[PDF] - Journal of Articial In telligence Researc h 12 (2000) PDF Document

SLIDE 1 Journal

f

Articial In telligence Researc h 12 (2000) 219-234 Submitted 5/99; published 5/00 Randomized Algorithms for the Lo

p

Cutset Problem Ann Bec k er anyut a@cs.technion.a c.il Reuv en Bar-Y eh uda reuven@cs.technion.a c.il Dan Geiger d ang@cs.technion.a c.il Computer Scienc e Dep artment T e chnion, Haifa, 32000, Isr ael Abstract W e sho w ho w to nd a minim um w eigh t lo

p

cutset in a Ba y esian net w

rk

with high probabilit y . Finding suc h a lo

p

cutset is the rst step in the metho d

f

conditioning for inference. Our randomized algorithm for nding a lo

p

cutset

utputs

a minim um lo

p

cutset after O (c 6 k k n) steps with probabilit y at least 1

(1
1

6 k ) c6 k , where c > 1 is a constan t sp ecied b y the user, k is the minima l size

f

a minim um w eigh t lo

p

cutset, and n is the n um b er

f

v ertices. W e also sho w empirically that a v arian t

f

this algorithm

ften

nds a lo

p

cutset that is closer to the minim um w eigh t lo

p

cutset than the

nes

found b y the b est deterministic algorithms kno wn. 1. In tro duction The metho d

f

conditioning is a w ell kno wn inference metho d for the computation

f

p

s-

terior probabilities in general Ba y esian net w

rks

(P earl, 1986, 1988; Suermondt & Co

p

er, 1990; P eot & Shac h ter, 1991) as w ell as for nding MAP v alues and solving constrain t satisfaction problems (Dec h ter, 1999). This metho d has t w

conceptual

phases. First to nd an

ptimal
r

close to

ptimal

lo

p

cutset and then to p erform a lik eliho

d

computation for eac h instance

f

the v ariables in the lo

p

cutset. This metho d is routinely used b y geneticists via sev eral genetic link age programs (Ott, 1991; Lang, 1997; Bec k er, Geiger, & Sc haer, 1998). A v arian t

f

this metho d w as dev elop ed b y Lange and Elston (1975). Finding a minim um w eigh t lo

p

cutset is NP-complete and th us heuristic metho ds ha v e

ften

b een applied to nd a reasonable lo

p

cutset (Suermondt & Co

p

er, 1990). Most metho ds in the past had no guaran tee

f

p erformance and p erformed v ery badly when presen ted with an appropriate example. Bec k er and Geiger (1994, 1996)

ered

an algorithm that nds a lo

p

cutset for whic h the logarithm

f

the state space is guaran teed to b e at most a constan t factor

the
ptimal

v alue. An adaptation

f

these appro ximation algorithms has b een included in v ersion 4.0

f

F ASTLINK, a p

pular

soft w are for analyzing large p edigrees with small n um b er

f

genetic mark ers (Bec k er et al., 1998). Similar algorithms in the con text

f

undirected graphs are describ ed b y Bafna, Berman, and F ujito (1995) and F ujito (1996). While appro ximation algorithms for the lo

p

cutset problem are quite useful, it is still w

rth

while to in v est in nding a minim um lo

p

cutset rather than an appro ximation b e- cause the cost

f

nding suc h a lo

p

cutset is amortized

v

er the man y iterations

f

the conditioning metho d. In fact,

ne

ma y in v est an eort

f

complexit y exp

nen

tial in the size

f

the lo

p

cutset in nding a minim um w eigh t lo

p

cutset b ecause the second phase

f

the conditioning algorithm, whic h is rep eated for man y iterations, uses a pro cedure

f

suc h c 2000 AI Access F

undation

and Morgan Kaufmann Publishers. All righ ts reserv ed.

SLIDE 2 Becker, Bar-Yehud a, & Geiger complexit y . The same considerations apply also to constrain t satisfaction problems as w ell as

ther

problems in whic h the metho d

f

conditioning is useful (Dec h ter, 1990, 1999). In this pap er w e describ e sev eral randomized algorithms that compute a lo

p

cutset. As done b y Bar-Y eh uda, Geiger, Naor, and Roth (1994),

ur

solution is based

n

a reduction to the w eigh ted feedbac k v ertex set problem. A fe e db ack vertex set (FVS) F is a set

f

v ertices

f

an undirected graph G = (V ; E ) suc h that b y remo ving F from G, along with all the edges inciden t with F , a set

f

trees is

btained.

The Weighte d F e e db ack V ertex Set (WFVS) pr

blem

is to nd a feedbac k v ertex set F

f

a v ertex-w eigh ted graph with a w eigh t function w : V ! I R + , suc h that P v 2F w (v ) is minimized. When w (v )

1,

this problem is called the FVS problem. The decision v ersion asso ciated with the FVS problem is kno wn to b e NP-Complete (Garey & Johnson, 1979, pp. 191{192). Our randomized algorithm for nding a WFVS, called Repea tedW GuessI,

utputs

a minim um w eigh t FVS after O (c 6 k k n) steps with probabilit y at least 1

(1
1

6 k ) c6 k , where c > 1 is a constan t sp ecied b y the user, k is the minimal size

f

a minim um w eigh t FVS, and n is the n um b er

f

v ertices. F

r

un w eigh ted graphs w e presen t an algorithm that nds a minim um FVS

f

a graph G after O (c 4 k k n) steps with probabilit y at least 1

(1
1

4 k ) c4 k . In comparison, sev eral deterministic algorithms for nding a minim um FVS are describ ed in the literature. One has a complexit y O ((2k + 1) k n 2 ) (Do wney & F ello ws, 1995b) and

thers

ha v e a complexit y O ((17k 4 )!n) (Bo dlaender, 1990; Do wney & F ello ws, 1995a). A nal v arian t

f
ur

randomized algorithms, called WRA, has the b est p erformance b ecause it utilizes information from previous runs. This algorithm is harder to analyze and its in v estigation is mostly exp erimen tal. W e sho w empirically that the actual run time

