Lecture 19

P-Complete Problems

Log-space many-one reduction

. and

. and

space. clogarithmiin computable function a via if

log

logloglog

log

NCANCBBA

CACBBA

fBABA

m

mmm

mm

DEFINITION

PROPERTY

P-complete

n.computatio parallelfor hard are problems complete-

. ,any for and if complete- is log

P

ABPBPAPA m

Circuit Value Problem (CVP)

complete.- is

.1| whether determine , assignmentan and circuit aGiven

PCVP

CC

THEOREM

PROOF

SIMULATE DTM COMPUTATION BY A CIRCUIT

Planar Circuit Value (PCV)

PCVCVP

complete.- is PCV

.1| whether determine , assignmentan and circuit planar aGiven

logm

P

CC

THEOREM

PROOF

.for circuit planar A yx .crosserA

Monotone Circuit Value (MCV)

MCVCVP

complete.- is MCV

.1| whether determine , assignmentan and circuit monotone aGiven

logm

P

CC

THEOREM

PROOF

Odd Maximum Flow (OMF)

OMF.MCV

complete.- is OMF

number.

oddan is of flow maximum he whether tdetermine ,capacity edge

integer positive and sink , source a with ),(network aGiven

logm

P

Nc

tsEVN

PROOF

THEOREM

time.polynomialin computed becan flow Maximum

gate. OR is

gateoutput the(b)

two,

most at fanout has

gate ORevery (a)

:assumemay WeReduce fanout

Add OR gate as output gate

label.smaller a with oneanother frominput an gets node no

:order icalin topologly consequent gatesother and 0h output wit Label (1)

.2capacity assign we, label with node from coming edgeany For (2) ii

22

42 42

32 32

6252

2

capacity. going-out

capacity comming-in total

make ocapacity t with togate

ANDany from edge Assign the

. togateoutput

from edge the to1capacity Assign

).deg(

where label with node to from

edge the to2capacity Assign (4)

gates. AND all and gateoutput

from conneting sink Add

1. e with valuableevery vari

toconnecting source Add (3)

t

t

ioutd

is

d

t

s

i

edges. incomming

ofcapacity total toequal

capacity with , togate OReach

from edgeimaginary an Add

s

. allfor 0),(set , variablefalseeach For

. allfor ),(),(set , vertex variableeach trueFor )2(

).,(),( ),,( edgeeach For (1)

:follows as flow aConstruct

jjifi

jjicjifi

iscisfis

f

. allfor 0),( , gate OR falseeach For

.),(),(),(

for ),(),(, gates OR each trueFor

:follows as edges outgoing itsfor set is then

gate, a of edges incoming allfor set been has If (3)

jjifi

jicikfsif

sjjicjifi

f

f

sjk

.for 0),(

),(),( , gate AND falseeach For

. allfor ),(),( , gate AND each tureFor

tjjif

ikftifi

jjicjifi

k

).,(),(,any for then vertex, truea is If (b)

.0),(,any for then vertex,false a is If (a)

:properties following thehas flow This

jicjifsji

jiftji

vertex.falseany through

passnot does edgeimaginary an through passing flow the(a),By