Polynomial-Time Reductions - Computer Science

Polynomial-Time Reductions - Computer Science

Chapter 8 NP and Computational Intractability Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 8.5 Sequencing Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE, TSP. Partitioning problems: 3D-MATCHING, 3-COLOR.

Numerical problems: SUBSET-SUM, KNAPSACK. Hamiltonian Cycle HAM-CYCLE: given an undirected graph G = (V, E), does there exist a simple cycle that contains every node in V. YES: vertices and faces of a dodecahedron. 3 Hamiltonian Cycle HAM-CYCLE: given an undirected graph G = (V, E), does there exist a simple cycle that contains every node in V. 1 1' 2 2' 3 3' 4 4'

5 NO: bipartite graph with odd number of nodes. 4 Directed Hamiltonian Cycle DIR-HAM-CYCLE: given a digraph G = (V, E), does there exists a simple directed cycle that contains every node in V? Claim. DIR-HAM-CYCLE P HAM-CYCLE. Pf. Given a directed graph G = (V, E), construct an undirected graph G' with 3n nodes. aout a din d b v e c G bout

vin v vout ein cout G' 5 Directed Hamiltonian Cycle Claim. G has a Hamiltonian cycle iff G' does. Pf. Suppose G has a directed Hamiltonian cycle . Then G' has an undirected Hamiltonian cycle (same order). Pf. Suppose G' has an undirected Hamiltonian cycle '. ' must visit nodes in G' using one of following two orders: , B, G, R, B, G, R, B, G, R, B, , B, R, G, B, R, G, B, R, G, B, Blue nodes in ' make up directed Hamiltonian cycle in G, or reverse of one.

6 3-SAT Reduces to Directed Hamiltonian Cycle Claim. 3-SAT P DIR-HAM-CYCLE. Pf. Given an instance of 3-SAT, we construct an instance of DIRHAM-CYCLE that has a Hamiltonian cycle iff is satisfiable. Construction. First, create graph that has 2n Hamiltonian cycles which correspond in a natural way to 2n possible truth assignments. 7 3-SAT Reduces to Directed Hamiltonian Cycle Construction. Given 3-SAT instance with n variables xi and k clauses. Construct G to have 2n Hamiltonian cycles. Intuition: traverse path i from left to right set variable xi = 1. s x1 x2

x3 t 3k + 3 8 3-SAT Reduces to Directed Hamiltonian Cycle Construction. Given 3-SAT instance with n variables xi and k clauses. For each clause: add a node and 6 edges. clause node clause node s x1 x2 x3 t 9

3-SAT Reduces to Directed Hamiltonian Cycle Claim. is satisfiable iff G has a Hamiltonian cycle. Pf. Suppose 3-SAT instance has satisfying assignment x*. Then, define Hamiltonian cycle in G as follows: if x*i = 1, traverse row i from left to right if x*i = 0, traverse row i from right to left for each clause Cj , there will be at least one row i in which we are going in "correct" direction to splice node C j into tour 10 3-SAT Reduces to Directed Hamiltonian Cycle Claim. is satisfiable iff G has a Hamiltonian cycle. Pf. Suppose G has a Hamiltonian cycle . If enters clause node Cj , it must depart on mate edge. thus, nodes immediately before and after Cj are connected by an edge e in G removing Cj from cycle, and replacing it with edge e yields Hamiltonian cycle on G - { Cj }

Continuing in this way, we are left with Hamiltonian cycle ' in G - { C1 , C2 , . . . , Ck }. Set x*i = 1 iff ' traverses row i left to right. Since visits each clause node Cj , at least one of the paths is traversed in "correct" direction, and each clause is satisfied. 11 Longest Path SHORTEST-PATH. Given a digraph G = (V, E), does there exists a simple path of length at most k edges? LONGEST-PATH. Given a digraph G = (V, E), does there exists a simple path of length at least k edges? Claim. 3-SAT P LONGEST-PATH. Pf 1. Redo proof for DIR-HAM-CYCLE, ignoring back-edge from t to s. Pf 2. Show HAM-CYCLE P LONGEST-PATH. 12 The Longest Path t Lyrics. Copyright 1988 by Daniel J. Barrett. Music. Sung to the tune of The Longest Time by Billy Joel.

