Emergence of Algorithmically Hard Phasesin Transportation Networksc. H. Yeung

Department of Physics,The Hong Kong Universityof Science and Technology,

Hong Kong, ChinaEmail: phbill@ust.hk

Abstract-We study a model of transportation networks withnonlinear elements which represent local shortage of resources.Frustration arises from competition among the nodes to becomesatisfied. When the initial resources are uniform, algorithmicallyhard regimes emerge when the average availability of resourcesincreases. These regimes are characterized by discrete fractions ofsatisfied nodes, resembling the Devil's staircase. Behavior similarto those in the vertex cover or close packing problems arefound. When initial resources are bimodally distributed, suchalgorithmically hard regimes also emerge when the fraction ofrich nodes increases.

I. INTRODUCTION

The search for efficient algorithms to solve NP-completeproblems is an unsolved issue in computer science. Effortswere made to understand the nature of the algorithmicallyhard regime in these problems. Physicists found that someof these algorithmically hard problems are closely related tothe glassy nature in physical systems [1]. The phase transitionfrom an algorithmically easy regime to a hard one correspondsto the spin glass transition [2], [3]. Examples include theexistence of the hard-SAT and hard-COL phases in the K­satisfiability [4], [5] and the graph coloring problem [6], [7]respectively, and the multiple states of decoded messagesobtained from the transmission in a noisy channel using error­correcting codes [3].

This algorithmic hardness arises from frustrations typicalof glasses, which refers to the conflict between the competinginteractions in the system. This can create numerous low­lying metastable states in the solution space, separated byenergy barriers. In the glassy regime, failure to take intoaccount this physical picture may have drastic algorithmicconsequences. Specifically, the so-called belief propagation(BP) algorithm [8] may experience convergence problems inthe algorithmically hard regime. Since it is based on theassumption of a single ground state of the system, a morepowerful survey propagation (SP) algorithm [4], [5] wasdeveloped to deal with the algorithmical hardness, based onthe assumption of multiple ground states.

In this paper, we consider the emergence of algorithmichardness in the allocation of resources in transportation net­works. Nodes in these networks have either surpluses or short­ages of resources, and the task is to distribute them to achieve

K. Y. Michael WongDepartment of Physics,

The Hong Kong Universityof Science and Technology,

Hong Kong, ChinaEmail: phkywong@ust.hk

a networkwide satisfaction with a mmnnum transportationcost [9], [10]. This problem is relevant to load balancing incomputer networks and network flow of commodities [11].When there is a networkwide surplus of resources, and thetransportation cost is frictionless, the network behavior can besatisfactorily derived, without assuming the multiple groundstates arising from frustrations.

On the other hand, when there is a networkwide shortageof resources, competition among the nodes for the scarceresources induces systemwide frustrations. To reduce thenumber of unsatisfied nodes, resource allocation is optimizedby minimizing the sum of transportation and shortage costs.In applications such as communications networks, resourceshortage can be detrimental to the performance of a node, irre­spective of the extent of the shortage. Hence it is interesting toconsider step-like shortage costs. When the transportation andshortage costs are comparable, both satisfied and unsatisfiednode are present in the network, and the choices of decidingthe satisfied and unsatisfied nodes lead to frustrations. Numer­ous metastable states can emerge, leading to glassy behavior.As we shall see, the main results in this paper are that changesin the shortage per node induce phase transitions to glassybehavior, similar to those in the vertex cover problem [12],[13], one of the fundamental NP-complete problems. We findthe glassy phase line beyond which the BP algorithm stopsto converge, and the system enters an algorithmically hardregime.

II. THE MODEL

We consider a dilute network of N nodes, labelled i ==

1 ... N. Each node is connected randomly to c neighbors withthe symmetric connectivity matrix Ai j == 1, 0 for connectedand unconnected node pairs respectively. Each node i hasinitial resource or capacity Ai, randomly drawn from a distri­bution p(Ai). Positive and negative values of Ai correspond tosurplus and shortage of resources respectively. These resourcescan be transported through connected links. We denote byYij == -Yji the current of resources from node j to node

(I)

i. The system minimizes the cost function

u2 y 2 .