f

WRA is comparable to a Mo died Greedy Algorithm (MGA), describ ed b y Bec k er and Geiger (1996), whic h is the b est a v ailable deterministic algorithm for nding close to

ptimal

lo

p

cutsets, and y et, the

utput
f

WRA is

ften

closer to the minim um w eigh t lo

p

cutest than the

utput
f

MGA. The rest

f

the pap er is

rganized

as follo ws. In Section 2 w e

utline

the metho d

f

conditioning, explain the related lo

p

cutset problem and describ e the reduction from the lo

p

cutset problem to the WFVS Problem. In Section 3 w e presen t three randomized algorithms for the WFVS problem and their analysis. In Section 4 w e compare exp erimen tally WRA and MGA with resp ect to

utput

qualit y and run time. 2. Bac kground: The Lo

p

Cutset Problem A short

v

erview

f

the metho d

f

conditioning and denitions related to Ba y esian net w

rks

are giv en b elo w. See the b

k

b y P earl (1988) for more details. W e then dene the lo

p

cutset problem. Let P (u 1 ; : : : ; u n ) b e a probabilit y distribution where eac h v ariable u i has a nite set

f

p

ssible

v alues called the domain

f

u i . A directed graph D with no directed cycles is called a Bayesian network

f

P if there is a 1{1 mapping b et w een fu 1 ; : : : ; u n g and v ertices in D , suc h that u i is asso ciated with v ertex i and P can b e written as follo ws: P (u 1 ; : : : ; u n ) = n Y i=1 P (u i j u i 1 ; : : : ; u i j (i) ) (1) where i 1 ; : : : ; i j (i) are the source v ertices

f

the incoming edges to v ertex i in D . 220

SLIDE 3 Randomized Algorithms f

r

the Loop Cutset Pr

blem

Supp

se

no w that some v ariables fv 1 ; : : : ; v l g among fu 1 ; : : : ; u n g are assigned sp ecic v alues fv 1 ; : : : ; v l g resp ectiv ely . The up dating pr

blem

is to compute the probabilit y P (u i j v 1 = v 1 ; : : : ; v l = v l ) for i = 1; : : : ; n. A tr ail in a Ba y esian net w

rk

is a subgraph whose underlying graph is a simple path. A v ertex b is called a sink with resp ect to a trail t if there exist t w

consecutiv

e edges a ! b and b c

n

t. A trail t is active by a set

f

vertic es Z if (1) ev ery sink with resp ect to t either is in Z

r

has a descendan t in Z and (2) ev ery

ther

v ertex along t is

utside

Z . Otherwise, the trail is said to b e blo cke d (d-sep ar ate d) b y Z . V erma and P earl pro v ed that if D is a Ba y esian net w

rk
f

P (u 1 ; : : : ; u n ) and all trails b et w een a v ertex in fr 1 ; : : : ; r l g and a v ertex in fs 1 ; : : : ; s k g are blo c k ed b y ft 1 ; : : : ; t m g, then the corresp

nding

sets

f

v ariables fu r 1 ; : : : ; u r l g and fu s 1 ; : : : ; u s k g are indep enden t conditioned

n

fu t 1 ; : : : ; u t m g (V erma & P earl, 1988). F urthermore, Geiger and P earl pro v ed that this result cannot b e enhanced (Geiger & P earl, 1990). Both results w ere presen ted and extended b y Geiger, V erma, and P earl (1990). Using the close relationship b et w een blo c k ed trails and conditional indep endence, Kim and P earl dev elop ed an algorithm upd a te-tree that solv es the up dating problem

n

Ba y esian net w

rks

in whic h ev ery t w

v

ertices are connected with at most

ne

trail (Kim & P earl, 1983). P earl then solv ed the up dating problem

n

an y Ba y esian net w

rk

as follo ws (P earl, 1986). First, a set

f

v ertices S is selected suc h that an y t w

v

ertices in the net w

rk

are connected b y at most

ne

active trail in S [ Z , where Z is an y subset

f

v ertices. Then, upd a te-tree is applied

nce

for eac h com bination

f

v alue assignmen ts to the v ariables corresp

nding

to S , and, nally , the results are com bined. This algorithm is called the metho d

f

c

nditioning

and its complexit y gro ws exp

nen

tially with the size

f

S . The set S is called a lo

p

cutset. Note that when the domain size

f

the v ariables v aries, then upd a te-tree is called a n um b er

f

times equal to the pro duct

f

the domain sizes

f

the v ariables whose corresp

nding

v ertices participate in the lo

p

cutset. If w e tak e the logarithm

f

the domain size (n um b er

f

v alues) as the w eigh t

f

a v ertex, then nding a lo

p

cutset suc h that the sum

f

its v ertices w eigh ts is minim um

ptimizes

P earl's up dating algorithm in the case where the domain sizes ma y v ary . W e no w giv e an alternativ e denition for a lo

p

cutset S and then pro vide a probabilistic algorithm for nding it. This denition is b

rro

w ed from a pap er b y Bar-Y eh uda et al. (1994). The underlying graph G

f

a directed graph D is the undirected graph formed b y ignoring the directions

f

the edges in D . A cycle in G is a path whose t w

terminal

v ertices coincide. A lo

p

in D is a subgraph

f

D whose underlying graph is a cycle. A v ertex v is a sink with resp ect to a lo

p
if

the t w

edges

adjacen t to v in

are

directed in to v . Ev ery lo

p

m ust con tain at least

ne

v ertex that is not a sink with resp ect to that lo

p.

