Approximation Algorithm for the Broadcast Time in k-Path Graph

2019 ◽  
Vol 19 (04) ◽  
pp. 1950006
Author(s):  
PUSPAL BHABAK ◽  
HOVHANNES A. HARUTYUNYAN
Keyword(s):  
Approximation Algorithm ◽  
Information Dissemination ◽  
Arbitrary Graph ◽  
Worst Case ◽  
Large Classes ◽  
Path Graph ◽  
Broadcast Time ◽  
Np Complete ◽  
Fully Connected ◽  
Communication Lines

Broadcasting is an information dissemination problem in a connected network in which one node, called the originator, must distribute a message to all other nodes by placing a series of calls along the communication lines of the network. In every unit of time, the informed nodes aid the originator in distributing the message. Finding the broadcast time of any vertex in an arbitrary graph is NP-complete. The polynomial time solvability is shown only for certain graphs like trees, unicyclic graphs, tree of cycles, necklace graphs, fully connected trees and tree of cliques. In this paper we study the broadcast problem in k-path graphs. For any originator of the k-path graph we present a (4 – ϵ)-approximation algorithm in the worst case. The algorithm gives a better approximation ratio for some large classes of k-path graphs. Moreover, our algorithm generates the optimal broadcast time for some cases.

scholarly journals A note on compact and compact circular edge-colorings of graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Dariusz Dereniowski ◽  
Adam Nadolski
Keyword(s):  
Approximation Algorithm ◽  
Decision Problem ◽  
Edge Coloring ◽  
Exact Algorithm ◽  
Weighted Graphs ◽  
Edge Colorings ◽  
Odd Cycle ◽  
International Audience ◽  
Np Complete ◽  
Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

scholarly journals On Rectilinear Distance-Preserving Trees

VLSI Design ◽  
1998 ◽  
Vol 7 (1) ◽  
pp. 15-30
Author(s):  
Gustavo E. Téllez ◽  
Majid Sarrafzadeh
Keyword(s):  
Approximation Algorithm ◽  
Heuristic Algorithm ◽  
Time Complexity ◽  
High Performance ◽  
Exact Algorithm ◽  
Wire Length ◽  
Worst Case ◽  
Rectilinear Distance ◽  
Source To Sink ◽  
Linear Delay

Given a set of terminals on the plane N={s,ν1,…,νn}, with a source terminal s, a Rectilinear Distance-Preserving Tree (RDPT) T(V, E) is defined as a tree rooted at s, connecting all terminals in N. An RDPT has the property that the length of every source to sink path is equal to the rectilinear distance between that source and sink. A Min- Cost Rectilinear Distance-Preserving Tree (MRDPT) minimizes the total wire length while maintaining minimal source to sink linear delay, making it suitable for high performance interconnect applications.This paper studies problems in the construction of RDPTs, including the following contributions. A new exact algorithm for a restricted version of the problem in one quadrant with O(n2) time complexity is proposed. A novel heuristic algorithm, which uses optimally solvable sub-problems, is proposed for the problem in a single quadrant. The average and worst-case time complexity for the proposed heuristic algorithm are O(n3/2) and O(n3), respectively. A 2-approximation of the quadrant merging problem is proposed. The proposed algorithm has time complexity O(α2T(n)+α3) for any constant α > 1, where T(n) is the time complexity of the solution of the RDPT problem on one quadrant. This result improves over the best previous quadrant merging solution which has O(n2T(n)+n3) time complexity.We test our algorithms on randomly uniform point sets and compare our heuristic RDPT construction against a Minimum Cost Rectilinear Steiner (MRST) tree approximation algorithm. Our results show that RDPTs are competitive with Steiner trees in total wire-length when the number of terminals is less than 32. This result makes RDPTs suitable for VLSI routing applications. We also compare our algorithm to the Rao-Shor RDPT approximation algorithm obtaining improvements of up to 10% in total wirelength. These comparisons show that the algorithms proposed herein produce promising results.

CONNECTED LIAR'S DOMINATION IN GRAPHS: COMPLEXITY AND ALGORITHMS

2013 ◽  
Vol 05 (04) ◽  
pp. 1350024 ◽  
Author(s):  
B. S. PANDA ◽  
S. PAUL
Keyword(s):  
Dominating Set ◽  
Chordal Graphs ◽  
Hardness Of Approximation ◽  
Induced Subgraph ◽  
Path Graph ◽  
Block Graphs ◽  
Domination Problem ◽  
Np Complete ◽  
Domination In Graphs ◽  
General Graphs

A subset L ⊆ V of a graph G = (V, E) is called a connected liar's dominating set of G if (i) for all v ∈ V, |NG[v] ∩ L| ≥ 2, (ii) for every pair u, v ∈ V of distinct vertices, |(NG[u]∪NG[v])∩L| ≥ 3, and (iii) the induced subgraph of G on L is connected. In this paper, we initiate the algorithmic study of minimum connected liar's domination problem by showing that the corresponding decision version of the problem is NP-complete for general graph. Next we study this problem in subclasses of chordal graphs where we strengthen the NP-completeness of this problem for undirected path graph and prove that this problem is linearly solvable for block graphs. Finally, we propose an approximation algorithm for minimum connected liar's domination problem and investigate its hardness of approximation in general graphs.