E = 2l:8(-~i) +l: ~J .i (i j )

~i == Ai + 2:j A ijYij is the final resources of node i, and8(x) = 1 when x > 0, and 0 otherwise. The secondterm corresponds to the transportation cost of resources. Thefirst term corresponds to a shortage cost of u2 /2 for eachunsatisfied node.

We first look for phase transitions in the model by numer­ical simulations. To formulate an algorithm, we introduce avariable Si = ± 1 for each node i and restrict its resources bysi ( A i + 2:j A i jYi j) ~ O. Introducing Lagrange multipliers J..L i

for the resource constraint, we minimize the Lagrangian

if> 1.2<lJ"0oI:"0<lJ

t.::if>

.~ 0.8I:

'"'00.6I:.:2

~ 0.4e,.~ 0.2l<l

D O E 'INx···x Fraction of unsatisfied nodesG-E) max cluster size of sat nodes

- 1.5 -IAlu

if><lJ"0oI:

50 3 "0<lJ

t.::if>

.~

~~~~+-J O if>

-2 - r:-s -I -0.5 0 '02<lJ

N.iii...<lJt'i

I-§~

::E

-0.5

with the Kiihn-Tucker conditions J..LiSi~i 0 and J..Li < o.Optimizing L with respect to Y ij , one obta ins

oA/u-J21 125-/3 -!3i2

Fig. I. Simulation results of the average energy per node, the fractionof unsatisfied nodes and the maximum cluster size of satisfied nodes afteroptimization, as a function of A/ u . E ' = 2E / u 2 . Parameters: c = 3,N = 100, 100 samples and 1000 flips. Inset: the maximum cluster sizeof satisfied nodes shown with a larger vertical scale.

Fig. 2. The onset of different types of clusters of satisfied nodes for c = 3,with filled and unfilled circles represent ing satisfied and unsatisfied nodesrespectively.

(2)

(4)

(3)J..L jS j - J..L iSi,

[

(A . - ~ . A . Ji . S . ) ]. 0 ' L.J 'J J JmIn , .S i C

Y i j =

Given a particular set of {s.] , we iterate these equations tofind the corresponding set of {lid. The set of optimal {s.]is found by an approach similar to the the Walksat algorithm[14], by comparing the Lagrangian in Eq. (2) for each choiceof {s.].

III. SIM ULATION R ES ULTS

We first consider networks with uniform capacity (Ai = Afor all i). As shown in Fig. I for C = 3, three phases canbe identified: (I) unsatisfied phase for A/u ::::: - J3, in whichall the nodes are unsatisfied and E / N = u2 /2; (2) partiallysatisfi ed phase for - J3 < A/u < 0, in which only somenodes are satisfied, and 0 < E / N < u 2 /2; (3) satisfi ed phasefor A/u ~ 0, in which all nodes are satisfied and E / N = O.

Unlike the relatively smooth decreasing trend of energy,the fraction of unsatisfied nodes is a discontinuous functionof A/ u, showing abrupt jumps at threshold values of A/u.The step size of the curve decreases as A/u increases, andgradually becomes unresolvable by the numerical experiments.This resembles the Devil 's staircase observed in the circle mapand other dynamical systems [15]. These threshold values ofA/ u mark the positions at which certain configurations of thesatisfied nodes become energetically stable.

To further confirm this picture, we measure the averagemaximum cluster size of satisfied nodes in the samples. Abruptjumps of the cluster size are observed at the same thresholdvalues as shown in Fig. 1. This indicates that new types ofclusters of satisfied nodes are formed at each jump, as shownin Fig. 2 for c = 3. We call the regime - J3 ::::: A/u < - fi72with isolated nodes the single-sat regime, and the regime- fi72 ::::: A/ u < - J 21/ 25 with clusters of satisfied

node pairs the double-sat regime. As shown in the inset ofFig. I, the maximum cluster size increases drastically to O(N)around A/u ~ 0.5. Isolated clusters of satisfied nodes becomeconnected, and an effectively infinite cluster spans the entirenetwork , giving rise to a percolation transition.

A more careful look at Fig. I reveals that jumps of thefraction of unsatisfied nodes are found in the double-sat regime(i.e. three small steps can be identified in Fig. I in the reangeof A/u between -1.22 and -0.92). They mark the positionsat which configurations increasingly dominated by two-nodeclusters become energetically stable, when A increases.