Eac h v ertex that is not a sink with resp ect to a lo

p
is

called an al lowe d vertex with r esp e ct to . A lo

p

cutset

f

a directed graph D is a set

f

v ertices that con tains at least

ne

allo w ed v ertex with resp ect to eac h lo

p

in D . The w eigh t

f

a set

f

v ertices X is denoted b y w (X ) and is equal to P v 2X w (v ) where w (x) = log(jxj) and jxj is the size

f

the domain asso ciated with v ertex x. A minimum weight lo

p

cutset

f

a w eigh ted directed graph D is a lo

p

cutset F

f

D for whic h w (F

)

is minim um

v

er all lo

p

cutsets

f

G. The L

p

Cutset Pr

blem

is dened as nding a minim um w eigh t lo

p

cutset

f

a giv en w eigh ted directed graph D . 221

SLIDE 4 Becker, Bar-Yehud a, & Geiger The approac h w e tak e is to reduce the lo

p

cutset problem to the w eigh ted feedbac k v ertex set problem, as done b y Bar-Y eh uda et al. (1994). W e no w dene the w eigh ted feedbac k v ertex set problem and then the reduction. Let G = (V ; E ) b e an undirected graph, and let w : V ! I R + b e a w eigh t function

n

the v ertices

f

G. A fe e db ack vertex set

f

G is a subset

f

v ertices F

V

suc h that eac h cycle in G passes through at least

ne

v ertex in F . In

ther

w

rds,

a feedbac k v ertex set F is a set

f

v ertices

f

G suc h that b y remo ving F from G, along with all the edges inciden t with F , w e

btain

a set

f

trees (i.e., a forest). The w eigh t

f

a set

f

v ertices X is denoted (as b efore) b y w (X ) and is equal to P v 2X w (v ). A minimum fe e db ack vertex set

f

a w eigh ted graph G with a w eigh t function w is a feedbac k v ertex set F

f

G for whic h w (F

)

is minim um

v

er all feedbac k v ertex sets

f

G. The Weighte d F e e db ack V ertex Set (WFVS) Pr

blem

is dened as nding a minim um feedbac k v ertex set

f

a giv en w eigh ted graph G ha ving a w eigh t function w . The reduction is as follo ws. Giv en a w eigh ted directed graph (D ; w ) (e.g., a Ba y esian net w

rk),

w e dene the splitting w eigh ted undirected graph D s with a w eigh t function w s as follo ws. Split eac h v ertex v in D in to t w

v

ertices v in and v

ut

in D s suc h that all incoming edges to v in D b ecome undirected inciden t edges with v in in D s , and all

utgoing

edges from v in D b ecome undirected inciden t edges with v

ut

in D s . In addition, connect v in and v

ut

in D s b y an undirected edge. No w set w s (v in ) = 1 and w s (v

ut

) = w (v ). F

r

a set

f

v ertices X in D s , w e dene (X ) as the set

btained

b y replacing eac h v ertex v in

r

v

ut

in X b y the resp ectiv e v ertex v in D from whic h these v ertices

riginated.

Note that if X is a cycle in D s , then (X ) is a lo

p

in D , and if Y is a lo

p

in D , then 1 (Y ) = S v 2Y 1 (v ) is a cycle in D s where 1 (v ) = 8 > < > : v in v is a sink

n

Y v

ut

v is a source

n

Y fv in ; v

ut

g

therwise

(A v ertex v is a sour c e with resp ect to a lo

p

Y if the t w

edges

adjacen t to v in Y

riginate

from v ). This mapping b et w een lo

ps

in D and cycles in D s is

ne-to-one

and

n

to. Our algorithm can no w b e easily stated. ALGORITHM Lo

pCutset

Input: A Bayesian network D Output: A lo

p

cutset

f

D 1. Construct the splitting graph D s with w eigh t function w s 2. Find a feedbac k v ertex set F for (D s ; w s ) using the W eigh ted Randomized Algorithm (WRA) 3. Output (F ). It is immediately seen that if WRA (dev elop ed in later sections)

utputs

a feedbac k v ertex set F

f

D s whose w eigh t is minim um with high probabilit y , then (F ) is a lo

p

cutset

f

D with minim um w eigh t with the same probabilit y . This

bserv

ation holds due to the

ne-to-one

and

n

to corresp

ndence

b et w een lo

ps

in D and cycles in D s and b ecause WRA nev er c ho

ses

a v ertex that has an innite w eigh t. 222

SLIDE 5 Randomized Algorithms f

r

the Loop Cutset Pr

blem

3. Algorithms for the WFVS Problem Recall that a feedbac k v ertex set

f

G is a subset

f

v ertices F

V

suc h that eac h cycle in G passes through at least

ne

v ertex in F . In Section 3.1 w e address the problem

f

nding a FVS with a minim um n um b er

f

v ertices and in Sections 3.2 and 3.3 w e address the problem

f

nding a FVS with a minim um w eigh t. Throughout, w e allo w G to ha v e parallel edges. If t w

v

ertices u and v ha v e parallel edges b et w een them, then ev ery FVS

f

G includes either u, v ,

r

b

th.

3.1 The Basic Algorithms In this section w e presen t a randomized algorithm for the FVS problem. First w e in tro duce some additional terminology and notation. Let G = (V ; E ) b e an undirected graph. The degree

f

a v ertex v in G, denoted b y d(v ), is the n um b er

f

v ertices adjacen t to v . A self-lo

p

is an edge with t w

endp
in

ts at the same v ertex. A le af is a v ertex with degree less

r

equal 1, a linkp

int

is a v ertex with degree 2 and a br anchp

int

is a v ertex with degree strictly higher than 2. The cardinalit y

f

a set X is denoted b y jX j. A graph is called rich if ev ery v ertex is a branc hp

in

t and it has no self-lo

ps.

Giv en a graph G, b y rep eatedly remo ving all lea v es, and b ypassing with an edge ev ery linkp

in

t, a graph G is

btained

suc h that the size

f

a minim um FVS in G and in G are equal and ev ery minim um FVS

f

G is a minim um FVS

f

G. Since ev ery v ertex in v

lv

ed in a self-lo

p

b elongs to ev ery FVS, w e can transform G to a ric h graph G r b y adding the v ertices in v

lv

ed in self lo

ps

to the

utput
f

the algorithm. Our algorithm is based

n

the

bserv

ation that if w e pic k an edge at random from a ric h graph there is a probabilit y

f

at least 1=2 that at least

ne

endp

in

t

f

the edge b elongs to an y giv en FVS F . A precise form ulation

f

this claim is giv en b y Lemma 1 whose pro

f

is giv en implicitl y b y V

ss

(1968, Lemma 4). Lemma 1 L et G = (V ; E ) b e a rich gr aph, F b e a fe e db ack vertex set

f

G and X = V n F . L et E X denote the set

f

e dges in E whose endp

ints

ar e al l vertic es in X and E F ;X denote the set

f

e dges in G that c

nne

ct vertic es in F with vertic es in X . Then, jE X j

jE

F ;X j. Pro

f.

