Solving Large Instances of the RSA Problem in Flexgrid Elastic Optical Networks

Mirosław Klinkowski; Mateusz Żotkiewicz; Krzysztof Walkowiak; Michał Pióro; Marc Ruiz; Luis Velasco

doi:10.1364/JOCN.8.000320

Journal of Optical Communications and Networking
Vol. 8,
Issue 5,
pp. 320-330
(2016)
•https://doi.org/10.1364/JOCN.8.000320

Solving Large Instances of the RSA Problem in Flexgrid Elastic Optical Networks

Mirosław Klinkowski, Mateusz Żotkiewicz, Krzysztof Walkowiak, Michał Pióro, Marc Ruiz, and Luis Velasco

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Mirosław Klinkowski, Mateusz Żotkiewicz, Krzysztof Walkowiak, Michał Pióro, Marc Ruiz, and Luis Velasco, "Solving Large Instances of the RSA Problem in Flexgrid Elastic Optical Networks," J. Opt. Commun. Netw. 8, 320-330 (2016)

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Abstract

We present an optimization procedure that mixes advanced large-scale optimization methods and heuristics to solve large instances (with over 1.7 million integer variables) of the routing and spectrum allocation (RSA) problem—a basic optimization problem in flexgrid elastic optical networks. We formulate the problem as a mixed-integer program for which we develop a branch-and-price algorithm enhanced with such techniques as problem relaxations and cuts for improving lower bounds (LBs) for the optimal objective value, and an RSA heuristic for improving the upper bounds. All these elements are combined into an effective optimization procedure. The results of numerical experiments run on network topologies of different dimensions and with large demand sets show that the algorithm performs well and can be applied to the problem instances that are difficult to solve using commercial solvers such as CPLEX.

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Sets and Parameters
$V$	set of nodes
$E$	set of links
$D$	set of demands
$S$	set of all frequency slices, $S = {1,2, \dots, S}$
$L (d)$	set of lightpaths allowable for demand $d$
$L$	set of all allowable lightpaths, $L = ⋃_{d \in D} L (d)$
$L (e, s)$	set of lightpaths routed through link $e$ and slice $s$
$P (d)$	set of routing paths allowable for demand $d$
$P$	set of all allowable paths, $P = ⋃_{d \in D} P (d)$
$P (e)$	set of paths routed through link $e$
$E (l)$	set of links of lightpath $l$
$S (l)$	set of slices of lightpath $l$
$d (l)$	demand realized by lightpath $l$
$d (p)$	demand realized by path $p$
$n (d, p)$	number of slices requested by demand $d$ on path $p$
$R (R_{+})$	set of real (non-negative real) numbers
Variables
$x_{d l}$	binary, $x_{d l} = 1$ when demand $d$ uses lightpath $l$ ; $x_{d l} = 0$ otherwise
$y_{e s}$	binary, $y_{e s} = 1$ when slice $s$ is allocated in link $e$ ; $y_{e s} = 0$ otherwise
$y_{s}$	binary, $y_{s} = 1$ when slice $s$ is allocated in any network link; $y_{s} = 0$ otherwise

		RMP		MIP		MIP & Cuts
	$\| D \|$	$z^{LB}$	$T^{BP}$	$z^{LB}$	$T^{BP}$	$z^{LB}$	$T^{BP}$	$z^{UB}$
$i 1$	50	117.667	3600	118	78	120	58	120
i2	100	242.879	3600	246	115	251	64	251
$i 3$	100	245.667	3600	246	3600	247	1111	247

	FF-SA $(P)$			FF-SA (a Limited Set of Paths)
$\| D \|$	$T^{BP}$	$Δ$	$k^{heur}$	$T^{BP}$	$Δ$	$k^{heur}$
60	500	0.14%	30	74	0%	1.56
80	506	0.34%	30	206	0%	1.56
100	614	0%	30	144	0%	1.73
150	3478	0.55%	30	933	0.12%	1.78

		Variables		Constraints
	$\| D \|$	$k = 10$	$k = 30$	$k = 10$	$k = 30$
DT12	50	35854	99870	3122	3094
	100	143790	411280	6048	6044
	200	606104	1738912	12524	12484
BT22	50	34315	95682	5006	4957
	100	137381	398672	9865	9837
	200	554845	1635236	19772	19716
EURO28	100	162457	203087	19608	19608
EURO28	150	384502	485315	30375	30326

		Solution and Its Status					Computation Time and Its Distribution
Network	$\| D \|$	$z^{LB}$	$z^{UB}$	$Δ$	$δ$	optimal/feasible/unknown	$T [s]$	$T^{init}$	$T^{RMP}$	$T^{LB}$	$T^{UB}$
DT12	50	74.6	74.6	0.0%	0	100%/0%/0%	115	9.1%	8.1%	5.8%	76.9%
	100	146.3	146.3	0.0%	0	100%/0%/0%	253	14.4%	11.3%	1.5%	72.8%
	150	232.8	232.8	0.0%	0	100%/0%/0%	920	9.8%	15.4%	1.0%	73.8%
	200	302.6	302.9	0.1%	0.3	70%/30%/0%	2107	7.2%	14.9%	1.1%	76.8%
BT22	50	68.4	68.6	0.4%	0.2	90%/10%/0%	461	2.1%	8.0%	19.9%	70.0%
	100	134.5	134.5	0.0%	0	100%/0%/0%	143	26.6%	25.8%	1.6%	45.9%
	150	187.3	187.5	0.1%	0.2	90%/10%/0%	927	8.1%	18.2%	1.5%	72.2%
	200	269.7	269.8	0.0%	0.1	90%/10%/0%	1557	9.2%	22.6%	1.5%	66.7%
EURO28	50	119.7	120.2	0.4%	0.5	80%/20%/0%	744	0.8%	16.9%	1.4%	80.8%
	100	225	225.6	0.3%	0.6	80%/20%/0%	1446	1.5%	30.4%	0.4%	67.6%
	150	355.8	356.2	0.1%	0.4	70%/30%/0%	2143	2.2%	39.5%	0.3%	58.1%

DCs	$\| D \|$	$z^{LB}$	$z^{UB}$	$Δ$	$δ$	$T$
2	50	33.2	33.3	0.3%	0.1	362
	80	51.8	51.8	0%	0	55
	100	62.7	62.7	0%	0	310
	150	96.7	96.8	0.1%	0.1	1001
3	50	23.8	23.8	0%	0	6
3	100	45	45	0%	0	47

Solving Large Instances of the RSA Problem in Flexgrid Elastic Optical Networks

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Journal of Optical Communications and Networking