To confirm this picture, we conduct simulations in thisregime using an improved version of the algorithm in sectionII. The simulation results are shown in Fig. 3 with an enlargedscale. More jumps are identified at threshold values A/u =- J (3m - 3)/ (2m - 1) where m = 2, 3, ... and some otherrational numb ers. An example is shown in Fig. 4, with anenergetically stable configuration below the threshold valueat m = 2 switching to a different one above the threshold.The two configurations are bistable when A/u = - 1, thusthere is an intermediate value of fraction between the twocascade values in Fig 3. Only some of those thresholds arevisible in Fig. 3, probably due to the absence of configurationswith larger m in networks of finite size, small occurrenceprobabilities and the limitations of the search algorithm.

IV. TH E CAVI TY ApPROACH

Fig. 4. (a) A five-node configuration composing of 3 isolated satisfied nodes(filled circles). It is stable when A/u < - 1. (b) A five-node configurationcomposing of 2 two-node clusters. It replaces the configuration in (a) whenA/u > - 1.

)'jk

Fig. 5. The tree-like assumption in cavity approaches. Node i and nodes kare respectively the ancestor and descendents of node j. Yij denotes the flowof resources from j to i . Negative Yij and Yj k correspond to flows in theopposite din

£/)'jk) ~(yjk)

Ej (Yij ) = {min [~ Ek(Yjk) + ~2 8( -';j(Yij)) + Y2fj] , (5)

YJd k=l

where ';j(Yij) = Aj +L: k Yjk -Yij· We call Ek(Yjk) the cavityenergy functions which is sent from the vertices to the nextlayer as shown in Fig. 5.

Due to the quadratic form of transportation cost in Eq. (I),we will demonstrate that the energy Ek (Yjk ) is a continuouspiecewise-quadratic function. Each segment of the functionhas the quadratic form

f~k (Yjk) = a~k (Yjk - Y~k)2 + d~k ' (6)

where nk = 0,1,2, ... is the label of the nk -th compositefunction of Ek (Yjk ), and is characterized by the coefficientsa~k ' Y~k and d~k' An example of the cavity energy functionEk (Yjk) is shown in Fig. 6, composing of the three quadraticfunctions labelled by n k = 0, 1, 2. The composite functionwith nk = °is given by a~ = 0, Y~ = 1/2, and correspondsto the unsatisfied case. Other composite functions can begenerated by plugging the composite functions into Eq. (5)recursively. The first two resultant composite functions, for thecase c = 3 and obtained with the constraint Yj = Aj +Yk +Ylactive , are given in Table 1.

To determine the composite functions selected by the min­imization in Eq. (5) , we also need to compare the functionsin Table I with the unsatisfied composite function, labelled bynj = .0 and obtained from Eq. (5) with Yb = 0, ab = 1/2

and db = u2/2+ d~in + d~nin where d~n i n = min{ nd d~k'

energies of its descendents k = 1, ... , c - 1,

tE/Yij)

/' j

E k(Yj k) Yj k

k

Fig. 6. An example of E k( Yj k ) composed of three quadratic functionsf~k (Yjk ) labelled by nk = 0, 1,2. Each quadratic function is characterizedby the values of (j)~ , a~) and its "height" d~ in the E k( Yj k ).

-0.9

'V'c!Y

m = 4 m=3 m=2.5 m = 2

-1.2 -1.1 -1A/u

;S:r:ilA::D: m :m::ll:rzc= = ;:X;.,

~OJI[ m:m~1

.,

~

I r·····\I , : i i ~~

0.46

r;0.54vocv~0.52.~

ce</)

§ 0.5'­oco'B 0.48eu,

Since the networks have a tree-like structure locally, wecarry out the analysis [16] using the Bethe approximation.As shown in Fig. 5, we identify node i to be the ancestorof node j , and nodes labelled by k to be the descendents ofj. We call this a sub-tree terminated at vertex node j in theabsence of i. A new sub-tree is constructed by merging c - 1sub-trees to a newly added vertex. We describe the physicalproperties of a node in the absence of its ancestor, leading toa cavity on the node. Energy functions of the sub-trees arepassed from the vertices of the sub-trees to the next layer,resulting in their