scholarly journals The Complexity of Pattern Matching for $321$-Avoiding and Skew-Merged Permutations

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Michael H. Albert ◽  
Marie-Louise Lackner ◽  
Martin Lackner ◽  
Vincent Vatter
Keyword(s):  
Pattern Matching ◽  
Polynomial Time ◽  
Matching Problem ◽  
Polynomial Time Algorithms ◽  
Permutation Pattern ◽  
Special Cases ◽  
Np Complete

The Permutation Pattern Matching problem, asking whether a pattern permutation $\pi$ is contained in a permutation $\tau$, is known to be NP-complete. In this paper we present two polynomial time algorithms for special cases. The first algorithm is applicable if both $\pi$ and $\tau$ are $321$-avoiding; the second is applicable if $\pi$ and $\tau$ are skew-merged. Both algorithms have a runtime of $O(kn)$, where $k$ is the length of $\pi$ and $n$ the length of $\tau$.

scholarly journals The Complexity Landscape of Resource-Constrained Scheduling

2020 ◽  
Author(s):  
Robert Ganian ◽  
Thekla Hamm ◽  
Guillaume Mescoff
Keyword(s):  
Artificial Intelligence ◽  
Polynomial Time ◽  
Project Scheduling ◽  
Scheduling Problem ◽  
Comprehensive Understanding ◽  
Resource Constrained ◽  
Special Cases ◽  
Project Scheduling Problem ◽  
Np Complete ◽  
New Algorithms

The Resource-Constrained Project Scheduling Problem (RCPSP) and its extension via activity modes (MRCPSP) are well-established scheduling frameworks that have found numerous applications in a broad range of settings related to artificial intelligence. Unsurprisingly, the problem of finding a suitable schedule in these frameworks is known to be NP-complete; however, aside from a few results for special cases, we have lacked an in-depth and comprehensive understanding of the complexity of the problems from the viewpoint of natural restrictions of the considered instances. In the first part of our paper, we develop new algorithms and give hardness-proofs in order to obtain a detailed complexity map of (M)RCPSP that settles the complexity of all 1024 considered variants of the problem defined in terms of explicit restrictions of natural parameters of instances. In the second part, we turn to implicit structural restrictions defined in terms of the complexity of interactions between individual activities. In particular, we show that if the treewidth of a graph which captures such interactions is bounded by a constant, then we can solve MRCPSP in polynomial time.

scholarly journals Internet shopping optimization problem

2010 ◽  
Vol 20 (2) ◽  
pp. 385-390 ◽  
Author(s):  
JACEK B£A ZÿEWICZ ◽  
Mikhail Kovalyov ◽  
Jędrzej Musiał ◽  
Andrzej Urbanski ◽  
Adam Wojciechowski
Keyword(s):  
Polynomial Time ◽  
Optimization Problem ◽  
Partial Solution ◽  
The Internet ◽  
Internet Shopping ◽  
Single Product ◽  
Polynomial Time Algorithms ◽  
Special Cases ◽  
Multiple Item ◽  
Np Hardness

Internet shopping optimization problemA high number of Internet shops makes it difficult for a customer to review manually all the available offers and select optimal outlets for shopping. A partial solution to the problem is brought by price comparators which produce price rankings from collected offers. However, their possibilities are limited to a comparison of offers for a single product requested by the customer. The issue we investigate in this paper is a multiple-item multiple-shop optimization problem, in which total expenses of a customer to buy a given set of items should be minimized over all available offers. In this paper, the Internet Shopping Optimization Problem (ISOP) is defined in a formal way and a proof of its strong NP-hardness is provided. We also describe polynomial time algorithms for special cases of the problem.

Parallel Algorithms for the Boxed-Mesh Permutation Pattern Matching Problem

2019 ◽  
Vol 46 (4) ◽  
pp. 299-307
Author(s):  
Jihyo Choi ◽  
Youngho Kim ◽  
Joong Chae Na ◽  
Jeong Seop Sim
Keyword(s):  
Parallel Algorithms ◽  
Pattern Matching ◽  
Matching Problem ◽  
Permutation Pattern

Parallel-Machine Scheduling to Minimize Flowtime, Holding, and Batch Delivery Costs

2014 ◽  
Vol 31 (06) ◽  
pp. 1450044 ◽  
Author(s):  
Yunqiang Yin ◽  
Shuenn-Ren Cheng ◽  
Chin-Chia Wu
Keyword(s):  
Polynomial Time ◽  
Parallel Machines ◽  
Parallel Machine Scheduling ◽  
Optimal Number ◽  
Batch Size ◽  
Batch Delivery ◽  
Delivery Costs ◽  
Polynomial Time Algorithms ◽  
Special Cases ◽  
Delivery Scheduling

