scholarly journals New Algorithms for Mixed Dominating Set

2021 ◽  
Vol vol. 23 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Louis Dublois ◽  
Michael Lampis ◽  
Vangelis Th. Paschos
Keyword(s):  
Lower Bound ◽  
Dominating Set ◽  
Exact Algorithm ◽  
Polynomial Space ◽  
Parameterized Algorithms ◽  
Exponential Time ◽  
Open Questions ◽  
Fpt Algorithm ◽  
Exponential Space ◽  
New Algorithms

A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parameterized by treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$ (improving the current best $O^*(6^{tw})$), as well as a lower bound showing that our algorithm cannot be improved under the Strong Exponential Time Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far overlooked observation on the structure of minimal solutions, we obtain branching algorithms which improve both the best known FPT algorithm for this problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to $O^*(1.912^n)$ and polynomial space. Comment: This paper has been accepted to IPEC 2020

scholarly journals Finding and Counting Permutations via CSPs

Algorithmica ◽  
2021 ◽  
Author(s):  
Benjamin Aram Berendsohn ◽  
László Kozma ◽  
Dániel Marx
Keyword(s):  
Constraint Satisfaction ◽  
Polynomial Space ◽  
Permutation Patterns ◽  
Counting Problem ◽  
Exponential Time ◽  
Running Time ◽  
Algorithmic Question ◽  
Stack Sorting ◽  
Exponential Time Algorithms ◽  
New Algorithms

AbstractPermutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k. In this work we give two new algorithms for this well-studied problem, one whose running time is $$n^{k/4 + o(k)}$$ n k / 4 + o ( k ) , and a polynomial-space algorithm whose running time is the better of $$O(1.6181^n)$$ O ( 1 . 6181 n ) and $$O(n^{k/2 + 1})$$ O ( n k / 2 + 1 ) . These results improve the earlier best bounds of $$n^{0.47k + o(k)}$$ n 0.47 k + o ( k ) and $$O(1.79^n)$$ O ( 1 . 79 n ) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when $$k \in \varOmega (\log {n})$$ k ∈ Ω ( log n ) . We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time $$f(k) \cdot n^{o(k/\log {k})}$$ f ( k ) · n o ( k / log k ) would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing (4321-avoiding) and 3-decreasing (1234-avoiding) permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that a sub-exponential running time is unlikely with the current techniques, even for patterns from these restricted classes.

Parameterized Algorithms for Total p-Dominating Set

2014 ◽  
Vol 36 (9) ◽  
pp. 1868-1879
Author(s):  
Wei-Zhong LUO ◽  
Qi-Long FENG ◽  
Jian-Xin WANG ◽  
Jian-Er CHEN
Keyword(s):  
Dominating Set ◽  
Parameterized Algorithms ◽  
Total P

Recent Advances in Lower Bound Shakedown Analysis

2009 ◽  
Author(s):  
Dieter Weichert ◽  
Abdelkader Hachemi
Keyword(s):  
Finite Element ◽  
Lower Bound ◽  
Process Engineering ◽  
Operating Conditions ◽  
Structural Elements ◽  
Shakedown Analysis ◽  
Material Models ◽  
Practical Engineering ◽  
Safe Operating ◽  
New Algorithms

The special interest in lower bound shakedown analysis is that it provides, at least in principle, safe operating conditions for sensitive structures or structural elements under fluctuating thermo-mechanical loading as to be found in power- and process engineering. In this paper achievements obtained over the last years to introduce more sophisticated material models into the framework of shakedown analysis are developed. Also new algorithms will be presented that allow using the addressed numerical methods as post-processor for commercial finite element codes. Examples from practical engineering will illustrate the potential of the methodology.

k-Efficient domination: Algorithmic perspective

2022 ◽  
Author(s):  
Mohsen Alambardar Meybodi
Keyword(s):  
Decision Problem ◽  
Dominating Set ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Sparse Graphs ◽  
Fpt Algorithm ◽  
Domination Problem ◽  
Efficient Dominating Set ◽  
Np Complete