The graph

btained

b y deleting a feedbac k v ertex set F

f

a graph G(V ; E ) is a forest with v ertices X = V n F . Hence, jE X j < jX j. Ho w ev er, eac h v ertex in X is a branc hp

in

t in G, and so, 3 jX j

X

v 2X d(v ) = jE F ;X j + 2 jE X j: Th us, jE X j

jE

F ;X j. 2 Lemma 1 implies that when pic king an edge at random from a ric h graph, it is at least as lik ely to pic k an edge in E F ;X than an edge in E X . Consequen tly , selecting a v ertex at random from a randomly selected edge has a probabilit y

f

at least 1=4 to b elong to a minim um FVS. This idea yields a simple algorithm to nd a FVS. 223

SLIDE 6 Becker, Bar-Yehud a, & Geiger ALGORITHM SingleGuess(G,j) Input: A n undir e cte d gr aph G and an inte ger j > 0. Output: A fe e db ack vertex set F

f

size

j

,

r

"F ail"

therwise.

F

r

i = 1; : : : ; j 1. Reduce G i1 to a ric h graph G i while placing self lo

p

v ertices in F . 2. If G i is the empt y graph Return F 3. Pic k an edge e = (u; v ) at random from E i 4. Pic k a v ertex v i at random from (u; v ) 5. F F [ fv i g 6. V V n fv i g Return "F ail" Due to Lemma 1, when SingleGuess(G; j ) terminates with a FVS

f

size j , there is a probabilit y

f

at least 1=4 j that the

utput

is a minim um FVS. Note that steps 3 and 4 in SingleGuess determine a v ertex v b y rst selecting an arbitrary edge and then selecting an arbitrary endp

in

t

f

this edge. An equiv alen t w a y

f

ac hieving the same selection rule is to c ho

se

a v ertex with probabilit y prop

rtional

to its degree: p(v ) = d(v ) P u2V d(u) = d(v ) 2

jE

j T

see

the equiv alence

f

these t w

selection

metho ds, dene (v ) to b e a set

f

edges whose

ne

endp

in

t is v , and note that for graphs without self-lo

ps,

p(v ) = X e2(v ) p(v je)

p(e)

= 1 2 X e2(v ) p(e) = d(v ) 2

jE

j This equiv alen t phrasing

f

the selection criterion is easier to extend to the w eigh ted case and will b e used in the follo wing sections. An algorithm for nding a minim um FVS with high probabilit y , whic h w e call Repea t- edGuess, can no w b e describ ed as follo ws: Start with j = 1. Rep eat SingleGuess c 4 j times where c > 1 is a parameter dened b y the user. If in

ne
f

the iterations a FVS

f

size

j

is found, then

utput

this FVS,

therwise,

increase j b y

ne

and con tin ue. ALGORITHM Rep eatedGuess(G,c) Input: A n undir e cte d gr aph G and a c

nstant

c > 1. Output: A fe e db ack vertex set F . F

r

j = 1; : : : ; jV j Rep eat c 4 j times 1. F SingleGuess(G; j ) 2. If F is not "F ail" then Return F End fRep eatg End fF

rg

224

SLIDE 7 Randomized Algorithms f

r

the Loop Cutset Pr

blem

The main claims ab

ut

these algorithms are giv en b y the follo wing theorem. Theorem 2 L et G b e an undir e cte d gr aph and c

1

b e a c

nstant.

Then, SingleGuess(G; k )

utputs

a FVS whose exp e cte d size is no mor e than 4k , and Repea tedGuess(G; c)

utputs,

after O (c 4 k k n) steps, a minimum FVS with pr

b

ability at le ast 1

(1
1

4 k ) c4 k , wher e k is the size

f

a minimum FVS and n is the numb er

f

vertic es. The claims ab

ut

the probabilit y

f

success and n um b er

f

steps follo w immediately from the fact that the probabilit y

f

success

f

SingleGuess(G; j ) is at least (1=4) j and that, in case

f

success, O (c 4 j ) iterations are p erformed eac h taking O (j n) steps. The result follo ws from the fact that P k j =1 j 4 j is

f
rder

O (k 4 k ). The pro

f

ab

ut

the exp ected size

f

a single guess is presen ted in the next section. Theorem 2 sho ws that eac h guess pro duces a FVS whic h,

n

the a v erage, is not to

far

from the minim um, and that after enough iterations, the algorithm con v erges to the minim um with high probabilit y . In the w eigh ted case, discussed next, w e managed to ac hiev e eac h

f

these t w

guaran

tees in separate algorithms, but w e w ere unable to ac hiev e b

th

guaran tees in a single algorithm. 3.2 The W eigh ted Algorithms W e no w turn to the w eigh ted FVS problem (WFVS)

f

size k whic h is to nd a feedbac k v ertex set F

f

a v ertex-w eigh ted graph (G; w ), w : V ! I R + ,

f

size less

r

equal k suc h that w (F ) is minimized. Note that for the w eigh ted FVS problem w e cannot replace eac h linkp

in

t v with an edge b ecause if v has w eigh t ligh ter than its branc hp

in

t neigh b

rs

then v can participate in a minim um w eigh t FVS

f

size k . A graph is called br anchy if it has no endp

in

ts, no self lo

ps,

and, in addition, eac h linkp

in

t is connected

nly

to branc hp

in

ts (Bar-Y eh uda, Geiger, Naor, & Roth, 1994). Giv en a graph G, b y rep eatedly remo ving all lea v es, and b ypassing with an edge ev ery linkp

