Efficient MIP formulation for optimal 1-for-Ndiversity coding

Mateusz Zotkiewicz 1,2

Institute of TelecommunicationsWarsaw University of Technology

Warszawa, Poland

Abstract

In the paper we consider a communication network that uses diversity coding inorder to achieve reliability. Having a set of demands and a network topology weface a problem of optimal routing of the demands and backup trees, and associationsbetween the demands and the backup trees. We present a compact mixed integerprogramming (MIP) formulation for the optimization problem, which proves to bemore efficient than other approaches that can be found in the literature.

Keywords: diversity coding, valid inequalities, mixed integer programming

1 Introduction

The idea of diversity coding was presented in [4,5] as a mechanism for providingself-healing and fault-tolerant communication networks. In its general formdiversity coding can be difficult in operation and management. Therefore, itssimplified version, i.e., 1-for-N diversity coding, became of interest, and willbe considered in this paper. The mechanism is briefly explained below.

1 Work sponsored by National Science Centre, Poland, under grant 2011/01/B/ST7/02967.2 Email: mzotkiew@tele.pw.edu.pl

Available online at www.sciencedirect.com

Electronic Notes in Discrete Mathematics 41 (2013) 133–140

1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

http://dx.doi.org/10.1016/j.endm.2013.05.085

Assume that N demands of the same size are destined to a single sinknode, and they all use failure-disjoint primary paths, i.e., each pair of primarypaths is failure-disjoint. The idea behind 1-for-N diversity coding is that allthose demands are also assigned to a backup tree rooted at the sink node,thus the sink node also receives a backup signal which is a modulo sum ofall the received primary signals. In such a situation, when a single demandsuffers a failure and the backup signal remains untouched, then the sink nodeis still able to decode the lost signal by modulo adding all the surviving signalsincluding the backup signal.

Summarizing, assuming possible single failures of all links, in order tosuccessfully use the mechanism of 1-for-N diversity coding in communicationnetworks, a returned feasible solution has to meet the following conditions.

• Each demand reserves resources for its primary path.

• Each demand is protected by a single backup tree.

• If two demands are protected by the same backup tree, then their primarypaths have to be link-disjoint.

• A primary path of a demand has to be link-disjoint with the whole backuptree that protects the demand.

We also make some additional assumptions that are not directly induced bythe idea of diversity coding, but are made to keep the paper consistent withthe recent published work on optimization of communication networks thattake advantage of diversity coding. The assumptions are listed below.

• All links are directed.

• All demands are directed and of the same size.

• Parallel demands between the same pair of nodes are allowed.

• A single failure between a pair of nodes is defined as failures of links betweenthose nodes in both directions (if both links exist).

• The problem is uncapacitated.

• The objective is to minimize the sum of capacity costs of all links.

The problem described above can be solved by heuristic methods presented in[3], suboptimally using an algorithm presented in [1] by adapting an availablesolution to a binary path restoration problem [7], or optimally using a MIP ap-proach presented in [2]. In this paper we present another exact MIP approachthat proves to be more efficient than the previously published methods.

The paper is organized as follows. In Section 2 we present a notation, an

M. Zotkiewicz / Electronic Notes in Discrete Mathematics 41 (2013) 133–140134

optimization model, and valid inequalities that strengthen the formulation.The formulation is numerically evaluated in Section 3. The results are followedby conclusions in Section 4.

2 Model

In this section we present a novel compact MIP formulation modeling andoptimizing a communication network that uses the idea of 1-for-N diversitycoding for protection. First, a notation is presented. It is followed by theMIP formulation. The section ends with valid inequalities strengthening themodel.

2.1 Notation

In order to formally present the optimization problem we introduce the fol-lowing notation.