scholarly journals COMPUTING SIGNED PERMUTATIONS OF POLYGONS

2011 ◽  
Vol 21 (01) ◽  
pp. 87-100
Author(s):  
GREG ALOUPIS ◽  
PROSENJIT BOSE ◽  
ERIK D. DEMAINE ◽  
STEFAN LANGERMAN ◽  
HENK MEIJER ◽  
...  
Keyword(s):  
Mirror Symmetry ◽  
Time Algorithm ◽  
Worst Case ◽  
Signed Permutations ◽  
Np Complete ◽  
Edge List ◽  
The Given ◽  
Special Case ◽  
Planar Polygon

Given a planar polygon (or chain) with a list of edges {e1, e2, e3, …, en-1, en}, we examine the effect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edge-swaps which are a special case and involve interchanging two consecutive edges. When each edge of the given polygon has also been assigned a direction we say that the polygon is signed. In this case any edge involved in a reversal changes direction. We show that a star-shaped polygon can be convexified using O(n2) edge-swaps, while maintaining simplicity, and that this is tight in the worst case. We show that determining whether a signed polygon P can be transformed to one that has rotational or mirror symmetry with P, using transpositions, takes Θ(n log n) time. We prove that the problem of deciding whether transpositions can modify a polygon to fit inside a rectangle is weakly NP-complete. Finally we give an O(n log n) time algorithm to compute the maximum endpoint distance for an oriented chain.

scholarly journals Chamberlin--Courant Rule with Approval Ballots: Approximating the MaxCover Problem with Bounded Frequencies in FPT Time

2017 ◽  
Vol 60 ◽  
pp. 687-716 ◽  
Author(s):  
Piotr Skowron ◽  
Piotr Faliszewski
Keyword(s):  
Approximation Algorithm ◽  
Exact Algorithms ◽  
The Other ◽  
Winner Determination ◽  
Exponential Time ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Voting Rule ◽  
Np Complete ◽  
Better Than

We consider the problem of winner determination under Chamberlin--Courant's multiwinner voting rule with approval utilities. This problem is equivalent to the well-known NP-complete MaxCover problem and, so, the best polynomial-time approximation algorithm for it has approximation ratio 1 - 1/e. We show exponential-time/FPT approximation algorithms that, on one hand, achieve arbitrarily good approximation ratios and, on the other hand, have running times much better than known exact algorithms. We focus on the cases where the voters have to approve of at most/at least a given number of candidates.

scholarly journals Some Things are Easier for the Dumb and the Bright Ones (Beware the Average!)

2019 ◽  
Author(s):  
Wojciech Jamroga ◽  
Michał Knapik
Keyword(s):  
Model Checking ◽  
Multi Agent Systems ◽  
Worst Case ◽  
Case Complexity ◽  
Special Cases ◽  
Worst Case Complexity ◽  
Multi Agent ◽  
Np Complete ◽  
Epistemic Structure ◽  
Verification Of Models

Model checking strategic abilities in multi-agent systems is hard, especially for agents with partial observability of the state of the system. In that case, it ranges from NP-complete to undecidable, depending on the precise syntax and the semantic variant. That, however, is the worst case complexity, and the problem might as well be easier when restricted to particular subclasses of inputs. In this paper, we look at the verification of models with "extreme" epistemic structure, and identify several special cases for which model checking is easier than in general. We also prove that, in the other cases, no gain is possible even if the agents have almost full (or almost nil) observability. To prove the latter kind of results, we develop generic techniques that may be useful also outside of this study.

scholarly journals Vertex-Partitioning into Fixed Additive Induced-Hereditary Properties is NP-hard

10.37236/1799 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Alastair Farrugia
Keyword(s):  
Planar Graph ◽  
Line Graph ◽  
Perfect Graph ◽  
Np Hard ◽  
Arbitrary Graph ◽  
Np Completeness ◽  
Graph Properties ◽  
Special Cases ◽  
Np Complete ◽  
Hereditary Properties

Can the vertices of an arbitrary graph $G$ be partitioned into $A \cup B$, so that $G[A]$ is a line-graph and $G[B]$ is a forest? Can $G$ be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are special cases of our result: if ${\cal P}$ and ${\cal Q}$ are additive induced-hereditary graph properties, then $({\cal P}, {\cal Q})$-colouring is NP-hard, with the sole exception of graph $2$-colouring (the case where both ${\cal P}$ and ${\cal Q}$ are the set ${\cal O}$ of finite edgeless graphs). Moreover, $({\cal P}, {\cal Q})$-colouring is NP-complete iff ${\cal P}$- and ${\cal Q}$-recognition are both in NP. This completes the proof of a conjecture of Kratochvíl and Schiermeyer, various authors having already settled many sub-cases.

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