CSci 5403 Lecture 15

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CSci 5403
COMPLEXITY THEORY
LECTURE XXVI APPROXIMATION ALGORITHMS
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Let R(x,y) be a polynomial time relation, and
define F(x) y R(x,y), i.e. the Feasible
Set.
The maximization problem for R and value
function val 0,1 0,1 ? R is to find Y ?
F(x) such that val(x,Y) OPT(x) maxy?F(x)
val(x,y)
The minimization problem for R and cost
function cost 0,1 0,1 ? R is to find Y
? F(x) such that cost(x,Y) OPT(x) miny?F(x)
cost(x,y)
Example. Let R ((V,E),S) S ? V and
c((V,E),S) (u,v) ? E u ? S ? v ? S
. Then the maximization problem is MAX-CUT, and
the minimization problem in MIN-CUT.
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• Definition. Let A be an algorithm such that for
  all x,
• A(x) ? F(x). Then A has approximation ratio a
  for
• - the minimization problem (R,cost) if
• ?x, cost(x,A(x)) a OPT(x)
• the maximization problem (R,val) if
• ?x, val(x,A(x)) OPT(x)/a

Example. We know an efficient algorithm with
approximation ratio 2 for MAX-CUT. (HW1.4)
Is there an efficient algorithm with
approximation ratio 1e?
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Example.
R ((w1,,wn,v1,vn,W),S) S ? n ? ?i?S wi
W. val((w,v,W),S) ?i?S vi.
KNAPSACK
1lb, 20
½ lb 15
3 lbs 2000
15 lbs 500
50
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Theorem. ?e gt 0, there is an efficient algorithm
Ae for KNAPSACK with approximation ratio 1e.
We start with a O(n(?i vi))-time exact algorithm

• Let V ?i vi.
• ?v V, set S(0,v) Ø, W(0,v) 8 set W(0,0)0.
  
• for each 1 i n, 0 v V
• If W(i-1,v-vi) wi lt W(i-1,v)
• set S(i,v) S(i-1,v-vi)?i
• else set S(i,v) S(i-1,v)
• Set W(i,v) min W(i-1,v-vi)wi, W(i-1,v)
• 4. Find (i,v) that maximizes v s.t. W(i,v)  W.

Here S(i,v) minimum weight S?i with value v.
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To get poly(n)-time, 1e approximation Ae(w,v,W)
1. Pick k so that 2n/nk lt e, i.e. k (log
2n/e)/log n
2. Let l log maxi vi wi W
3. Set vi ?vi / 2l - k lg n ?
4. Return exact-alg(w,v,W)
If there is a subset S ? n with val((w,v,W),S)
V, then val((w,v,W),S) V n 2l1 - k lg n
Thus val(Ae(w,v,W))/OPT(w,v,W)  (2l -
2l2n/nk)/2l (1-e)
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Definition. A Polynomial-Time Approximation
Scheme (PTAS) for optimization problem (R,v)
is a PPT A such that A(x,e) is a 1e
approximation of OPT(x), for all x and e, and
TIMEA(x,e) pe(x) is a polynomial for fixed
e. A Fully Polynomial-Time Approximation Scheme
(FPTAS) is a PTAS where TIMEA(x,e) p(x,1/e).
Example. The previous algorithm is a FPTAS
for KNAPSACK.
Example. Unless PNP, there is no PTAS
for GRAPH-COLOR.
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We begin our study of approximation
complexity with familiar-looking problems
R (?,x) ??kCNF and ?(x)?0,1 cost(?,x)
true clauses in ?(x)
MAX-kSAT
R (?,x) ? (?1,,?m), x?0,1n
vars(?i)k, vars(?j)?x1xn cost(?,x) j
?j(x)
MAX-G-kSAT
R (?,x) ? (?1,,?m), x?Wn ?j, ?j
Wk ? 0,1. ?j(x) j(xj1,,xjk)
cost(?,x) ? j(x)
k-CSP-W
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Approximation algorithm for k-CSP-W
Let Yi ?i(x), ? ?i ExYi, and d mini
ExYi. We compute a 1/d-approximation A(?)
1. Compute ? ?i Ex Yi in time mWk.
2. ?w?W, compute ?w ?i ExYi x1w let
w1 argmaxw?W ?w and set x1 w1.
3. Repeat for x2,,xn.
Claim 1. We always satisfy ? constraints.
Claim 2. Let m i ?x ?i(x), then ? dm
Claim 3. OPT(?)/A(?) 1/d.
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Over the next four lectures we will prove
the following theorem.
Theorem. ?e gt 0 such that there is a polytime
reduction R from SAT to MAX-3SAT such that ? ?
SAT ? OPT(R(?)) m ? ? SAT ? OPT(R(?)) lt
(1-e)m.
Corollary. ?e gt 0 such that there is no
efficient 1e approximation for MAX-3SAT unless
PNP.
Corollary. There is no PTAS for MAX-3SAT
unless PNP.
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Definition. Let ? and ? be maximization
problems. A (c,?), (c,?) gap preserving
reduction from ? to ? is a polynomial-time
algorithm such that OPT?(x) c ? OPT?((x))
c OPT?(x) lt c/? ? OPT?((x)) lt c/?. A
gap producing reduction is a (1,1), (c,?)
gap preserving reduction for constants c,?.
Proposition. If there is a gap-producing
reduction from SAT to ? and a gap-preserving
reduction from ? to ?, then there is a
gap-producing reduction from SAT to ?.
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Definition. An L-reduction from A to B is a
pair (R,S) of logspace functions such that ?a,ß
OPTB(R(x))  a OPTA(x) OPTA(x)
vA(x,S(y)) ß OPTB(R(x)) vB(R(x),y)
Proposition. If (R,S) is an L-reduction from A to
B and (R,S) is an L-reduction from B to C,
then (RR, SS) is an L-reduction from A to C.
Proposition. If there is an (a,ß) L-reduction
from A to B and a polytime 1e approximation
algorithm for B, then there is a polytime 1aße
approximation algorithm for A.
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Definition. An optimization problem ? is in
the class MAX-SNP if ?k gt 0 such that ?
is L-reducible to MAX-G-kSAT.
? is MAX-SNP hard if ?P ? MAX-SNP, ?c, ? ?c,?
so there is a (c,?),(c,?) gap
preserving reduction from P to ?.
Claim. MAX-3SAT, MAX-CUT ? MAX-SNP.
Theorem. ??? MAX-SNP, ?egt0 such that ? has a
polynomial-time 1e approximation algorithm.
Proof. Approximation algorithm for MAX-G-kSAT.
Theorem. PNP iff there is a PTAS for some MAXSNP
hard problem.
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