyperreconfigurable architectures and illustrate the introduced concepts for corresponding application problems.

On the number of multilinear partitions and the computing capacity of multiple-valued multiple-threshold perceptrons

We introduce the concept of multilinear partition of a point set V/spl sub/R/sup n/ and the concept of multilinear separability of a function f:Vtwo head right arrowK={0,...,k-1}. Based on well-known relationships between linear partitions and minimal pairs, we derive formulae for the number of multilinear partitions of a point set in general position and of the set K(2). The (n,k,s)-perceptrons partition the input space V into s+1 regions with s parallel hyperplanes. We obtain results on the capacity of a single (n,k,s)-perceptron, respectively, for V subset R(n) in general position and for V=K(2). Finally, we describe a fast polynomial-time algorithm for counting the multilinear partitions of K(2).

Algebraic factoring algorithm to recognise read-once functions

A fast polynomial-time algorithm was recently proposed to determine whether a logic function expressed as a unate DNF (disjunctive normal form) can be expressed as a read-once formula where each variable appears no more than once. The paper uses a combinatorial characterisation of read-once formulas to achieve its objective. It is shown that one can use the well-known and more intuitive notion of algebraic factoring to devise a recognition algorithm for read-once formulas of slightly worse complexity than the best known polynomial-time algorithm for this problem.

A New Semidefinite Programming Bound for Indefinite Quadratic Forms Over a Simplex

The paper describes a method for computing a lower bound of the global minimum of an indefinite quadratic form over a simplex. The bound is derived by computing an underestimator of the convex envelope by solving a semidefinite program (SDP). This results in a convex quadratic program (QP). It is shown that the optimal value of the QP is a lower bound of the optimal value of the original problem. Since there exist fast (polynomial time) algorithms for solving SDP‘s and QP‘s the bound can be computed in reasonable time. Numerical experiments indicate that the relative error of the bound is about 10 percent for problems up to 20 variables, which is much better than a known SDP bound.

Approximate Max k-Cut with subgraph guarantee

We study the following variant of the Max k-Cut problem. Given an input graph G with positively weighted edges and k colors, we wish to compute a subgraph H of G containing “lots” of heavy edges and a color assignment c : V → [ k] such that: 1. (a) all edges in H are properly colored and 2. (b) a “large fraction” of edges in G n0 ⊮ is properly colored. We give several definitions of “lots” and “large fraction” and give fast polynomial time algorithms to compute such color assignments. This problem is related to the frequency allocation problems for cellular telephone networks but could be useful in other scenarios too.

River Routing with a Generalized Model

Traditional restrictions on river routing confine the connecting wires to the channel between the terminal rows. In (J. Comput. System Sci.28(1984), 420–438) these restrictions were somewhat relaxed, thereby permitting a limited type of routing outside of the channel. Here, the constraints are further relaxed to allow for a new class of “generalized” river routings. We show that this new class of generalized river routings contains routings that are significantly more compact than those previously considered. In addition, we give a fast polynomial time algorithm for producing optimal routings in this new class. The run time of our algorithm is the best possible and is identical to the time required to produce optimal river routings under the traditional model.

Optimal folding of standard and custom cells

We study the problem of folding an ordered list of standard and custom cells into rows of a chip so as to minimize either the routing area or the total chip area. Nine versions of the folding problem are formulated and fast polynomial time algorithms are obtained for each. Two of our formulations correspond to problems formulated in Paik and Sahni [1993] for the folding of a stack of bit-slice components. Our algorithms for these two formulations are asymptotically superior to those of Paik and Sahni [1993].

A Linear Recognition Algorithm for Cographs

Cographs are the graphs formed from a single vertex under the closure of the operations of union and complement. Another characterization of cographs is that they are the undirected graphs with no induced paths on four vertices. Cographs arise naturally in such application areas as examination scheduling and automatic clustering of index terms. Furthermore, it is known that cographs have a unique tree representation called a cotree. Using the cotree it is possible to design very fast polynomial time algorithms for problems which are intractable for graphs in general. Such problems include chromatic number, clique determination, clustering, minimum weight domination, isomorphism, minimum fill-in and Hamiltonicity. In this paper we present a linear time algorithm for recognizing cographs and constructing their cotree representation.

Threshold Dimension of Graphs

This paper examines the problem of determining the threshold dimension of a graph. There exists numerous characterizations of threshold graphs, those graphs of threshold dimension one, as well as fast polynomial time algorithms to test if a graph is a threshold graph. Yannakakis [1982] proved that, in general, determining the threshold dimension is a hard problem by proving that for fixed $k\geqq 3$, determining if the threshold dimension of a graph is less than or equal to k is an NP-complete problem. In this paper we compute the threshold dimension of several classes of graphs and obtain upper and lower bounds on the threshold dimension of a graph. Counterexamples to a conjecture on threshold dimension are provided.

Preemptive Scheduling of Independent Jobs with Release and Due Times on Open, Flow and Job Shops

We study the problem of obtaining feasible preemptive schedules for independent jobs. It is assumed that each job has associated with it a release and due time. No job can begin before its release time. All jobs must be completed by their respective due times. It is shown that determining the existence of feasible preemptive schedules for two processor flow and job shops is NP-hard in the strong sense even when all jobs have the same due time. A linear programming formulation for the open shop problem is obtained. Also, a fast polynomial time algorithm is obtained for a restricted class of open shop problems.