Minimum vertex cover problems on random hypergraphs: replica symmetric solution and a leaf removal algorithm

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by  Satoshi Takabe

Minimum vertex cover problems on random hypergraphs:
replica symmetric solution and a leaf removal algorithm
Satoshi Takabe∗ and Koji Hukushima
Graduate School of Arts and Sciences, The University of Tokyo,
3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
(Dated: January 25, 2013)
We study minimum vertex cover problems on random α-uniform hypergraphs using two differ-
ent approaches, a replica method in statistical mechanics of random systems and a leaf removal
algorithm. It is found that there exists a phase transition at the critical average degree e/(α−1).
Belowthecriticaldegree,areplicasymmetricansatzinthestatistical-mechanicalmethodholdsand
thealgorithm estimates asolution oftheproblemwhich coincideswith thatbythereplicamethod.
In contrast, above the critical degree, the replica symmetric solution becomes unstable and these
methodsfail toestimate theexact solution. Theseresults strongly suggest aclose relation between
3 thereplica symmetry and theperformance of approximation algorithm.
1
0 PACSnumbers: 75.10.Nr,02.60.Pn,05.20.-y,89.70.Eg
2

n
a Themorecrucialpartofeverydaylifecomputersbear, rithms has been also studied [14–16], suggesting that
J the more significance computer science and information thereisanon-trivialrelationbetweenthereplicasymme-

4 theory seem to have. In particular, the computational try and the performance of approximationalgorithms.
2 complexity theory shows the difficulty, the limit of im-
In this Letter, we study the minimum vertex cover

proving algorithms, to solve theoretical computational problem on a random hypergraph. The random graph
]
n problems. It has revealed that the problems belong to is defined by two distributions, the degree distribution
n several classes such as P and NP and there are many
and the edge size distribution. The degree means the
- inclusion relations between these classes. For example,
s numberofedgesconnectingtoavertexandtheedgesize
i 2-satisfiability problems (2-SAT) belong to a class of
d represents the number of vertices connected to an edge.
P guaranteed to be solved in polynomial time. 3-SAT
. As the former distribution, the Poisson distribution and
t
a and the vertex cover problems belong to a class of NP- the delta function are often used and they are called an
m complete [1]. These problems are deeply related to the
Er¨odos-R´enyirandomgraphandaregularrandomgraph,
- well-knownPversusNPproblemplaguingthetheoretical respectively [5]. As the latter distribution, one uses the
d computerscientists,whohavestudiedtheworst-caseper-
delta function with a mean α, which yields a random
n
formance to solve the computational problems. Among
o graph with the same edge size as α called a random α-
many types of combinatorial optimization problems, the
c uniform hypergraph. In general, a statistical-mechanical
[ minimum vertex cover problem (min-VC) belongs to a
model defined on a hypergraph has multi-body interac-

class of NP-hard. The approximation algorithm for the
1 tions determined by its edge size. In contrast to a con-
min-VC and its performance have been studied [2]. The
v ventional two-body interaction, the higher-order multi-
9 application of the problem is to search a file on a file body interactions often change a type of phase transi-
6 storage [3] and to improve the group testing [4].
tion and a breaking pattern of the replica symmetry as
7
shown in the p-body spin glass model [17]. From this
5 In addition to the worst-case analysis, an important
. alternative is the study of typical-case behavior on a viewpoint, influence of an edge size on the typical esti-
1
matesofrandomcomputationalproblemsisinvestigated
0 class of random instances of the computational prob-
3 lems. Recently, statistical-mechanical methods of ran- bystatistical-mechanicalapproaches. Infact,ithasbeen
1 dom spin systems have been applied to the problems revealedthattheedgesizechangesthepropertiesofsome
: problems such as K-SAT [5, 7], q-coloring [18] and min-
v such as K-SAT and constraint-satisfaction problems [5].
