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Reductions
2
Desiderata
Desiderata. Classify problems according to their computational
requirements.
Frustrating news. Huge number of fundamental problems have defied
classification for decades.
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Desiderata
Desiderata. Classify problems according to their computational
requirements.
Desiderata'. Suppose we could (couldn't) solve problem X efficiently.
What else could (couldn't) we solve efficiently?
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. - Archimedes 4
Reduction
Def. Problem X reduces to problem Y if given a subroutine for Y,
can solve X.
! Cost of solving X = cost of solving Y + cost of reduction.
Ex. X = Euclidean MST, Y = Voronoi.
don't confuse with reduces from instance I (of X) Algorithm for X Algorithm for Y solution to I
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Reduction
Def. Problem X reduces to problem Y if given a subroutine for Y,
can solve X.
! Cost of solving X = cost of solving Y + cost of reduction.
Consequences.
! Classify problems: establish relative difficulty between two problems.
! Design algorithms: given algorithm for Y, can also solve X.
! Establish intractability: if X is hard, then so is Y.
don't confuse with reduces from 6
Linear Time Reductions
7
Linear Time Reductions
Def. Problem X linear reduces to problem Y if X can be solved with:
! Linear number of standard computational steps.
! One call to subroutine for Y.
! Notation: X! L Y.
Some familiar examples.
! Median! L sorting.
! Element distinctness! L sorting.
! Closest pair! L Voronoi.
! Euclidean MST! L Voronoi.
! Arbitrage! L Negative cycle detection.
! Linear programming! L Linear programming in std form.
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Linear Time Reductions
Def. Problem X linear reduces to problem Y if X can be solved with:
! Linear number of standard computational steps.
! One call to subroutine for Y.
Consequences.
! Design algorithms: given algorithm for Y, can also solve X.
! Establish intractability: if X is hard, then so is Y.
! Classify problems: establish relative difficulty between two problems.
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Sorting and Convex Hull: Lower Bound
Theorem. In quadratic decision tree model of computation,
sorting N integers requires "(N log N) steps.
Claim. Sorting linear reduces to convex hull.
Corollary. Any ccw-based convex hull algorithm requires "(N log N) steps.
allow tests of the form xi < xj or (xj - xi) (yk - yi) - (yj - yi) (xk - xi) < 0 see next slide convex hull 1251432 2861534 3988818 4190745 13546464 89885444 sorting 14
Sorting Linear Reduces to Convex Hull
Sorting instance.
Convex hull instance.
Observation. Region {x : x^2 # x} is convex $ all points are on hull.
Consequence. Starting at point with most negative x,
counter-clockwise order of hull points yields items in ascending order.
( x 1 , x 12 ), ( x 2 , x 22 ),K, ( xN , xN^2 ) ! x 1 , x 2 ,K, xN ! ( xi , xi^2 ) ! ( x (^) j , x^2 j^ )
f ( x ) = x^2 15
3 - SUM Reduces to 3 - COLLINEAR
3 - SUM. Given N distinct integers, are there three that sum to 0?
3 - COLLINEAR. Given N distinct points in the plane,
are there 3 that all lie on the same line?
Claim. 3 - SUM! L 3-COLLINEAR.
Conjecture. Any algorithm for 3 - SUM requires "(N^2 ) time.
Corollary. Sub-quadratic algorithm for 3 - COLLINEAR unlikely.
recall Assignment 2 your N^2 log N algorithm from Assignment 2 was pretty good see next two slides 16
3 - SUM Reduces to 3 - COLLINEAR
Claim. 3 - SUM! L 3-COLLINEAR.
! 3-SUM instance:
! 3-COLLINEAR instance:
( 1 , 1 ) ( 2 , 8 ) (- 3 , - 27 )
- 3 + 2 + 1 = 0
( x 1 , x 13 ), ( x 2 , x 23 ),K, ( xN , xN^3 ) ! x 1 , x 2 ,K, xN
y = x^3
17
3 - SUM Reduces to 3 - COLLINEAR
Lemma. If a, b, and c are distinct then a + b + c = 0 if and only if
(a, a^3 ), (b, b^3 ), (c, c^3 ) are collinear.
Pf. Three points (a, a^3 ), (b, b^3 ), (c, c^3 ) are collinear iff:
a^3 " b^3 a " b =^ b^3 " c^3 b " c #^ ( a " b ) ( a^2 + ab + b^2 ) a " b =^ ( b " c ) ( b^2 + bc + c^2 ) b " c
c^2 + bc " a^2 " ab = 0
( c " a ) ( c + a + b ) = 0
c = a or a + b + c = 0
not distinct d ife nao, mb,i naantdo rcs a arree d niostniznecrto 18
Linear Time Reductions
Def. Problem X linear reduces to problem Y if X can be solved with:
! Linear number of standard computational steps.
! One call to subroutine for Y.
Consequences.
! Design algorithms: given algorithm for Y, can also solve X.
! Establish intractability: if X is hard, then so is Y.
! Classify problems: establish relative difficulty between two problems.
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Primality and Compositeness
PRIME. Given an integer x (represented in binary), is x prime?
COMPOSITE. Given an integer x, does x have a nontrivial factor?
