scholarly journals The knapsack problem with special neighbor constraints

2021 ◽  
Author(s):  
Steffen Goebbels ◽  
Frank Gurski ◽  
Dominique Komander
Keyword(s):  
Knapsack Problem ◽  
Directed Graphs ◽  
Upper Bounds ◽  
Knapsack Problems ◽  
Vertex Set ◽  
Hard Problems ◽  
Graph Constraints ◽  
The One ◽  
Additional Constraints ◽  
Np Hard Problems

AbstractThe knapsack problem is one of the simplest and most fundamental NP-hard problems in combinatorial optimization. We consider two knapsack problems which contain additional constraints in the form of directed graphs whose vertex set corresponds to the item set. In the one-neighbor knapsack problem, an item can be chosen only if at least one of its neighbors is chosen. In the all-neighbors knapsack problem, an item can be chosen only if all its neighbors are chosen. For both problems, we consider uniform and general profits and weights. We prove upper bounds for the time complexity of these problems when restricting the graph constraints to special sets of digraphs. We discuss directed co-graphs, minimal series-parallel digraphs, and directed trees.

scholarly journals Extended Formulation for CSP that is Compact for Instances of Bounded Treewidth

10.37236/5474 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Petr Kolman ◽  
Martin Koutecký
Keyword(s):  
Constraint Satisfaction ◽  
Upper Bounds ◽  
Constraint Satisfaction Problems ◽  
Np Hard ◽  
Extended Formulation ◽  
Bounded Treewidth ◽  
Extension Complexity ◽  
Constraint Graph ◽  
Hard Problems ◽  
Np Hard Problems

In this paper we provide an extended formulation for the class of constraint satisfaction problems and prove that its size is polynomial for instances whose constraint graph has bounded treewidth. This implies new upper bounds on extension complexity of several important NP-hard problems on graphs of bounded treewidth.

scholarly journals Meta-Heuristics Approach to Knapsack Problem in Memory Management

2019 ◽  
pp. 1-10 ◽  
Author(s):  
Emmanuel Ofori Oppong ◽  
Stephen Opoku Oppong ◽  
Dominic Asamoah ◽  
Nuku Atta Kordzo Abiew
Keyword(s):  
Genetic Algorithm ◽  
Simulated Annealing ◽  
Knapsack Problem ◽  
Memory Management ◽  
Knapsack Problems ◽  
Computer Memory ◽  
Integer Programs ◽  
Hard Problems ◽  
With Memory ◽  
Better Than

The Knapsack Problems are among the simplest integer programs which are NP-hard. Problems in this class are typically concerned with selecting from a set of given items, each with a specified weight and value, a subset of items whose weight sum does not exceed a prescribed capacity and whose value is maximum. The classical 0-1 Knapsack Problem arises when there is one knapsack and one item of each type. This paper considers the application of classical 0-1 knapsack problem with a single constraint to computer memory management. The goal is to achieve higher efficiency with memory management in computer systems. This study focuses on using simulated annealing and genetic algorithm for the solution of knapsack problems in optimizing computer memory. It is shown that Simulated Annealing performs better than the Genetic Algorithm for large number of processes. 

scholarly journals On the Packing Partitioning Problem on Directed Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3148
Author(s):  
Babak Samadi ◽  
Ismael G. Yero
Keyword(s):  
Directed Graphs ◽  
Complete Solution ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Vertex Coloring ◽  
Lower And Upper Bounds ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Partitioning Problem

This work is aimed to continue studying the packing sets of digraphs via the perspective of partitioning the vertex set of a digraph into packing sets (which can be interpreted as a type of vertex coloring of digraphs) and focused on finding the minimum cardinality among all packing partitions for a given digraph D, called the packing partition number of D. Some lower and upper bounds on this parameter are proven, and their exact values for directed trees are given in this paper. In the case of directed trees, the proof results in a polynomial-time algorithm for finding a packing partition of minimum cardinality. We also consider this parameter in digraph products. In particular, a complete solution to this case is presented when dealing with the rooted products.

scholarly journals C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

Algorithmica ◽  
2021 ◽  
Author(s):  
Giordano Da Lozzo ◽  
David Eppstein ◽  
Michael T. Goodrich ◽  
Siddharth Gupta
Keyword(s):  
Dual Graph ◽  
Graph Visualization ◽  
Closed Disk ◽  
Running Time ◽  
Planar Embedding ◽  
Bounded Treewidth ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
The One ◽  
Clustered Planarity

AbstractFor a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time $$O(r(n^2) + n^2)$$ O ( r ( n 2 ) + n 2 ) for general, possibly non-flat, instances.

scholarly journals More about solar g modes

2018 ◽  
Vol 612 ◽  
pp. L1 ◽  
Author(s):  
E. Fossat ◽  
F. X. Schmider
Keyword(s):  
Cross Correlation ◽  
Previous Analysis ◽  
Model Parameters ◽  
Real Spectrum ◽  
Space Observatory ◽  
Correlation Peak ◽  
Similar Frequency ◽  
Precise Estimation ◽  
The One ◽  
Additional Constraints

Context. The detection of asymptotic solar g-mode parameters was the main goal of the GOLF instrument onboard the SOHO space observatory. This detection has recently been reported and has identified a rapid mean rotation of the solar core, with a one-week period, nearly four times faster than all the rest of the solar body, from the surface to the bottom of the radiative zone. Aim. We present here the detection of more g modes of higher degree, and a more precise estimation of all their parameters, which will have to be exploited as additional constraints in modeling the solar core. Methods. Having identified the period equidistance and the splitting of a large number of asymptotic g modes of degrees 1 and 2, we test a model of frequencies of these modes by a cross-correlation with the power spectrum from which they have been detected. It shows a high correlation peak at lag zero, showing that the model is hidden but present in the real spectrum. The model parameters can then be adjusted to optimize the position (at exactly zero lag) and the height of this correlation peak. The same method is then extended to the search for modes of degrees 3 and 4, which were not detected in the previous analysis.Results. g-mode parameters are optimally measured in similar-frequency bandwidths, ranging from 7 to 8 μHz at one end and all close to 30 μHz at the other end, for the degrees 1 to 4. They include the four asymptotic period equidistances, the slight departure from equidistance of the detected periods for l = 1 and l = 2, the measured amplitudes, functions of the degree and the tesseral order, and the splittings that will possibly constrain the estimated sharpness of the transition between the one-week mean rotation of the core and the almost four-week rotation of the radiative envelope. The g-mode periods themselves are crucial inputs in the solar core structure helioseismic investigation.

NP vs QMA_\log(2)

2010 ◽  
Vol 10 (1&2) ◽  
pp. 141-151
Author(s):  
S. Beigi
Keyword(s):  
Quantum Algorithms ◽  
Np Hard ◽  
Complete Problems ◽  
Hard Problems ◽  
Np Complete ◽  
Np Hard Problems

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve NP-complete problems given a "short" quantum proof; more precisely, NP\subseteq QMA_{\log}(2) where QMA_{\log}(2) denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion NP\subseteq QMA_{\log}(2) has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap 1/24n^6. Moreover, Aaronson et al. have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if QMA_{\log}(2) with a constant gap contains NP. In this paper, we show that 3-SAT admits a QMA_{\log}(2) protocol with the gap 1/n^{3+\epsilon}} for every constant \epsilon>0.

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