Download - Lecture 19 P-Complete Problems
Lecture 19
P-Complete Problems
Log-space many-one reduction
. and
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space. clogarithmiin computable function a via if
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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
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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
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for ),(),(, gates OR each trueFor
:follows as edges outgoing itsfor set is then
gate, a of edges incoming allfor set been has If (3)
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.0),(,any for then vertex,false a is If (a)
:properties following thehas flow This
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vertex.falseany through
passnot does edgeimaginary an through passing flow the(a),By