The Minimal Logically-Defined NP-Complete Problem

AbstractWe consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

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Dealing with Hardness

The Golden Ticket ◽

10.23943/princeton/9780691175782.003.0006 ◽

2017 ◽

Author(s):

Lance Fortnow

Keyword(s):

Brute Force ◽

Complete Problem ◽

Hard Problems ◽

Np Complete ◽

Np Problems ◽

Single Technique

This chapter demonstrates several approaches for dealing with hard problems. These approaches include brute force, heuristics, and approximation. Typically, no single technique will suffice to handle the difficult NP problems one needs to solve. For moderate-sized problems one can search over all possible solutions with the very fast computers available today. One can use algorithms that might not work for every problem but do work for many of the problems one cares about. Other algorithms may not find the best possible solution but still a solution that's good enough. Other times one just cannot get a solution for an NP-complete problem. One has to try to solve a different problem or just give up.

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Quality of Service for Multimedia and Real-Time Services

International Journal of Information Technology and Web Engineering ◽

10.4018/jitwe.2010100101 ◽

2010 ◽

Vol 5 (4) ◽

pp. 1-22

Author(s):

F. W. Albalas ◽

B. A. Abu-Alhaija ◽

A. Awajan ◽

Khalid Al-Begain

Keyword(s):

Decision Making ◽

Quality Of Service ◽

Real Time ◽

Qos Routing ◽

Routing Algorithms ◽

Complete Problem ◽

Filtering Technique ◽

Np Complete ◽

Multiple Metrics

New web technologies have encouraged the deployment of various network applications that are rich with multimedia and real-time services. These services demand stringent requirements are defined through Quality of Service (QoS) parameters such as delay, jitter, loss, etc. To guarantee the delivery of these services QoS routing algorithms that deal with multiple metrics are needed. Unfortunately, QoS routing with multiple metrics is considered an NP-complete problem that cannot be solved by a simple algorithm. This paper proposes three source based QoS routing algorithms that find the optimal path from the service provider to the user that best satisfies the QoS requirements for a particular service. The three algorithms use the same filtering technique to prune all the paths that do not meet the requirements which solves the complexity of NP-complete problem. Next, each of the three algorithms integrates a different Multiple Criteria Decision Making method to select one of the paths that have resulted from the route filtering technique. The three decision making methods used are the Analytic Hierarchy Process (AHP), Multi-Attribute Utility Theory (MAUT), and Kepner-Tregoe KT. Results show that the algorithms find a path using multiple constraints with a high ability to handle multimedia and real-time applications.

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Towards Efficient Bounds on Completion Time and Resource Provisioning for Scheduling Workflows on Heterogeneous Processing Systems

International Journal of Grid and High Performance Computing ◽

10.4018/ijghpc.2017070105 ◽

2017 ◽

Vol 9 (3) ◽

pp. 60-82 ◽

Cited By ~ 2

Author(s):

D. Sirisha ◽

G. Vijayakumari

Keyword(s):

Completion Time ◽

High Performance ◽

Turnaround Time ◽

Resource Provisioning ◽

Optimal Solutions ◽

Complete Problem ◽

Heterogeneous Processing ◽

Heuristic Scheduling ◽

Np Complete ◽

Workflow Tasks

Compute intensive applications featured as workflows necessitate Heterogeneous Processing Systems (HPS) for attaining high performance to minimize the turnaround time. Efficient scheduling of the workflow tasks is paramount to attain higher potentials of HPS and is a challenging NP-Complete problem. In the present work, Branch and Bound (BnB) strategy is applied to optimally schedule the workflow tasks. The proposed bounds are tighter, simpler and less complex than the existing bounds and the upper bound is closer to the exact solution. Moreover, the bounds on the resource provisioning are devised to execute the workflows in the minimum possible time and optimally utilize the resources. The performance of the proposed BnB strategy is evaluated on a suite of benchmark workflows. The experimental results reveal that the proposed BnB strategy improved the optimal solutions compared to the existing heuristic scheduling algorithms for more than 20 percent of the cases and generated better schedules over 7 percent for 82.6 percent of the cases.