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

scholarly journals Finding Points in General Position

2017 ◽  
Vol 27 (04) ◽  
pp. 277-296 ◽  
Author(s):  
Vincent Froese ◽  
Iyad Kanj ◽  
André Nichterlein ◽  
Rolf Niedermeier
Keyword(s):  
Lower Bound ◽  
General Position ◽  
Subset Selection ◽  
Selection Problem ◽  
Fixed Parameter Tractability ◽  
Maximum Cardinality ◽  
Exponential Time ◽  
Running Time ◽  
Fixed Parameter ◽  
Set Of Points

We study the General Position Subset Selection problem: Given a set of points in the plane, find a maximum-cardinality subset of points in general position. We prove that General Position Subset Selection is NP-hard, APX-hard, and present several fixed-parameter tractability results for the problem as well as a subexponential running time lower bound based on the Exponential Time Hypothesis.

scholarly journals Optimal Bounds for the k -cut Problem

2022 ◽  
Vol 69 (1) ◽  
pp. 1-18
Author(s):  
Anupam Gupta ◽  
David G. Harris ◽  
Euiwoong Lee ◽  
Jason Li
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Connected Components ◽  
Small Integer ◽  
Main Ingredient ◽  
Fine Grained ◽  
Optimal Bounds ◽  
Sunflower Lemma ◽  
General Graphs ◽  
New Algorithms

In the k -cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger and Stein can solve this in roughly O ( n 2k ) time. However, lower bounds from conjectures about the k -clique problem imply that Ω ( n (1- o (1)) k ) time is likely needed. Recent results of Gupta, Lee, and Li have given new algorithms for general k -cut in n 1.98k + O(1) time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show that the Contraction Algorithm of Karger outputs any fixed k -cut of weight α λ k with probability Ω k ( n - α k ), where λ k denotes the minimum k -cut weight. This also gives an extremal bound of O k ( n k ) on the number of minimum k -cuts and an algorithm to compute λ k with roughly n k polylog( n ) runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight k -clique. The first main ingredient in our result is an extremal bound on the number of cuts of weight less than 2 λ k / k , using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—in the Karger process.

Cluster-Based Online Routing Protocols for Ad Hoc Network

2014 ◽  
Vol 9 (4) ◽  
pp. 54-66
Author(s):  
Alaa E. Abdallah ◽  
Mohammad Bsoul ◽  
Emad E. Abdallah ◽  
Ibrahim Al–Oqily ◽  
George Kao
Keyword(s):  
Ad Hoc ◽  
Dominating Set ◽  
Routing Algorithm ◽  
Routing Algorithms ◽  
Greedy Routing ◽  
Mobile Nodes ◽  
3D Space ◽  
Geographical Position ◽  
2D Space ◽  
New Algorithms

In geographical routing algorithms, mobile nodes rely on geographical position to make routing judgments. Researchers frequently discuss such routing algorithms in (2D) space. However, in reality, mobile nodes spread in (3D) space. In this paper the authors present four new 3D geographical-based routing algorithms Cylinder, Greedy-Cylinder, Cluster-Cylinder, and Greedy-cluster-Cylinder. In Cylinder routing, the nodes are locally projected on the inner surface of a cylinder, perimeter routing is executed after that. Greedy-Cylinder starts with Greedy routing algorithm until a local minimum is reached. The algorithm then switches to Cylinder routing. Cluster-Cylinder elects a dominating set for all nodes and then uses this set for projection and routing. The fourth algorithm Greedy-cluster-Cylinder is a combination between Greedy-Cylinder and Cluster-Cylinder. The authors evaluate their new algorithms and compare them with many classical known algorithms. The simulation outcomes show the substantial enhancement in delivery rate over other algorithms.

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