in

t that has a neigh b

r

with equal

r

ligh ter w eigh t, a graph G is

btained

suc h that the w eigh t

f

a minim um w eigh t FVS (of size k ) in G and in G are equal and ev ery minim um WFVS

f

G is a minim um WFVS

f

G. Since ev ery v ertex with a self-lo

p

b elongs to ev ery FVS, w e can transform G to a branc h y graph without self-lo

ps

b y adding the v ertices in v

lv

ed in self lo

ps

to the

utput
f

the algorithm. T

address

the WFVS problem w e

er

t w

sligh

t mo dications to the algorithm Sin- gleGuess presen ted in the previous section. The rst algorithm, whic h w e call Sin- gleW GuessI, is iden tical to SingleGuess except that in eac h iteration w e mak e a reduction to a branc h y graph instead

f

a reduction to a ric h graph. It c ho

ses

a v ertex with probabilit y prop

rtional

to the degree using p(v ) = d(v )= P u2V d(u). Note that this probabilit y do es not tak e the w eigh t

f

a v ertex in to accoun t. A second algorithm, whic h w e call SingleW GuessI I, c ho

ses

a v ertex with probabilit y prop

rtional

to the ratio

f

its degree

v

er its w eigh t, p(v ) = d(v ) w (v ) = X u2V d(u) w (u) : (2) 225

SLIDE 8 Becker, Bar-Yehud a, & Geiger ALGORITHM SingleW GuessI(G,j) Input: A n undir e cte d weighte d gr aph G and an inte ger j > 0. Output: A fe e db ack vertex set F

f

size

j

,

r

"F ail"

therwise.

F

r

i = 1; : : : ; j 1. Reduce G i1 to a branc h y graph G i (V i ; E i ) while placing self lo

p

v ertices in F . 2. If G i is the empt y graph Return F 3. Pic k a v ertex v i 2 V i at random with probabilit y p i (v ) = d i (v )= P u2V i d i (u) 4. F F [ fv i g 5. V V n fv i g Return "F ail" The second algorithm uses Eq. 2 for computing p(v ) in Line 3. These t w

algorithms

ha v e remark ably dieren t guaran tees

f

p erformance. V ersion I guaran tees that c ho

sing

a v ertex that b elongs to an y giv en FVS is larger than 1=6, ho w ev er, the exp ected w eigh t

f

a FVS pro duced b y v ersion I cannot b e b

unded

b y a constan t times the w eigh t

f

a minim um WFVS. V ersion I I guaran tees that the exp ected w eigh t

f

its

utput

is b

unded

b y 6 times the w eigh t

f

a minim um WFVS, ho w ev er, the probabilit y

f

con v erging to a minim um after an y xed n um b er

f

iterations can b e arbitrarily small. W e rst demonstrate via an example the negativ e claims. The p

sitiv

e claims are phrased more precisely in Theorem 3 and pro v en thereafter. Consider the graph sho wn in Figure 1 with three v ertices a,b and c, and corresp

nding

w eigh ts w (a) = 6, w (b) = 3 and w (c) = 3m, with three parallel edges b et w een a and b, and three parallel edges b et w een a and c. The minim um WFVS F

with

size 1 consists

f

v ertex a. According to V ersion I I, the probabilit y

f

c ho

sing

v ertex a is (Eq. 2): p(a) =

(1

+ 1=m)

+

1 So if

is

arbitrarily small and m is sucien tly large, then the probabilit y

f

c ho

sing

v ertex a is arbitrarily small. Th us, the probabilit y

f

c ho

sing

a v ertex from some F

b

y the criterion d(v )=w (v ), as done b y V ersion I I, can b e arbitrarily small. If,

n

the

ther

hand, V ersion I is used, then the probabilit y

f

c ho

sing

a; b,

r

c is 1=2; 1=4; 1=4 , resp ectiv ely . Th us, the exp ected w eigh t

f

the rst v ertex to b e c hosen is 3=4

(

+ m + 4), while the w eigh t

f

a minim um WFVS is 6. Consequen tly , if m is sucien tly large, the exp ected w eigh t

f

a WFVS found b y V ersion I can b e arbitrarily larger than a minim um WFVS. The algorithm for rep eated guesses, whic h w e call Repea tedW GuessI(G; c; j ) is as follo ws: rep eat SingleW GuessI(G; j ) c 6 j times, where j is the minimal n um b er

f

v ertices

f

a minim um w eigh t FVS w e seek. If no FVS is found

f

size

j

, the algorithm

utputs

that the size

f

a minim um WFVS is larger than j with high probabilit y ,

therwise,

it

utputs

the ligh test FVS

f

size less

r

equal j among those explored. The follo wing theorem summarizes the main claims. 226

SLIDE 9 Randomized Algorithms f

r

the Loop Cutset Pr

blem

c a b w (c) = 3 w (a) = 6 w (b) = 3m Figure 1: The minim um WFVS F

=

fag. Theorem 3 L et G b e a weighte d undir e cte d gr aph and c

1

b e a c

nstant.

a) The algorithm Repea tedW GuessI(G; c; k )

utputs

after O (c 6 k k n) steps a minimum WFVS with pr

b

ability at le ast 1

(1
1

6 k ) c6 k , wher e k is the minimal size

f

a minimum weight FVS

f

G and n is the numb er

f

vertic es. b) The algorithm SingleW GuessI I(G,k )

utputs

a fe e db ack vertex set whose exp e cte d weight is no mor e than six times the weight

f

a minimum weight FVS. The pro

f
f

eac h part requires a preliminary lemma. Lemma 4 L et G = (V ; E ) b e a br anchy gr aph, F b e a fe e db ack vertex set

f

G and X = V n F . L et E X denote the set

f

e dges in E whose endp

ints

ar e al l vertic es in X and E F ;X denote the set

f

e dges in G that c

nne

ct vertic es in F with vertic es in X . Then, jE X j

2
jE

F ;X j. Pro

f.