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Applying Neural Networks to Find the Minimum Cost Coverage of a Boolean Function

VLSI Design ◽

10.1155/1995/64953 ◽

1995 ◽

Vol 3 (1) ◽

pp. 13-19 ◽

Cited By ~ 1

Author(s):

Pong P. Chu

Keyword(s):

Neural Network ◽

Boolean Function ◽

Input Data ◽

Selection Process ◽

Optimal Solution ◽

Minimum Cost ◽

Hopfield Network ◽

Neural Network Approach ◽

Complete Problem ◽

Np Complete

To find a minimal expression of a boolean function includes a step to select the minimum cost cover from a set of implicants. Since the selection process is an NP-complete problem, to find an optimal solution is impractical for large input data size. Neural network approach is used to solve this problem. We first formalize the problem, and then define an “energy function” and map it to a modified Hopfield network, which will automatically search for the minima. Simulation of simple examples shows the proposed neural network can obtain good solutions most of the time.

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A scalable photonic computer solving the subset sum problem

Science Advances ◽

10.1126/sciadv.aay5853 ◽

2020 ◽

Vol 6 (5) ◽

pp. eaay5853 ◽

Cited By ~ 3

Author(s):

Xiao-Yun Xu ◽

Xuan-Lun Huang ◽

Zhan-Ming Li ◽

Jun Gao ◽

Zhi-Qiang Jiao ◽

...

Keyword(s):

Three Dimensional ◽

Propagation Speed ◽

Direct Writing ◽

Laser Direct Writing ◽

Time Consumption ◽

Subset Sum Problem ◽

Complete Problem ◽

Strong Robustness ◽

Subset Sum ◽

Np Complete

The subset sum problem (SSP) is a typical nondeterministic-polynomial-time (NP)–complete problem that is hard to solve efficiently in time with conventional computers. Photons have the unique features of high propagation speed, strong robustness, and low detectable energy level and therefore can be promising candidates to meet the challenge. Here, we present a scalable chip built-in photonic computer to efficiently solve the SSP. We map the problem into a three-dimensional waveguide network through a femtosecond laser direct writing technique. We show that the photons sufficiently dissipate into the networks and search for solutions in parallel. In the case of successive primes, our approach exhibits a dominant superiority in time consumption even compared with supercomputers. Our results confirm the ability of light to realize computations intractable for conventional computers, and suggest the SSP as a good benchmarking platform for the race between photonic and conventional computers on the way toward “photonic supremacy.”

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GRAPH BISECTION MODELED AS CARDINALITY CONSTRAINED BINARY QUADRATIC TASK ALLOCATION

International Journal of Information Technology & Decision Making ◽

10.1142/s0219622013500119 ◽

2013 ◽

Vol 12 (02) ◽

pp. 261-276 ◽

Cited By ~ 1

Author(s):

MARK LEWIS ◽

GARY KOCHENBERGER

Keyword(s):

Task Allocation ◽

Graph Partitioning ◽

State Of The Art ◽

Quadratic Model ◽

Complete Problem ◽

Binary Model ◽

Comparison Results ◽

Graph Bisection ◽

Strategic Oscillation ◽

Np Complete

In this paper, the cardinality constrained quadratic model for binary quadratic programming is used to model and solve the graph bisection problem as well as its generalization in the form of the task allocation problem with two processors (2-TAP). Balanced graph bisection is an NP-complete problem which partitions a set of nodes in the graph G = (N, E) into two sets with equal cardinality such that a minimal sum of edge weights exists between the nodes in the two separate sets. 2-TAP is graph bisection with the addition of node preference costs in the objective function. We transform the general linear k-TAP model to the cardinality constrained quadratic binary model so that it may be efficiently solved using tabu search with strategic oscillation. On a set of benchmark graph bisections, we improve the best known solution for several problems. Comparison results with the state-of-the-art graph partitioning program METIS, as well as Cplex and Gurobi are presented on a set of randomly generated graphs. This approach is shown to also work well with 2-TAP, comparing favorably to Cplex and Gurobi, providing better solutions in a much shorter time.

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