recursion relations. We note the equivalenceof this cavity approach with message-passing algorithms [9],[10], in which a message passed from a vertex describes theconditional probability of the vertex states in the absenceof the receiving node. In the cavity approach for discretevariables [4]-[7], these messages are usually the local fieldsin the absence of the ancestors of the vertices, known as thecavity fields. Here, since the variables in the cost function ofEq. (I) are continuous, the messages become energy functions[9], [10]. Let Ej(Yij) be the energy of the tree terminated ata link from vertex j to its ancestor i, when a current Yij isdrawn from j to i . One can express E j (Yij) in terms of the

Fig. 3. Simulation results of the fraction of unsatisfied nodes in the double­sat regime. Parameters: c = 3, N = 300, 200 samples and 60000 flips. Errorbars arc shown. The dashed lines correspond to the analytical threshold valuesat A/u = - J (3m - 3)/(2m - 1) for the indicated values of m.

-0.33 0yl lAI

V. TH E CASE OF NON-U NIF ORM C APACITY

Next, we consider a simple case with quenched disorder ofthe node capacities. In this case, Ai is drawn from a bimodaldistribution, namely, Ai = A± with probabilities f ±, whereA_ Iu < - V3 < A+l u < - fili and 1+ + f - = 1for c = 3. A± are referred to as the rich and poor nodesrespectively. The recursion relation s for the rich nodes fo1I0wthe close-packing rule of Eq. (8), whereas the poor nodes arealways in the U-state. Note that the end points of this range are

of satisfied nodes is equivalent to the minimi zation of thecovered set size in the vertex cover problem. Alternatively,the model can be considered as the close packing limit of alattice glass model [17], [18]. Both the vertex cover problemand the lattice glass model exhibit glassy behavior, and bothphases with single ground state and multiple metastable statesare found therein , corresponding to the computationally easyand hard phases respectively.

The description of the glassy behavior can be approached bythe replica analysis [2]. In the so-called replica symmetric (RS)ansatz , one assume s that the network behavior is dominatedby a single ground state. However, this ansatz is unstable inthe entire single-sat phase , as can be seen from the violationof Eq. (l0) to be discussed in the following section. Instead,the network behavior is described by the so-called full repl icasymmetry-breaking (RSB) ansatz, which assumes an infinitehierarchy of metastabl e states. Indeed, we have computed theoptimized energy in the one-step RSB (l RSB) approximation,assuming that the number r(E) of optimized states withenergy E is distributed according to r(E) ex exp[NI:(E )].Since apart from a constant, I:(E) = In f(E) IN , it is referredto as the configurational entropy [19]. Futhermore, the 1RSBansatz predicts that I:(E) exists up to an energy Ed abovethe ground state energy E s , and the numerous metastabl estates prevent practical algorithm s from converging to the trueground state, result ing in a dynamical transition.

Fig. 7. (a) A typical set of cavity energy functions Ek(Y) at c = 3, A/ u =

- 1.22 > - J372. (b-d) The closed set of Ek(Y) at c = 3, A/u = - 5/ 3,corresponding to (b) the u-state, (c) the s-state and (d) the b-state.

Descendents I fj~ . ~7 7

IAt 3 AL k 1Ink = 0, ti ; = 0 - ::l A h + do + do

IAt 4 A'l. k 12nk = 1, nl = 0 - ') ~ 'I + d j + do

TABLE IG EN ERATION O F TH E CO MPOSIT E FUN CTIONS LA BELLE D BY n j = 1 AND

2 FROM DES CEND EN TS k AND 1 FO R c = 3.

(For negative A, the over-satisfied composite function withYj < Aj + Yk + Yl does not contribute.) Since the networkbehavior only depends on the energy differences among thecomposite functions, we consider recursion relations for theenergy differences E~ == d~ - dJ

o·. In particular, in the single-J J

sat regime where the composite functions with nj = 0 and 1have the lowest energies , we have Eb= 0,

