Lower and upper bounds for the linear arrangement problem on interval graphs

2018 ◽  
Vol 52 (4-5) ◽  
pp. 1123-1145
Author(s):  
Alain Quilliot ◽  
Djamal Rebaine ◽  
Hélène Toussaint
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Feasible Solution ◽  
Interval Graph ◽  
Upper Bounds ◽  
Unit Interval ◽  
Lower And Upper Bounds ◽  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

We deal here with theLinear Arrangement Problem(LAP) onintervalgraphs, any interval graph being given here together with its representation as theintersectiongraph of some collection of intervals, and so with relatedprecedenceandinclusionrelations. We first propose a lower boundLB, which happens to be tight in the case ofunit intervalgraphs. Next, we introduce the restriction PCLAP of LAP which is obtained by requiring any feasible solution of LAP to be consistent with theprecedencerelation, and prove that PCLAP can be solved in polynomial time. Finally, we show both theoretically and experimentally that PCLAP solutions are a good approximation for LAP onintervalgraphs.

scholarly journals Outerplanar Crossing Numbers, the Circular Arrangement Problem and Isoperimetric Functions

10.37236/1834 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Éva Czabarka ◽  
Ondrej Sýkora ◽  
László A. Székely ◽  
Imrich Vrťo
Keyword(s):  
Lower Bound ◽  
General Setting ◽  
Crossing Number ◽  
Linear Arrangement ◽  
Crossing Numbers ◽  
Arrangement Problem ◽  
Isoperimetric Functions ◽  
Circular Arrangement ◽  
Graph Families ◽  
Linear Arrangement Problem

We extend a lower bound due to Shahrokhi, Sýkora, Székely and Vrťo for the outerplanar crossing number (in other terminologies also called convex, circular and one-page book crossing number) to a more general setting. In this setting we can show a better lower bound for the outerplanar crossing number of hypercubes than the best lower bound for the planar crossing number. We exhibit further sequences of graphs, whose outerplanar crossing number exceeds by a factor of $\log n$ the planar crossing number of the graph. We study the circular arrangement problem, as a lower bound for the linear arrangement problem, in a general fashion. We obtain new lower bounds for the circular arrangement problem. All the results depend on establishing good isoperimetric functions for certain classes of graphs. For several graph families new near-tight isoperimetric functions are established.

scholarly journals Full-Duplex Relay with Delayed CSI Elevates the SDoF of the MIMO X Channel

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1484
Author(s):  
Tong Zhang ◽  
Gaojie Chen ◽  
Shuai Wang ◽  
Rui Wang
Keyword(s):  
Lower Bound ◽  
Channel State Information ◽  
Interference Alignment ◽  
Upper Bounds ◽  
Full Duplex ◽  
Multiple Antenna ◽  
Lower And Upper Bounds ◽  
Channel State ◽  
State Information ◽  
Full Duplex Relay

In this article, the sum secure degrees-of-freedom (SDoF) of the multiple-input multiple-output (MIMO) X channel with confidential messages (XCCM) and arbitrary antenna configurations is studied, where there is no channel state information (CSI) at two transmitters and only delayed CSI at a multiple-antenna, full-duplex, and decode-and-forward relay. We aim at establishing the sum-SDoF lower and upper bounds. For the sum-SDoF lower bound, we design three relay-aided transmission schemes, namely, the relay-aided jamming scheme, the relay-aided jamming and one-receiver interference alignment scheme, and the relay-aided jamming and two-receiver interference alignment scheme, each corresponding to one case of antenna configurations. Moreover, the security and decoding of each scheme are analyzed. The sum-SDoF upper bound is proposed by means of the existing SDoF region of two-user MIMO broadcast channel with confidential messages (BCCM) and delayed channel state information at the transmitter (CSIT). As a result, the sum-SDoF lower and upper bounds are derived, and the sum-SDoF is characterized when the relay has sufficiently large antennas. Furthermore, even assuming no CSI at two transmitters, our results show that a multiple-antenna full-duplex relay with delayed CSI can elevate the sum-SDoF of the MIMO XCCM. This is corroborated by the fact that the derived sum-SDoF lower bound can be greater than the sum-SDoF of the MIMO XCCM with output feedback and delayed CSIT.

scholarly journals Tractable Parameterizations for the Minimum Linear Arrangement Problem

2016 ◽  
Vol 8 (2) ◽  
pp. 1-12 ◽  
Author(s):  
Michael R. Fellows ◽  
Danny Hermelin ◽  
Frances Rosamond ◽  
Hadas Shachnai
Keyword(s):  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

scholarly journals On minimum cuts and the linear arrangement problem

2000 ◽  
Vol 103 (1-3) ◽  
pp. 127-139 ◽  
Author(s):  
S.B. Horton ◽  
R.Gary Parker ◽  
R.B. Borie
Keyword(s):  
Linear Arrangement ◽  
Minimum Cuts ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

On Bipartite Drawings and the Linear Arrangement Problem

2001 ◽  
Vol 30 (6) ◽  
pp. 1773-1789 ◽  
Author(s):  
Farhad Shahrokhi ◽  
Ondrej Sýkora ◽  
László A. Székely ◽  
Imrich Vrto
Keyword(s):  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

scholarly journals A Computational Perspective on Network Coding

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Qin Guo ◽  
Mingxing Luo ◽  
Lixiang Li ◽  
Yixian Yang
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Lower Bound ◽  
Network Coding ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Multicast Networks ◽  
Computational Perspective ◽  
Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

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