Let X b b e the set

f

branc hp

in

ts in X . W e replace ev ery linkp

in

t in X b y an edge b et w een its neigh b

rs,

and denote the resulting set

f

edges b et w een v ertices in X b b y E b X b and b et w een v ertices in X b and F b y E b F ;X b . The pro

f
f

Lemma 1 sho ws that jE b X b j

jE

b F ;X b j: Since ev ery linkp

in

t in X has b

th

neigh b

rs

in the set X b [ F , the follo wing holds: jE X j

2
jE

b X b j and jE F ;X j = jE b F ;X b j: Hence, jE X j

2
jE

F ;X j. 2 An immediate consequence

f

Lemma 4 is that the probabilit y

f

randomly c ho

sing

an edge that has at least

ne

endp

in

t that b elongs to a FVS is greater

r

equal 1=3. Th us, selecting a v ertex at random from a randomly selected edge has a probabilit y

f

at least 1=6 to b elong to a FVS. Consequen tly , if the algorithm terminates after c 6 k iterations, with a WFVS

f

size k , there is a probabilit y

f

at least 1

(1
1

6 k ) c6 k that the

utput

is a minim um WFVS

f

size at most k . This pro v es part (a)

f

Theorem 3. Note that since k is not kno wn in adv ance, w e use Repea tedW GuessI(G; c; j ) with increasing v alues

f

j un til a FVS is found, sa y when j=J. When suc h a set is found it is still p

ssible

that there exists a WFVS with more than J v ertices that has a smaller w eigh t than the

ne

found. This happ ens when k > J . Ho w ev er, among the WFVSs

f

size at most J , the algorithm nds

ne

with minim um w eigh t with high probabilit y . The second part requires the follo wing lemma. 227

SLIDE 10 Becker, Bar-Yehud a, & Geiger Lemma 5 L et G b e a br anchy gr aph and F b e a FVS

f

G. Then, X v 2V d(v )

6

X v 2F d(v ): Pro

f.

Denote b y d Y (v ) the n um b er

f

edges b et w een a v ertex v and a set

f

v ertices Y . Then, X v 2V d(v ) = X v 2X d(v ) + X v 2F d(v ) = X v 2X d X (v ) + X v 2X d F (v ) + X v 2F d(v ): Due to Lemma 4, X v 2X d X (v ) = 2jE X j

4jE

F ;X j = 4 X v 2X d F (v ): (3) Consequen tly , X v 2V d(v )

4

X v 2X d F (v )+ X v 2X d F (v ) + X v 2F d(v )

6

X v 2F d(v ) as claimed. 2 W e can no w pro v e part (b)

f

Theorem 3 analyzing SingleW GuessI I(G,k ). Recall that V i is the set

f

v ertices in graph G i in iteration i, d i (v ) is the degree

f

v ertex v in G i , and v i is the v ertex c hosen in iteration i. F urthermore, recall that p i (v ) is the probabilit y to c ho

se

v ertex v in iteration i. The exp ected w eigh t E i (w (v )) = P v 2V i w (v )

p

i (v )

f

a c hosen v ertex in iteration i is denoted with a i . Th us, due to the linearit y

f

the exp ectation

p

erator, E (w (F )) = P k i=1 a i , assuming jF j = k . W e dene a normalization constan t for iteration i as follo ws:

i

= " X u2V i d i (u) w (u) # 1 Then, p i (v ) =

i
d

i (v ) w (v ) and a i = X v 2V i w (v )

d

i (v ) w (v )

i

=

i
X

v 2V i d i (v ) Let F

b

e a minim um FVS

f

G and F

i

b e minim um w eigh t FVS

f

the graph G i . The exp ected w eigh t E i (w (v )jv 2 F

i

))

f

a v ertex c hosen from F

i

in iteration i is denoted with b i . W e ha v e, b i = X v 2F

i

w (v )

p

i (v ) =

i
X

v 2F

i

d i (v ) By Lemma 5, a i =b i

6

for ev ery i. 228

SLIDE 11 Randomized Algorithms f

r

the Loop Cutset Pr

blem

Recall that b y denition F

2

is the minim um FVS in the branc h y graph G 2

btained

from G 1 n fv 1 g. W e get, E (w (F

))
E

1 (w (v )jv 2 F

1

)) + E (w (F

2

)) b ecause the righ t hand side is the exp ected w eigh t

f

the

utput

F assuming the algorithm nds a minim um FVS

n

G 2 and just needs to select

ne

additional v ertex, while the left hand side is the unrestricted exp ectation. By rep eating this argumen t w e get, E (w (F

))
b

1 + E (w (F

2

))

k

X i=1 b i Using P i a i = P i b i

max

i a i =b i

6,

w e

btain

E (w (F ))

6
E

(w (F

)):

Hence, E (w (F ))

6
w

(F

)

as claimed. 2 The pro

f

that SingleGuess(G; k )

utputs

a FVS whose exp ected size is no more than 4k (Theorem 2) where k is the size

f

a minim um FVS is analogous to the pro

f
f

Theorem 3 in the follo wing sense. W e assign a w eigh t 1 to all v ertices and replace the reference to Lemma 5 b y a reference to the follo wing claim: If F is a FVS

f

a ric h graph G, then P v 2V d(v )

4

P v 2F d(v ). The pro

f
f

this claim is iden tical to the pro

f
f

Lemma 5 except that instead

f

using Lemma 4 w e use Lemma 1. 3.3 The Practical Algorithm In previous sections w e presen ted sev eral algorithms for nding minim um FVS with high probabilit y . The description

f

these algorithms w as geared to w ards analysis, rather than as a prescription to a programmer. In particular, the n um b er

f

iterations used within Repea tedW GuessI(G; c; k ) is not c hanged as the algorithm is run with j < k . This feature allo w ed us to regard eac h call to SingleW GuessI(G; j ) made b y Repea tedW GuessI as an indep enden t pro cess. F urthermore, there is a small probabilit y for a v ery long run ev en when the size

f

the minim um FVS is small. W e no w sligh tly mo dify Repea tedW GuessI to

btain

an algorithm, termed WRA, whic h do es not suer from these deciencies. The new algorithm w

rks

as follo ws. Rep eat SingleW GuessI(G; jV j) for min(Max; c 6 w (F ) ) iterations, where w (F ) is the w eigh t

f

the ligh test WFVS found so far and Max is some sp ecied constan t determining the maxim um n um b er

f

iterations

f

SingleW GuessI. ALGORITHM WRA(G; c; Max ) Input: A n undir e cte d weighte d gr aph G(V ; E ) and c

nstants

Max and c > 1 Output: A fe e db ack vertex set F F SingleW GuessI (G; jV j) M min(Max ; c 6 w (F ) ); i 1; While i