Claim. PRIME! L COMPOSITE.
public static boolean isPrime(BigInteger x) { if (isComposite(x)) return false; else return true; } 20
Primality and Compositeness
PRIME. Given an integer x (represented in binary), is x prime?
COMPOSITE. Given an integer x, does x have a nontrivial factor?
Claim. COMPOSITE! L PRIME.
Conclusion. COMPOSITE and PRIME have same complexity.
public static boolean isComposite(BigInteger x) { if (isPrime(x)) return false; else return true; }
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Assignment Problem
Assignment problem. Assign n jobs to n machines to minimize total
cost, where cij = cost of assigning job j to machine i.
Applications. Match jobs to machines, match personnel to tasks,
match Princeton students to writing seminars.
1 2 3 4 5 1 ' 2 ' 3 ' 4 ' 5 ' 3 8 9 15 10 4 10 7 16 14 9 13 11 19 10 8 13 12 20 13 1 7 5 11 9 cost = 3 + 10 + 11 + 20 + 9 = 53 1 2 3 4 5 1 ' 2 ' 3 ' 4 ' 5 ' 3 8 9 15 10 4 10 7 16 14 9 13 11 19 10 8 13 12 20 13 1 7 5 11 9 cost = 8 + 7 + 20 + 8 + 11 = 44 26
Assignment Problem Reduces to Linear Programming
LP formulation. xij = 1 if job j assigned to machine i.
Theorem. [Birkhoff 1 946, von Neumann 1953] All extreme points
of the above polytope are {0-1}-valued.
Corollary. Assignment problem reduces to LP; can solve in poly-time.
min 1 " i " n
# cij xij
1 " j " n
s. t. xij 1 " j " n
# =^1 1 "^ i^ "^ n
xij 1 " i " n
# =^1 1 "^ j^ "^ n
xij $ 0 1 " i , j " n we assume LP returns an extreme point solution 27
Literal: A Boolean variable or its negation.
Clause. A disjunction of 3 distinct literals.
Conjunctive normal form. A propositional
formula % that is the conjunction of clauses.
3 - SAT. Given a CNF formula % consisting of k clauses over n literals,
does it have a satisfying truth assignment?
Key application. Electronic design automation (EDA).
3 - Satisfiability
C j = x 1 " x 2 " x 3
xi or xi
" = C 1 # C 2 # C 3 # C 4
(^ x 1 "^ x 2 "^ x 3 ) #^ ( x 1 "^ x 2 "^ x 3 ) #^ ( x 1 "^ x 2 "^ x 3 ) #^ ( x 1 "^ x 2 "^ x 4 ) #^ ( x 2 "^ x 3 "^ x 4 ) ! Solution : x 1 = true, x 2 = true, x 3 = false, x 4 = true 28
Graph 3 - Colorability
3 - COLOR. Given a graph, is there a way to color the vertices
red, green, and blue so that no adjacent vertices have the same color?
yes instance
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Graph 3 - Colorability
3 - COLOR. Given a graph, is there a way to color the vertices
red, green, and blue so that no adjacent vertices have the same color?
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Graph 3 - Colorability
Claim. 3 - SAT! P 3 - COLOR.
Pf. Given 3 - SAT instance %, we construct an instance of 3 - COLOR
that is 3-colorable iff % is satisfiable.
Construction.
i. Create one vertex for each literal.
ii. Create 3 new vertices T, F, and B; connect them in a triangle,
and connect each literal to B.
iii. Connect each literal to its negation.
iv. For each clause, attach a gadget of 6 vertices and 13 edges.
to be described next 31
Graph 3 - Colorability
Claim. Graph is 3-colorable iff % is satisfiable.
Pf. $ Suppose graph is 3-colorable.
! Consider assignment that sets all T literals to true.
! (ii) ensures each literal is T or F.
T B F ! x 1 ! x 1 ! x 2 ! x 2 ! xn ! xn ! x 3 ! x 3 true false base 32
Graph 3 - Colorability
Claim. Graph is 3-colorable iff % is satisfiable.
Pf. $ Suppose graph is 3-colorable.
! Consider assignment that sets all T literals to true.
! (ii) ensures each literal is T or F.
! (iii) ensures a literal and its negation are opposites.
T B F ! x 1 ! x 1 ! x 2 ! x 2 ! xn ! xn ! x 3 ! x 3 true false base
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Cook + Karp
3-SAT 3DM VERTEX COVER CLIQUE HAM-CYCLE INDEPENDENT SET 3-COLOR EXACT COVER PLANAR-3-COLOR SUBSET-SUM HAM-PATH PARTITION INTEGER PROGRAMMING KNAPSACK 3 - rCedOuLcOeRs to^3 - SA T TSP BIN-PACKING A ofll oonfe^ t "hreesael^ lpy rhoabrldem" ps^ raorbel^ edmif: feP r=e nNtP^ m?anifestations (^3) r-SedAuTces to^3 - CO LOR 42
Summary
Reductions are important in theory to:
! Establish tractability.
! Establish intractability.
! Classify problems according to their computational requirements.
Reductions are important in practice to:
! Design algorithms.
! Design reusable software modules.
- stack, queue, sorting, priority queue, symbol table, set, graph
shortest path, regular expressions, linear programming
! Determine difficulty of your problem and choose the right tool.
- use exact algorithm for tractable problems
- use heuristics for intractable problems e.g., bin packing