i Thesemethods,developedinthespin-glasstheory[6],en- VCs on K-uniform regular random hypergraphs [19]. It
X
isalsofoundthatthereexistsaP/NPtransitionbetween
ableustostudythe typicalpropertiesoftherandomized
r 2-SAT and 3-SAT [20]. Here we study the typical case
a problems. For example, the statistical-mechanical ap-
of the size of the min-VC, explained later,on randomα-
proachesfindaSAT/UNSAT transitionofK-SAT[7], p-
uniform hypergraphs and focus on the relation between
XOR-SAT [8], q-coloring [9] and min-VC [10–13]. These
the replicasymmetryandthe performanceofanapprox-
results clarify that there is a so-called replica symmet-
imation algorithm called a leaf removal algorithm.
ric (RS) phase where a replica symmetry ansatz pro-
vides correct estimates of the typical properties, and a Let us suppose that an α-uniform hypergraph G =
replicasymmetrybreaking(RSB)phasewherethoseesti- (HV,HE) consists of N vertices i ∈ HV = {1,··· ,N}
matesbecomeunstable. Togetherwiththeseapproaches, and (hyper)edges (i ,··· ,i ) ∈ HE ⊂ HVα(i < ··· <
1 α 1
a typical-case performance of some approximation algo- i ). We define covered vertices as a subset HV′ ⊂ HV
α
2
and covered edges as a subset of edges connecting to at of the replica method, the original problem is reduced
least a covered vertex. The vertex cover problem on the to solving a saddle-point equation of a replicated order
hypergraphGistofindasetofthecoveredverticesHV′ parameterfunctional. To proceedthe calculation,weas-
bywhichalledgesarecovered. We define the coverratio sume the RSansatzthatthe solutionofthe saddle-point
onGas|HV′|/N with|HV′| beingthe sizeofthevertex equationhasareplicasymmetry. Introducingalocalfield
cover problem. The min-VC on G is to search a set of onavertexassociatedtotheorderparameteranditsdis-
the covered vertices with the minimum cover ratio. In tribution function, we obtain the saddle-point equation
the random α-uniform hypergraph all the edges are set of the distribution. Finally, under the RS ansatz, the
independentlyfromallα-tuplesofverticeswithprobabil- averageminimum-coverratioisobtainedasafunctionof
ity p. The degree distribution of the graph converges to the averagedegree c,
thePoissondistributionwiththeaveragedegreec,which
1
is given as c=pNα−1/(α−1)! for large N. In this Let- W((α−1)c) α−1 W((α−1)c)
x (c)=1− 1+ ,
ter, we focus on an average of the minimum cover ratio c (α−1)c α
(cid:20) (cid:21) (cid:18) (cid:19)
x overthe sparserandomhypergraphswith the average (4)
c
degree c being O(1). where W(x) is the Lambert W function defined as
The vertex cover problems are mapped on the lattice W(x)exp(W(x)) = x. We call this estimate the RS so-
gas model [10, 11, 21] on the random hypergraphs. We lution of min-VCs. This solution is also obtained by an
define a variable ν on each vertex, representing the ex- alternative cavity method [12]. Although the instability
i
istence of a gas particle, which takes 0 if a vertex i is oftheRSsolutionsuchasthedeAlmeida-Thoulessinsta-
coveredand1 ifuncovered. Ancoverededgehas atleast bility [23] must be examined to validate the solution, we
a vertex with ν = 0 in its connecting vertices. Thus, here naively study an instability condition of the saddle-
i
an indicator function for a given particle configuration point equation against a perturbation of the local field
ν ={ν }={0,1}N is defined as distribution within the RS sector. The analysis leads to
i
acriticalvalueoftheaveragedegreec∗ =e/(α−1)above
χ(ν)= (1−ν ···ν ), (1) which the RS solution becomes unstable. These results,
i1 iα
(i1,···Y,iα)∈HE xc and c∗, include the case of α = 2 [10]. The obtained
x givesacorrectvaluebelowthecriticalaveragedegree,
c
which takes1 if ν is a solutions ofthe vertex coverprob- while a RSB solution for x is required above it.