This paper considers a batch delivery scheduling problem in which n independent and simultaneously available jobs are to be processed on m unrelated or uniform parallel machines. The jobs scheduled on the same machine are delivered in batches to customers and the delivery date of a batch equals the completion time of the last job in the batch. The number of jobs in each delivery batch is constrained by the batch size, and the cost of delivering a batch depends not only on the number of jobs in the batch but also on the machine on which the batch is processed. The objective is to find jointly the optimal number of batches on each machine, the optimal assignment of jobs to the batches, and the optimal job processing sequence to minimize the sum of total flowtime, total holding time, and delivery costs. When the number of batches has a fixed upper bound, we present polynomial-time algorithms to solve the problems with unrelated and uniform parallel machines. If the bound constraint on the number of batches is relaxed, we provide polynomial-time algorithms to solve two special cases of the problem with uniform parallel machines.

scholarly journals Polynomial Time Algorithms for Variants of Graph Matching on Partial k-Trees

2016 ◽  
Vol 41 (3) ◽  
pp. 163-181
Author(s):  
Takayuki Nagoya
Keyword(s):  
Polynomial Time ◽  
Graph Matching ◽  
Graph Isomorphism ◽  
Exact Algorithms ◽  
Polynomial Time Algorithms ◽  
Np Complete

AbstractIn this paper, we deal with two variants of graph matching, the graph isomorphism with restriction and the prefix set of graph isomorphism. The former problem is known to be NP-complete, whereas the latter problem is known to be GI-complete. We propose polynomial time exact algorithms for these problems on partial k-trees.

scholarly journals Permutation Pattern matching in (213, 231)-avoiding permutations

2017 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Both Neou ◽  
Romeo Rizzi ◽  
Stéphane Vialette
Keyword(s):  
Pattern Matching ◽  
Related Problem ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Matching Problem ◽  
Permutation Pattern ◽  
Special Case ◽  
Longest Subsequence

Given permutations σ of size k and π of size n with k < n, the permutation pattern matching problem is to decide whether σ occurs in π as an order-isomorphic subsequence. We give a linear-time algorithm in case both π and σ avoid the two size-3 permutations 213 and 231. For the special case where only σ avoids 213 and 231, we present a O(max(kn 2 , n 2 log log n)-time algorithm. We extend our research to bivincular patterns that avoid 213 and 231 and present a O(kn 4)-time algorithm. Finally we look at the related problem of the longest subsequence which avoids 213 and 231.

scholarly journals 3-Colorability of Pseudo-Triangulations

2015 ◽  
Vol 25 (04) ◽  
pp. 283-298
Author(s):  
Oswin Aichholzer ◽  
Franz Aurenhammer ◽  
Thomas Hackl ◽  
Clemens Huemer ◽  
Alexander Pilz ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Plane Graphs ◽  
Np Completeness ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Complexity Results ◽  
The Family ◽  
General Plane ◽  
Np Complete

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.

scholarly journals Some algorithmic results for finding compatible spanning circuits in edge-colored graphs

2020 ◽  
Vol 40 (4) ◽  
pp. 1008-1019
Author(s):  
Zhiwei Guo ◽  
Hajo Broersma ◽  
Ruonan Li ◽  
Shenggui Zhang
Keyword(s):  
Polynomial Time ◽  
Sufficient Conditions ◽  
Complete Graphs ◽  
Hamilton Cycles ◽  
Colored Graph ◽  
Colored Graphs ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Np Complete ◽  
Euler Tours

Abstract A compatible spanning circuit in a (not necessarily properly) edge-colored graph G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. Sufficient conditions for the existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and Euler tours), and polynomial-time algorithms for finding compatible Euler tours have been considered in previous literature. More recently, sufficient conditions for the existence of more general compatible spanning circuits in specific edge-colored graphs have been established. In this paper, we consider the existence of (more general) compatible spanning circuits from an algorithmic perspective. We first show that determining whether an edge-colored connected graph contains a compatible spanning circuit is an NP-complete problem. Next, we describe two polynomial-time algorithms for finding compatible spanning circuits in edge-colored complete graphs. These results in some sense give partial support to a conjecture on the existence of compatible Hamilton cycles in edge-colored complete graphs due to Bollobás and Erdős from the 1970s.

EDGE-DELETION GRAPH PROBLEMS WITH FIRST-ORDER EXPRESSIBLE SUBGRAPH PROPERTIES

1991 ◽  
Vol 02 (02) ◽  
pp. 83-99
Author(s):  
V. ARVIND ◽  
S. BISWAS
Keyword(s):  
Normal Form ◽  
Polynomial Time ◽  
Edge Deletion ◽  
Present Evidence ◽  
First Order ◽  
Graph Problems ◽  
Polynomial Time Algorithms ◽  
Complete Problems ◽  
Np Complete ◽  
New Perspective

In this paper edge-deletion problems are studied with a new perspective. In general an edge-deletion problem is of the form: Given a graph G, does it have a subgraph H obtained by deleting zero or more edges such that H satisfies a polynomial-time verifiable property? This paper restricts attention to first-order expressible properties. If the property is expressed by π, which in prenex normal form is Q(Φ) where Q is the quantifier-prefix, then we prove results on the quantifier structure that characterize the complexity of the edge-deletion problem. In particular we give polynomial-time algorithms for problems for which Q is ‘simple’ and in other cases we encode certain NP-complete problems as edge-deletion problems, essentially using the quantifier structure of π. We also present evidence that Q alone cannot capture the complexity of the edge-deletion problem.

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