Sets:

E set of links; a single link is denoted by e

V set of nodes; a single node is denoted by v

D set of demands; a single demand is denoted by d

T set of backup trees; a single backup tree is denoted by t

δ+(v) set of links originating at node v

δ−(v) set of links ending at node v

Parameters:

we weight of link e; we > 0 for all e ∈ Ee link opposite to link e (if exists)

s(e) source node of link e

d(e) destination node of link e

s(d) source node of demand d

d(d) destination node of demand d

r(t) root node of backup tree t

Variables:

xte 1 if a primary path of any demand protected by backup tree t uses link e;0 otherwise

M. Zotkiewicz / Electronic Notes in Discrete Mathematics 41 (2013) 133–140 135

ste 1 if backup tree t uses link e; 0 otherwise

yde 1 if a backup path of demand d uses link e; 0 otherwise

ntd 1 if demand d is protected by backup tree t; 0 otherwise

2.2 Optimization model

Having the notation defined, the problem can be presented as follows.

min∑e∈E

∑t∈T

we(ste + xt

e)∑e∈δ+(v)

xte −

∑e∈δ−(v)

xte =

∑d∈D

ntd ∀t ∈ T , ∀v ∈ V : v �= r(t) (1a)

∑e∈δ+(v)

yde −∑

e∈δ−(v)

yde =

⎧⎪⎨⎪⎩1 if s(d) = v

−1 if d(d) = v

0 otherwise

∀v ∈ V , ∀d ∈ D (1b)

∑t∈T

ntd = 1 ∀d ∈ D (1c)

ste ≥ ntd + yde − 1 ∀e ∈ E , ∀t ∈ T , ∀d ∈ D (1d)∑

e∈E(ste + xt

e + ste + xte) ≤ 1 ∀e ∈ E , ∀t ∈ T (1e)

ntd = 0 ∀d ∈ D, ∀t ∈ T : d(d) �= r(t) (1f)

The objective of the presented formulation is to minimize a sum of costsof capacities that have to be reserved on all links. Constraints (1a) and (1b)enforce conservation of flows with respect to variables xt

e and yde , respectively.In the first case, flows are considered per tree, and their sources are resultsof associating a demand with a given backup tree. In the second case, flowsrelated to different demands are independent of each other, and for each ofthem there is a set of variables yde . The fact that each demand has to beprotected by one backup tree is enforced by (1c). Constraint (1d) assuresthat if a demand uses a given backup tree, then this backup tree contains alllinks that are used by a backup path of the demand. Disjointness of primarypaths from their backup trees is assured by (1e). The fact that primary pathsof any two demands that are protected by a single backup tree are disjoint isassured by an upper bound on variable xt

e. Finally, constraint (1f) assures thata demand can be protected only by a tree that has its root in a destinationnode of the demand.

M. Zotkiewicz / Electronic Notes in Discrete Mathematics 41 (2013) 133–140136

2.3 Valid inequalities

The presented formulation can be strengthened by the following valid inequal-ities. The inequalities are not crucial from the point of view of the correctnessof the formulation, but can significantly improve performance of the MIPmodel. ∑

e′∈δ+(d(e)):e′ �=e

ste′ ≥ ste ∀t ∈ T , ∀e ∈ E \ δ−(r(t)) (2)

Inequality (2) enforces that if a backup tree enters a node, which is not itsroot node, then it also has to leave this node.∑

e∈δ+(s(d))

ste ≥ ntd ∀t ∈ T , ∀d ∈ D (3)

∑e∈δ−(r(t))

ste ≥ ntd ∀t ∈ T , ∀d ∈ D (4)

Inequality (3) assures that if a demand is protected by a backup tree, thenthis backup tree has to leave a source node of the demand, while inequality(4) assures that the same tree has to enter its root node. Notice that not allbackup tree have to enter their root nodes. If a backup tree is not used byany demand, then it does not have to use any link at all.∑

d∈Dntd ≤ |δ−(r(t))| − 1 ∀t ∈ T (5)

Inequality (5) limits a number of demands that can be supported by a singlebackup tree to a number of links entering a root of the backup tree minusone. The inequality can be directly derived from the assumptions of Section1. If N demands are protected by a single backup tree, then we need N + 1different links that enter a root of the backup tree in order to assure the neededdisjointness.

∑e∈δ+(v)

ste

{= 0 if v = r(t)

≤ 1 if v �= r(t)∀t ∈ T , ∀v ∈ V (6)

The last inequality, i.e., (6), helps to enforce that backup trees are actual treesby letting them use at most one link to leave a node. Notice that backup treesin a solution provided by (1) are actual trees even when (6) is not present.However, in case when weights of links can be equal to zero, then (6) has to

M. Zotkiewicz / Electronic Notes in Discrete Mathematics 41 (2013) 133–140 137

be added to (1) in order to assure that in an obtained solution backup treesare actual trees.