c
lem on the hypergraph,and 0 otherwise. Using the indi- Here we turn our attention to the estimate of x by
c
cator function, the grand canonical partition function of using an approximation algorithm. The leaf removal al-
the model reads gorithm has been proposed as an approximation algo-
rithm to solve a min-VC on a graph with α=2 [24] and
N
Ξ= exp µ ν χ(ν), (2) has also been applied to searchfor a k-core [25] and a 3-
i
! XOR-SATsolution[15]. Foramin-VC onagivengraph,
ν i=1
X X
this algorithm consists of iterative steps, where vertices
where µ is a chemical potential and the sum is over all calledaleaf,aswellastheedgesconnectingtotheleaves,
configurations of ν. In this formulation, only the solu- areremovedfromthe graphwithcoveredverticesappro-
tionsofthevertexcoverproblemcontributethepartition priately assigned to those vertices. This removal step
functionanditsgroundstatesinalargeµlimitaregiven makesnew leavesandthe algorithmcontinuesinaniter-
bythesolutionsofthemin-VC.Tostudythetypicalcase ative way until the leaf is empty. By this procedure, the
ofmin-VCsweneedtotaketheaverageovertherandom minimum cover ratio is estimated correctly at least for
hypergraphsandthelimitasN →∞. Then,theaverage the removed part of the graph. We consider the global
minimum-cover ratio is represented as leafremoval(GLR)algorithm[14],whichremovessimul-
taneously all the leaves found in a recursive step. We
x (c)=1− lim lim 1 E ν , (3) focus on the expansion of this algorithm for the min-VC
c i
µ→∞N→∞N * + on a hypergraph with α = 3, while it is straightforward
Xi µ to extend it to that on a hypergraph with α≥4. A cru-
where h···i is the grand canonical average and E is cial point in our algorithm is in definition of leaf, where
µ
the average over the random hypergraph ensemble. Our a leaf {i,j,k} ∈ HV3(i < j < k) is defined as a 3-tuple
aim is to obtain the theoretical estimate of the average of vertices connecting to an edge (i,j,k), at least two
minimum-cover ratio as a function of the averagedegree of which the degree is one. The definition of the GLR
c. algorithm is as follows:
The average minimum-cover ratio is derived from the
Step 1: The initial graph G is named G(0). Set k =0.
averaged grand potential density −(µN)−1ElnΞ, which
is obtained by using the replica method for finite con- Step 2: Search all leaves from the graph G(k). If there
nectivity graphs [22]. Following the standard procedure is no leaf, go to Step 6.
3
Step 3: Remove all the leaves except for the vertices
which belong to more than two leaves, named 0.7
bunch of leaves [14], and remove only one of leaves GLR N=10,000
0.6 GLR N=50,000
in each bunch. GLR N=100,000
y 0.5 GLR theory
Step 4: Assigncoveredverticestotheonewiththemax- sit
n
imal degree in each removed leaf from G(k). de 0.4
e
Step 5: The left graph is named G(k+1), and return to siz 0.3
e
Step 2 with k incresed by one. or 0.2
c
Step 6: If there exist connected vertices in the left 0.1
graph, assign all of them to covered vertices. Stop
0
the algorithm.
1 1.2 1.4 1.6 1.8 2
It is proven that the result of the algorithm is indepen- c
dent of order of removal and a selection of a leaf out
of a bunch of leaves in the removal process. When the FIG. 1. (Color Online). The core size density in the GLR
algorithm as a function of the average degree c. Open marks
recursive steps stop, the left graph consists of isolated
are thedataobtained bytheGLR algorithm with thevertex
vertices and a core, which is defined as a set of vertices size104,5×104,and105,whicharetakenanaverageover104
connecting to edges without leaves. Vertices in a bunch
random hypergraphs. The solid line is the core size density
of leaves which are not selected for the removal in Step predicted by our recursive analysis. The vertical dotted line
3 become isolated and the core of the order O(N) ex- represents thecritical average degree c∗ =e/2.