M

do 1. F SingleW GuessI(G; jV j) 2. If w (F )

w

(F ) then 229

SLIDE 12 Becker, Bar-Yehud a, & Geiger jV j jE j v alues size MGA WRA Eq. 15 25 2{6 3{6 12 81 7 15 25 2{8 3{6 7 89 4 15 25 2{10 3{6 6 90 4 25 55 2{6 7{12 3 95 2 25 55 2{8 7{12 3 97 25 55 2{10 7{12 100 55 125 2{10 17{22 100 31 652 17 Figure 2: Num b er

f

graphs in whic h MGA

r

WRA yield a smaller lo

p

cutset. The last column records the n um b er

f

graphs for whic h the t w

algorithms

pro duced lo

p

cutsets

f

the same w eigh t. Eac h line in the table is based

n

100 graphs. F F ; M min(Max; c 6 w (F ) ) 3. i i + 1; End fWhileg Return F Theorem 6 If Max

c6

k , wher e k is the minimal size

f

a minimum WFVS

f

an undi- r e cte d weighte d gr aph G, then WRA(G; c; Max )

utputs

a minimum WFVS

f

G with pr

b-

ability at le ast 1

(1
1

6 k ) c6 k . The pro

f

is an immediate corollary

f

Theorem 3. The c hoice

f

Max and c dep end

n

the application. A decision-theoretic approac h for selecting suc h v alues for an y-time algorithms is discussed b y Breese and Horvitz (1990). 4. Exp erime n tal Results The exp erimen ts compared the

utputs
f

WRA vis- a-vis a greedy algorithm GA and a mo died greedy algorithm MGA (Bec k er & Geiger, 1996) based

n

randomly generated graphs and

n

some real graphs con tributed b y the Hugin group (www.h ugin.com). The random graphs are divided in to three sets. Graphs with 15 v ertices and 25 edges where the n um b er

f

v alues asso ciated with eac h v ertex is randomly c hosen b et w een 2 and 6, 2 and 8, and b et w een 2 and 10. Graphs with 25 v ertices and 55 edges where the n um b er

f

v alues asso ciated with eac h v ertex is randomly c hosen b et w een 2 and 6, 2 and 8, and b et w een 2 and 10. Graphs with 55 v ertices and 125 edges where the n um b er

f

v alues asso ciated with eac h v ertex is randomly c hosen b et w een 2 and 10. Eac h instance

f

the three classes is based

n

100 random graphs generated as describ ed b y Suermondt and Co

p

er (1990). The total n um b er

f

random graphs w e used is 700. The results are summarized in the table

f

Figure 2. WRA is run with Max = 300 and c = 1. The t w

algorithms,

MGA and WRA,

utput

lo

p

cutsets

f

the same size in

nly

230

SLIDE 13 Randomized Algorithms f

r

the Loop Cutset Pr

blem

Name jV j jE j jF

j

GA MGA WRA W ater 32 123 16 40.7 42.7 29.5 Mildew 35 80 14 48.1 40.5 39.3 Barley 48 126 20 72.1 76.3 57.3 Munin1 189 366 59 159.4 167.5 122.6 Figure 3: Log size (base 2)

f

the lo

p

cutsets found b y GA, MGA, and WRA. 17 graphs and when the algorithms disagree, then in 95%

f

these graphs WRA p erformed b etter than MGA. The actual run time

f

W RA(G; 1; 300) is ab

ut

300 times slo w er than GA (or MGA)

n

G. On the largest random graph w e used, it to

k

4.5 min utes. Most

f

the time is sp end in the last impro v emen t

f

WRA. Considerable run time can b e sa v ed b y letting Max = 5. F

r

all 700 graphs, WRA(G,1,5) has already

btained

a b etter lo

p

cutset than MGA. The largest impro v emen t, with Max = 300, w as from a w eigh t

f

58.0 (l

g

2 scale) to a w eigh t

f

35.9. The impro v emen ts in this case w ere

btained

in iterations 1, 2, 36, 83, 189 with resp ectiv e w eigh ts

f

46.7, 38.8, 37.5, 37.3, 35.9 and resp ectiv e sizes

f

22, 18, 17, 18, and 17 no des. On the a v erage, after 300 iterations, the impro v emen t for the larger 100 graphs w as from a w eigh t

f

52 to 39 and from size 22 to 20. The impro v emen t for the smaller 600 graphs w as from a w eigh t

f

15 to 12.2 and from size 9 to 6.7. The second exp erimen t compared b et w een GA, MGA and WRA

n

four real Ba y esian net w

rks

sho wing that WRA

utp

erformed b

th

GA and MGA after a single call to Sin- gleW GuessI. The w eigh t

f

the

utput

con tin ued to decrease logarithmically with the n um b er

f

iterations. W e rep

rt

the results with Max = 1000 and c = 1. Run time w as b et w een 3 min utes for W ater and 15 min utes for Munin1

n

a P en tium 133 with 32M RAM. 5. Discussion Our randomized algorithm, WRA, has b een incorp

rated

in to the p

pular

genetic soft w are F ASTLINK 4.1 b y Alejandro Sc h aer who dev elops and main tains this soft w are at the National Institute

f

Health. WRA replaced previous appro ximation algorithms for nding FVS b ecause with a small Max v alue it already matc hed

r

impro v ed F ASTLINK 4.0

n

most datasets examined. The datasets used for comparison are describ ed b y Bec k er et al. (1998). The main c haracteristics

f

these datasets is that they w ere all collected b y geneticists, they ha v e a small n um b er

f

lo

ps,

and a large n um b er

f

v alues at eac h no de (tens to h undreds dep ending

n

the genetic analysis). F

r

suc h net w

rks

the metho d

f

conditioning is widely used b y geneticists. The leading inference algorithm, ho w ev er, for Ba y esian net w

rks

is the clique-tree algorithm (Lauritzen & Spiegelhalter, 1988) whic h has b een further dev elop ed in sev eral pap ers (Jensen, Lauritzen, & Olsen, 1990a; Jensen, Olsen, & Andersen, 1990b). F

r

the net w

rks

presen ted in T able 3 conditioning is not a feasible metho d while the clique tree algorithm can and is b eing used to compute p

sterior

probabilities in these net w

rks.