3 Numerical results

First we notice that the problem defined in Section 2 can be decomposed into|V| independent subproblems. As a result of constraint (1f), variables relatedto a single destination node are independent of variables related to otherdestination nodes. For that reason we validated the methods using test caseswith one destination only, as solving the problem with N different destinationscould be decomposed into solving N single destination problems (each of thesame complexity).

We tested three different approaches of solving the problem: the state-of-the-art MIP approach of [2] used as a benchmark in our research (refereed toas benchmark), our novel MIP approach (refereed to as novel), and the samenovel approach facilitated with the presented valid inequalities (refereed to asstrong novel). We tested the approaches on three different topologies fromSNDlib [6]. The problem was modeled using Visual Studio 2010 and solvedby CPLEX 12.3 on Intel 4-core 2.4 GHz CPU running Windows 7. For eachnetwork we randomly generated 140 test cases (20 for each considered numberof demands). A number of backup trees used for each test case is equal to thehalf of a number of demands. The obtained results can be found in Table 1.

The table consists of ten columns. In the first three columns we put basicdata about used topologies, i.e., their names in SNDlib, numbers of nodes,and numbers of links, while in the fourth column a number of demands ispresented. The last six columns contain actual results. In columns time onecan find average time spent by an approach on solving a problem. Notice thatfor all test cases we used a 10-minute time limit, so for configurations withexactly 20 timeouts the average time will be equal to 600 s. In columns TLEa number of timeouts is presented.

The results prove that the presented method, when facilitated by the de-veloped valid inequalities, outperforms the best approach to this problem pre-sented so far. The novel approach proves to be better in the wide spectrumof different configurations and sizes of the problem. The only test cases thathave been solved faster by the state-of-the-art approach were the smallest in-stances of the problem, but even in those cases our approach not facilitatedby the valid inequalities can be considered as an equally efficient alternative.

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Table 1Performance of different MIP approaches

benchmark novel strong novel

top. |V| |A| |D| time TLE time TLE time TLE

6 0.3 0 0.3 0 0.8 0

8 5.5 0 13.7 0 1.8 0

10 41.0 0 160.3 0 9.3 0

polska 12 36 12 227.1 2 443.7 11 166.1 2

14 521.6 17 537.3 17 439.8 9

16 420.0 14 420.5 14 420.1 14

18 480.0 16 480.4 16 480.1 16

18 480.0 16 480.4 16 480.1 16

6 5.8 0 0.5 0 0.8 0

8 93.1 2 11.9 0 3.9 0

10 178.4 2 164.5 2 17.5 0

atlanta 15 44 12 382.9 12 228.6 7 158.9 4

14 450.0 15 360.6 12 363.2 12

16 390.0 13 360.8 12 362.2 12

18 510.0 17 480.7 16 484.5 16

18 480.0 16 480.4 16 480.1 16

6 2.2 0 0.6 0 0.6 0

8 113.1 3 23.8 0 3.4 0

10 205.5 5 163.9 3 49.5 0

france 25 90 12 340.2 9 388.1 8 312.7 8

14 302.7 10 319.0 10 301.2 10

16 420.6 14 421.3 14 420.2 14

18 450.5 15 451.3 15 450.3 15

18 480.0 16 480.4 16 480.1 16

M. Zotkiewicz / Electronic Notes in Discrete Mathematics 41 (2013) 133–140 139

4 Conclusion

In the paper we considered a communication network that uses diversity codingin order to achieve reliability. We presented a novel compact mixed integerprogramming formulation for the related problem of routing optimization inthose networks. We implemented the proposed model, together with the set ofdeveloped valid inequalities, and evaluated it on a number of test case. We alsoimplemented the state-of-the-art approach presented in [2], and we comparedit with our novel approach. The results show that the proposed formulationis more efficient than other formulations that can be found in the literature.Particulary, it outperforms the mentioned state-of-the-art MIP approach tothis problem.

References

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[3] Avci, S. N., X. Hu, and E. Ayanoglu, Recovery from link failures innetworks with arbitrary topology via diversity coding, Proceedings IEEE GlobalCommunications Conference (Houston, TX, USA, 2011).

[4] Ayanoglu, E., I. Chih-Lin, R. D. Gitlin, and J. E. Mazo, Diversity coding:using error control for self-healing in communication networks, Proceedings IEEEConference on Computer Communications (San Francisco, CA, USA, 1990), 95–104.

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