ists in large c. We note that Step 4 can be omitted if
one is interested only in the minimum cover ratio, not
the covered vertices. Because the algorithm covers all
of the removed vertices r up to the n-th step is given
vertices in the core without searchingthe solution of the n
min-VC as shown in Step 6, the existence of the core by rn =1−in−cn. These fractions are governedby the
sequence of e and their values at the end of the algo-
of the order O(N) leads to overestimation of the aver- n
rithm are determined by the asymptotic behavior of the
age minimum-cover ratio. We study the core size at the
recursionrelation of {e }. It is found that there exists a
endoftheGLRalgorithmbynumericallyperformingthe n
above-mentionedprocedureforfinite-sizerandomhyper- criticalaveragedegreec∗ =e/2fortherecursionrelation.
Below the critical value, the sequence {e } converges to
graphs with α = 3. While the computational time for n
the unique value [W(2c)/(2c)]1/2 and consequently the
the GLRalgorithmisproportionalto the numberofver-
tices, it takes time of the order O(N3) for generating a core size c∞ is zero. Above the critical value, however,
a bifurcation occurs in the recursionrelation and the se-
random graph. To avoid it, we use the microcanonical
quence has a cycle with period two. This type of the
ensemble [14] with fixing the number of edges to the ex-
transition would occur above α = 3 at the critical aver-
pectation number of edges cN/3, ignoring fluctuation of
the average degree. We expect that such fluctuation is age degree c∗ = e/(α−1). Because e−1 = 0, an even
term e is larger than that at one-step later, that is
irrelevant in a large size N limit. In Fig. 1, the core size 2n
densityobtainedbynumericalsimulationsispresentedas e2n+1. We compute the limiting values limn→∞e2n+1
afunctionoftheaveragedegreecuptothesizeN =105. and limn→∞e2n numerically as a function of c. The dif-
Thedataaveragedover104randomgraphsconvergeswell ferencebetweenthemyieldsemergenceofthe coreofthe
orderofO(N). Wepresentthecoresizedensityobtained
forlargesizesandagiantcorewithO(N)emergesabove
fromthe asymptotic analysisofthe recursionrelationby
a certain value of c.
the solid line in Fig. 1, which coincides with the data by
We discuss the asymptotic behavior of the recursive
numericalsimulations. Thus,weconfirmthatacoreper-
procedure in the GLR algorithm. We introduce the av-
colationoccursatthe criticalaveragedegreein the GLR
erage fraction of the core c and the isolated vertices
n
algorithm, which coincides with that of the RS instabil-
i over random hypergraphs after n-th step of the algo-
n
ity. Fromtheanalysisnearthecriticaldegree,itisfound
rithm, and find
that the size of the core emerges linearly near above the
i =e +2e +2ce e2 −2, critical average degree. These findings, the bifurcation
n 2n+1 2n 2n 2n−1
(5)
c =e −e −2ce e2 +2ce3 , in the recursion relation and the core percolation, are
n 2n 2n+1 2n 2n−1 2n−1
commoninthe min-VCs onrandomgraphswith α=2.
where a parameter e obeys a recursion relation e = Asmentionedabove,theGLRalgorithmestimatesthe
n n
exp(−ce2n−1) with the initial condition e−1 = 0. A de- minimum cover ratio by the size of the removed part in
tailed derivation of the formulas will be reported in a the graphduring the recursive procedure, which is given
separate paper [26]. By definition, the average fraction asr∞ =1−i∞−c∞. Takingone-thirdofr∞ andadding
4
c∞ to the value, we obtain the estimate of the average
minimum-cover ratio by the algorithm. Thus, we find 0.3
EMC
that below the critical average degree e/2 the estimate
0.25 RS solution
r∞/3 coincides with the RS solution Eq. (4) estimated GLR N=10,000
by the replica method. In contrast,the sequence {e } of GLR N=100,000
n 0.2 GLR theory
the algorithmdoes not convergeto a unique value above
(removed part)
the critical value and the GLR algorithm could not give c 0.15
x
a precise estimate of x there.