E{ = - "Y - min(O, E1 ) - min(O, ED, (7)

and E~ = E{ + A213- u 2/2 + "Y + min( E~ ,ED, where "Y ==u2/ 2 - A2/6.

In general , the iteration of Eq. (5) results in a distributionof the cavity energy functions Ek (Y), as shown in Fig. 7(a).However, in the single-sat regime - V3 :::; Alu < - filifor c = 3, there are 3 functional forms of Ek (Y) forming aclosed set of solutions to the recursion relation in Eq. (5), eachcomposing of the composite functions with n k = 0 and 1 only.Indeed, since "Y ~ 0 in this regime, the values E~ = - "Y , 0, "Yform a closed set under the iteration of Eq. (7). Furthermore,since A213- u 2/2 > 0 in this regime, we have E~ > E~ , sothat higher composite function s do not contribute.

The three functions with E~ = "Y, - "Y and 0 are referred to asthe u-, s- and b-states, as shown in Fig. 7(b-d). Respectively,they correspond to states favoring unsatisfaction, satisfaction,and bistability, and have absolute minima at Y = 0, Y =-I AI /3, and both Y = 0 and -IAI /3. Furthermore, numericaliterations of Eq. (5) starting with random Ek(Y) show that thisset of solutions is stable. For general values of c, the single-satregime lies in the range - ..jC :::; Al u < - Jc(c - l) /( c + 1),and the closed set consists of more states having absoluteminima at Y = 0, and local minima at different energies atY = -IAI /3, similar to the u-state in Fig. 7 [16].

Considering the recursion relations among the c states, wefind that all states with absolute minima at Yk = 0 (i.e., E~ ~ 0)behave in the same way. Thus, we denote the u- and b-statesby the U-state, and the s-state by the 5 -state. Their recursionrelations are given by

U + · · · +U ----t 5 ,

all other combi nations ----t U. (8)

The closure property of the c states greatly simplifies thecalculation. The physical states of a node and its links can beobtained by feeding c vertices to a central node. The recursionrules imply that optimization is achieved by the close packingof satisfied nodes in the network, with the constraint that theydo not form clusters. This reduces the probl em to the vertexcover problem [12], [13]. Since there is at most one satisfiednode at the end of each link, the maximization of the number

Fig. 8. (a) The fraction of nodes with Pu ,k = 0, I, and 0 < Pu ,k < Ifor c = 3. (b) The phase diagram. Inset: Simulation results of the fractionof messages from rich nodes with BP convergence on dilute networks withc = 3.

[I] R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman , and L. Troyansky,Nature 400, 133 (1999).

[2] M. Mezard, G. Parisi , and M. A Virasoro, Spin Glass Theory and Beyond(World Scientific , 1987).

[3] H. Nishimori, Statistical Physics of Spin Glasses and Information Pro-cessing (Oxford University Press, Oxford, UK, 2001).

[4] M. Mezard, G. Parisi , and R. Zecchina, Science 297, 812 (2002).[5] M. Mezard and R. Zecchina, Phys, Rev. E 66, 056126 (2002).[6] R. Mulct, A. Pagnani , M. Weigt, and R. Zecchina, Phys. Rev. Lett. 89,

26870 I (2002).[7] A. Braunstein, R. Mulet , A. Pagnani, M. Weigt , and R. Zecchina, Phys .

Rev. E 68, 036702 (2003).[8] D. J. C. Mackay, Information Theory, Inference, and Learning Algorithms

(Cambridge University Press , Cambridge, UK, 2003).[9] K. Y .M. Wong and D. Saad, Phys. Rev. E 74, 010104 (2006).[10] K. Y .M. Wong and D. Saad, Phys, Rev. E 76, 011115 (2007).[11] S. Shenker, D. Clark , D. Estrin and S. Herzog, Com put. Commun.

Rev. 26. 19 (1996); R. L. Rard in Optimization in Operations Research(Prentice Hall, Englewood Cli ffs, NJ, 1998).

[12] M. Weigt and A. K. Hartmann, Phys. Rev. Lett . 84 , 6118 (2000).[13] M. Weigt and A K. Hartmann, Phys, Rev. E 63, 056127 (2001).[14] B. Selman , H. Kautz and B. Cohen DIMACS Series in Discrete Math­

ematics and Theoretical Computer Science 26, 521 (1996).[15] R. L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-