F urthermore, it has b een sho wn that the w eigh t

f

the largest clique is b

unded

b y the w eigh t

f

the lo

p

cutset union the largest paren t set

f

a v ertex in a Ba y esian net w

rk

implying that the 231

SLIDE 14 Becker, Bar-Yehud a, & Geiger clique tree algorithm is alw a ys sup erior in time p erformance

v

er the conditioning algorithm (Shac h ter, Andersen, & Szolo vits, 1994). The t w

metho

ds, ho w ev er, can b e com bined to strik e a balance b et w een time and space requiremen ts as done within the buc k et elimination framew

rk

(Dec h ter, 1999). The algorithmic ideas b ehind the randomized algorithms presen ted herein can also b e applied for constructing go

d

clique trees and initial exp erimen ts conrm that an impro v e- men t

v

er deterministic algorithms is

ften
btained.

The idea is that instead

f

greedily selecting the smallest clique when constructing a clique tree,

ne

w

uld

randomly select the next clique according to the relativ e w eigh ts

f

the candidate cliques. It remains to dev elop the theory b ehind random c hoices

f

clique trees b efore a solid assessmen t can b e presen ted. Curren tly , there is no algorithm for nding a clique tree suc h that its size is guaran teed to b e close to

ptimal

with high probabilit y . Horvitz et al. (1989) sho w that the metho d

f

conditioning can b e useful for appro ximate inference. In particular, they sho w ho w to rank the instances

f

a lo

p

cutset according to their prior probabilities assuming all v ariables in the cutset are marginally indep enden t. The conditioning algorithm can then b e run according to this ranking and the answ er to a query b e giv en as an in terv al that shrinks to w ards the exact solution as more instances

f

the lo

p

cutset are considered (Horvitz, Suermondt, & Co

p

er, 1989; Horvitz, 1990). Applying this idea without making indep endence assumptions is describ ed b y Darwic he (1994). So if the maximal clique is to

large

to store

ne

can still p erform appro ximate inferences using the conditioning algorithm. Ac kno wledgmen t W e thank Se Naor for fruitful discussions. P art

f

this w

rk

w as done while the third author w as

n

sabbatical at Microsoft Researc h. A v arian t

f

this w

rk

has b een presen ted at the fteen th conference

n

uncertain t y in articial in telligence, July 1999, Sw eden. References Bafna, V., Berman, P ., & F ujito, T. (1995). Constan t ratio appro ximations

f

the w eigh ted feedbac k v ertex set problem for undirected graphs. In Pr

c

e e dings

f

the Sixth A nnual Symp

sium
n

A lgorithms and Computation (ISAA C95), pp. 142{151. Bar-Y eh uda, R., Geiger, D., Naor, J., & Roth, R. (1994). Appro ximation algorithms for the feedbac k v ertex set problems with applications to constrain t satisfaction and Ba y esian inference. In Pr

c

e e dings

f

the 5th A nnual A CM-Siam Symp

sium

On Discr ete A l- gorithms, pp. 344{354. Bec k er, A., & Geiger, D. (1994). Appro ximation algorithms for the lo

p

cutset problem. In Pr

c

e e dings

f

the 10th c

nfer

enc e

n

Unc ertainty in A rticial Intel ligenc e, pp. 60{68. Bec k er, A., & Geiger, D. (1996). Optimization

f

p earl's metho d

f

conditioning and greedy- lik e appro ximation algorithms for the feedbac k v ertex set problem. A rticial Intel ligenc e, 83, 167{188. 232

SLIDE 15 Randomized Algorithms f

r

the Loop Cutset Pr

blem

Bec k er, A., Geiger, D., & Sc haer, A. (1998). Automatic selection

f

lo

p

break ers for genetic link age analysis. Human Her e dity, 48, 47{60. Bo dlaender, H. (1990). On disjoin t cycles. International Journal

f

F

undations
f

Com- puter Scienc e (IJF CS), 5, 59{68. Breese, J., & Horvitz, E. (1990). Ideal reform ulation

f

b elief net wroks. In Pr

c

e e dings

f

the 6th c

nfer

enc e

n

Unc ertainty in A rticial Intel ligenc e, pp. 64{72. Darwic he, A. (1994).

b
unded

conditioning: A metho d for the appro ximate up dating

f

causal net w

rks.

R ese ar ch note, R

ckwel

l Scienc e Center. Dec h ter, R. (1990). Enhancemen t sc hemes for constrain t pro cessing: bac kjumping, learning, and cutset decomp

sition.

A rticial Intel ligenc e, 41, 273{312. Dec h ter, R. (1999). Buc k et elimination: A unifying framew

rk

for structure-driv en inference. A rticial Intel ligenc e, T

app

e ar. Do wney , R., & F ello ws, M. (1995a). Fixed-parameter tractabilit y and completeness I: Basic results. SIAM Journal

n

Computing, 24 (4), 873{921. Do wney , R., & F ello ws, M. (1995b). P arameterized computational feasibilit y . In P . Clote, . J. R. (Ed.), F e asible Mathematics II, pp. 219{244. Birkhauser, Boston. F ujito, T. (1996). A note

n

appro ximation

f

the v ertex co v er and feedbac k v ertex set problems

unied

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