c
In order to confirm whether these analyses estimate 0.1
the average minimum-cover ratio x correctly, we also
c
0.05
evaluate the min-VCs by the Markov chain Monte Carlo
method. We use the replica exchange Monte Carlo
0
method (EMC) [27], for accelerating the dynamics of 0 0.5 1 1.5 2 2.5 3
the system, with 50 replicas in the range of the chem- c
ical potential from −2 to 10. In our Monte Carlo sim-
ulations, the smallest cover ratio found in typically 217 FIG. 2. (Color Online). The average minimum-cover ratio
Monte Carlo steps is used as the estimate of xc for each on random α-uniform hypergraphs with α = 3 as a function
random graph, which is averaged over 800 hypergraphs of the average degree c. Open marks are numerical results
randomlygenerated. Thenumberofverticesofthegraph by the exchange MC (diamonds) and by the GLR algorithm
for N = 104 (squares) and 105 (triangles). Lines represent
is up to N = 512. The average minimum-cover ratio is
analytical results by the replica method (solid), by the GLR
extrapolated from these numerical results for finite N.
algorithm(dashed)andontheremovedpartofthegraphsby
Fig.2showstheobtainedminimumcoverratioasafunc-
theGLRalgorithm (dashed-dotted). Theverticaldottedline
tion of the average degree c. Below the critical average is the critical average degree c∗ = e/2, below which all lines
degreee/2wheretheRSsolutionisconsideredtobecor- merge into a single line.
rect, we observe that the MC result is consistent with
thosebythetwoapproaches,the replicamethodandthe
GLR algorithm. Above the critical value, on the other
hand, the MC estimate stays slightly above that by the the leaf removal algorithm even when the edge size α is
replica method and considerably deviates from that by larger than two.
the GLR algorithm. The former is due to the instability It is noted that this relation is not always true for all
of the RS solution and the latter is the existence of the types of random graphs. For instance, the GLR algo-
core of the order O(N). rithm removes no vertex on regular random graphs with
To summarize, we consider the minimum vertex cover c≥2becausenoleafisfoundtherewhile,fromthepoint
problems on random α-uniform hypergraphs, and ana- of the statistical-mechanicalview, the min-VCs on regu-
lyze them by the statistical-mechanical method and the lar random 2-uniform graphs with degree 2 is described
approximation algorithm. The replica method estimates by the RS solution[19]. Thus, the relationdepends on a
the average minimum-cover ratio x as a function of the typeofrandomgraphsandapproximationalgorithms. In
c
average degree c under the replica symmetric assump- addition to the leaf removalalgorithm,a recentwork for
tion. We find that there is an RS/RSB phase transition the min-VCproblemwithα=2[16]suggeststhatlinear
atthecriticalaveragedegreec∗ =e/(α−1),whichiswell programmingalgorithms,whichareoneofthemostcom-
aboveapercolationthresholdc=1/(α−1)intherandom monlyusedtoolsforsolvingoptimizationproblems,have
graph. Wealsoperformthegloballeafremovalalgorithm therelationdiscussedinthepresentwork. Furtherstudy
and study the asymptotic behavior of the recursive pro- will need to establish the relation between the replica
cedureofthealgorithm,particularlyinthecaseofα=3. symmetry and the performance of numerous algorithms.
Ifthe averagedegreeisbelowthecriticalvaluewhichco- This research was supported by a Grants-in-Aid
incides with that in the replica theory, there is a core of for Scientific Research from the MEXT, Japan, No.
the orderO(1)inthe remainingpartofthegraph,which 22340109.
does not affect the estimate of the minimum coverratio.
In contrast, above the critical value, the core of the or-
der O(N) emerges, leading to a wrong estimation of the
minimum cover ratio. Comparing the results obtained
∗
E-mail: s [email protected]
by MC simulations, we confirm that these estimates are
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