Wesley, Redwood City, CA, 1989).[16] C. H. Yeung and K. Y. M. Wong, unpublished.[17] G. Biroli and M. Mezard, Phys. Rev. Lett. 88, 025501 (2002).[181 O. Rivoire , G. Biroli , O. C. Martin , and M. Mezard, Eur. Phys. 1. B 37,

55 (2004).[19] M. Mezard and G. Parisi , Eur. Phys, J. B 20, 217 (2001).[20] D. J. Thouless, Phys . Rev. Lett. 56, 1082 (1986).

REFERENCES

subnetwork of rich nodes. In the large e limit, this subnetworkhas a Poissonian connectivity distribution with an average el+.The result eltT = e agrees with the point of RS instabilityderived in [12], [13].

VI. CONCLUSION

We have studied a model of transportation networks with anonlinear cost function. In the case of uniform capacity, phasetransitions resembling the Devil's staircase reveal the cascadesof clustered flow patterns . In the single-sat regime with aclosed set of only a few cavity energy functions, the flowpattern has a correspondence with the vertex cover problemand the lattice glass models. In the case with quenched disor­der of the capacities, an increase in the fraction of rich nodesinduces a phase transition from an algorithmically easy phaseto a hard one. Incorporating the picture of multiple metastablestates typical of glasses, we find the glassy phase line beyondwhich the belief propagation algorithms stop to converge, andthe system becomes algorithmically hard. This transition ishighly relevant to vertex cover problem and general networkoptimization problems with nonlinear elements .

ACK NOWL EDGM EN T

We thank David Saad for meaningful discussions. This workis supported by the Research Grant Council of Hong Kong(grant numbers HKUST 603607 and 604008).

(9)

(10)

30

Easy

oo!c-~-----;~~---2lI..

Hard

10 20c

v<0>

... "o ~" .....g ~ 0.5<0>0e u"- 0..

co

0.4

0.2

Pu,j

\....: \\,,

: ,'. ,'. \

". "00" 0.2 080.4 0.6 .r.

(a) I -. ( (b) I

0.8" 0.8..g 1

o~0.6 -- PII-I : 0.6 '.~ ... PII= 0 \ ~

.g0.4 - 0 < PII< I

£0.2

In the RS regime, (p~) = tt« = (Pu) for Pu,k = 0 or 1. Fore = 3, this implies an AT transition at It T = 3/4.

When 1+ > It T, nodes with 0 < Pu,k < 1 exist, as shownin Fig. 8(a). This implies that nodes with free states start toexist, analogous to the free states in the non-trivial solution ofvertex cover [12], [13] and graph coloring [6], [7]. Anothersolution without any free states exists for all values of 1+, butis not stable. This characterizes the hard region with multiplestates. Algorithmically, decimation procedures, such as thoseused in SP algorithms [4], [5], are required.

As shown in the 1+-e space in Fig. 8(b), the easy and hardphases exist below and above the AT line respectively. In thelarge e limit, ItT approaches e]c. This result has an inter­esting connection with the vertex cover problem. Consideringthe cover set as the set of unsatisfied nodes, all links involvingpoor nodes are covered . This reduces the problem to one on the

(

c-l )

= bA j ,A_ + bA j,A+ 1 - II Pu,k ,k=l

This implies that the probability tt« of finding an unsatisfiednode can be obtained from the equation tl« = 1- + 1+(1 ­1]~-1). In the easy regime at low 1+, a stable fixed pointsolution with all Pu,k = 0 or 1 exists. In terms of algorithms ,optimal network states in this regime can be obtained by theBP algorithm, initializing the messages to 0 or 1.

The stability of the RS solution can be studied by con­sidering the propagation of fluctuations ((bPu,k )2) under therecursion relation Eq. (9) [20]. This leads to the Almeida­Thouless (AT) stability condition , which reads

1+ = 0 and 1, corresponding to the unsatisfied and partiallysatisfied phases respectively. Since these two phases are in theRS and RSB regimes respectively, we expect that there is aphase transition when 1+ increases from 0 to 1.

Applying the RS ansatz to the region of low 1+, we let Pu,kbe the probability that node k is in the U -state. Its recursionrelation can be written as