Woh-oh-oh-oh, find the longest path! Woh-oh-oh-oh, find the longest path! If you said P is NP tonight, There would still be papers left to write, I have a weakness, I'm addicted to completeness, And I keep searching for the longest path. The algorithm I would like to see Is of polynomial degree, But it's elusive: Nobody has found conclusive Evidence that we can find a longest path. I have been hard working for so long. I swear it's right, and he marks it wrong. Some how I'll feel sorry when it's done: GPA 2.1 Is more than I hope for. Garey, Johnson, Karp and other men (and women) Tried to make it order N log N. Am I a mad fool If I spend my life in grad school, Forever following the longest path? Woh-oh-oh-oh, find the longest path! Woh-oh-oh-oh, find the longest path! Woh-oh-oh-oh, find the longest path. t Recorded by Dan Barrett while a grad student at Johns Hopkins during a difficult algorithms final. 13

Traveling Salesperson Problem TSP. Given a set of n cities and a pairwise distance function d(u, v), is there a tour of length D? All 13,509 cities in US with a population of at least 500 Reference: http://www.tsp.gatech.edu 14 Traveling Salesperson Problem TSP. Given a set of n cities and a pairwise distance function d(u, v), is there a tour of length D? Optimal TSP tour Reference: http://www.tsp.gatech.edu 15 Traveling Salesperson Problem TSP. Given a set of n cities and a pairwise distance function d(u, v), is there a tour of length D? 11,849 holes to drill in a programmed logic array Reference: http://www.tsp.gatech.edu 16 Traveling Salesperson Problem TSP. Given a set of n cities and a pairwise distance function d(u, v), is there a tour of length D?

Optimal TSP tour Reference: http://www.tsp.gatech.edu 17 Traveling Salesperson Problem TSP. Given a set of n cities and a pairwise distance function d(u, v), is there a tour of length D? HAM-CYCLE: given a graph G = (V, E), does there exists a simple cycle that contains every node in V? Claim. HAM-CYCLE P TSP. Pf. Given instance G = (V, E) of HAM-CYCLE, create n cities with distance function 1 if (u, v) E d(u, v) = 2 if (u, v) E TSP instance has tour of length n iff G is Hamiltonian. Remark. TSP instance in reduction satisfies -inequality. 18 8.6 Partitioning Problems

Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE, TSP. Partitioning problems: 3D-MATCHING, 3-COLOR. Numerical problems: SUBSET-SUM, KNAPSACK. 3-Dimensional Matching 3D-MATCHING. Given n instructors, n courses, and n times, and a list of the possible courses and times each instructor is willing to teach, is it possible to make an assignment so that all courses are taught at different times? Instructor Course

Time Wayne COS 423 MW 11-12:20 Wayne COS 423 TTh 11-12:20 Wayne COS 226 TTh 11-12:20 Wayne COS 126 TTh 11-12:20 Tardos COS 523 TTh 3-4:20

Tardos COS 423 TTh 11-12:20 Tardos COS 423 TTh 3-4:20 Kleinberg COS 226 TTh 3-4:20 Kleinberg COS 226 MW 11-12:20 Kleinberg COS 423 MW 11-12:20

20 3-Dimensional Matching 3D-MATCHING. Given disjoint sets X, Y, and Z, each of size n and a set T X Y Z of triples, does there exist a set of n triples in T such that each element of X Y Z is in exactly one of these triples? Claim. 3-SAT P INDEPENDENT-COVER. Pf. Given an instance of 3-SAT, we construct an instance of 3Dmatching that has a perfect matching iff is satisfiable. 21 3-Dimensional Matching number of clauses Construction. (part 1) Create gadget for each variable xi with 2k core and tip elements. No other triples will use core elements. In gadget i, 3D-matching must use either both grey triples or set x = true set x = false both blue ones. i

i false clause 1 tips core true k = 2 clauses n = 3 variables x1 x2 x3 22 3-Dimensional Matching Construction. (part 2) For each clause Cj create two elements and three triples. Exactly one of these triples will be used in any 3D-matching. Ensures any 3D-matching uses either (i) grey core of x 1 or (ii)

blue core of x2 or (iii) grey core of x3. clause 1 gadget C j = x1 x2 x3 each clause assigned its own 2 adjacent tips false clause 1 tips core true x1 x2 x3 23 3-Dimensional Matching Construction. (part 3) For each tip, add a cleanup gadget. clause 1 gadget

cleanup gadget false clause 1 tips core true x1 x2 x3 24 3-Dimensional Matching Claim. Instance has a 3D-matching iff is satisfiable. Detail. What are X, Y, and Z? Does each triple contain one element from each of X, Y, Z? clause 1 gadget cleanup gadget false clause 1 tips core true

x1 x2 x3 25 3-Dimensional Matching Claim. Instance has a 3D-matching iff is satisfiable. Detail. What are X, Y, and Z? Does each triple contain one element from each of X, Y, Z? clause 1 gadget cleanup gadget clause 1 tips core x1 x2 x3 26 8.7 Graph Coloring Basic genres.

Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE, TSP. Partitioning problems: 3D-MATCHING, 3-COLOR. Numerical problems: SUBSET-SUM, KNAPSACK. 3-Colorability 3-COLOR: Given an undirected graph G does there exists a way to color the nodes red, green, and blue so that no adjacent nodes have the same color? yes instance 28 Register Allocation Register allocation. Assign program variables to machine register so that no more than k registers are used and no two program

variables that are needed at the same time are assigned to the same register. Interference graph. Nodes are program variables names, edge between u and v if there exists an operation where both u and v are "live" at the same time. Observation. [Chaitin 1982] Can solve register allocation problem iff interference graph is k-colorable. Fact. 3-COLOR P k-REGISTER-ALLOCATION for any constant k 3. 29 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. n For each literal, create a node. n Create 3 new nodes T, F, B; connect them in a triangle, and connect each literal to B. n Connect each literal to its negation. n For each clause, add gadget of 6 nodes and 13 edges. to be described next 30 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.

true false T F B x1 x 1 x2 x 2 base x3

x 3 xn x n 31 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. (iv) ensures at least one literal in each clause is T. B x x1

x3 2 6-node gadget true T F false 32 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. (iv) ensures at least one literal in each clause is T.

not 3-colorable if all are red B x x1 x3 2 contradiction true T F

false 33 3-Colorability Claim. Graph is 3-colorable iff is satisfiable. Pf. Suppose 3-SAT formula is satisfiable. Color all true literals T. Color node below green node F, and node below that B. Color remaining middle row nodes B. Color remaining bottom nodes T or F as forced. a literal set to true in 3-SAT assignment B x x1 true T

x3 2 F false 34 8.8 Numerical Problems Basic genres. Packing problems: SET-PACKING, INDEPENDENT SET. Covering problems: SET-COVER, VERTEX-COVER. Constraint satisfaction problems: SAT, 3-SAT. Sequencing problems: HAMILTONIAN-CYCLE, TSP.

Partitioning problems: 3-COLOR, 3D-MATCHING. Numerical problems: SUBSET-SUM, KNAPSACK. Subset Sum SUBSET-SUM. Given natural numbers w1, , wn and an integer W, is there a subset that adds up to exactly W? Ex: { 1, 4, 16, 64, 256, 1040, 1041, 1093, 1284, 1344 }, W = 3754. Yes. 1 + 16 + 64 + 256 + 1040 + 1093 + 1284 = 3754. Remark. With arithmetic problems, input integers are encoded in binary. Polynomial reduction must be polynomial in binary encoding. Claim. 3-SAT P SUBSET-SUM. Pf. Given an instance of 3-SAT, we construct an instance of SUBSET-SUM that has solution iff is satisfiable. 36 Subset Sum Construction. Given 3-SAT instance with n variables and k clauses, form 2n + 2k decimal integers, each of n+k digits, as illustrated below. Claim. is satisfiable iff there exists a subset that sums to W. C1 C 2 C 3 x y

z Pf. No carries possible. x 1 0 0 0 1 0 100,010 x 1 0 0 1 0 1

100,101 y 0 1 0 1 0 0 10,100 y 0 1 0 0 1

1 10,011 z 0 0 1 1 1 0 1,110 z 0 0 1 0 0

1 1,001 0 0 0 1 0 0 100 0 0 0 2 0 0

200 0 0 0 0 1 0 10 0 0 0 0 2 0 20 0

0 0 0 0 1 1 0 0 0 0 0 2 2 1 1

1 4 4 4 111,444 dummies to get clause columns to sum to 4 W 37 My Hobby Randall Munro http://xkcd.com/c287.html 38 Scheduling With Release Times SCHEDULE-RELEASE-TIMES. Given a set of n jobs with processing time ti, release time ri, and deadline di, is it possible to schedule all jobs on a single machine such that job i is processed with a contiguous slot of ti time units in the interval [ri, di ] ? Claim. SUBSET-SUM P SCHEDULE-RELEASE-TIMES.

Pf. Given an instance of SUBSET-SUM w1, , wn, and target W, Create n jobs with processing time ti = wi, release time ri = 0, and no deadline (di = 1 + j wj). Create job 0 with t0 = 1, release time r0 = W, and deadline d0 = W+1. Can schedule jobs 1 to n anywhere but [W, W+1] job 0 0 W W+1 S+1 39 8.10 A Partial Taxonomy of Hard Problems Polynomial-Time Reductions constraint satisfaction 3-SAT Dick Karp (1972) 1985 Turing Award

to es ET c u TS ed T r DEN A S N 3- EPE IND INDEPENDENT SET DIR-HAM-CYCLE GRAPH 3-COLOR SUBSET-SUM VERTEX COVER HAM-CYCLE PLANAR 3-COLOR SCHEDULING SET COVER

TSP packing and covering sequencing partitioning numerical 41 Extra Slides Subset Sum (proof from book) Construction. Let X Y Z be a instance of 3D-MATCHING with triplet set T. Let n = |X| = |Y| = |Z| and m = |T|. Let X = { x1, x2, x3 x4 }, Y = { y1, y2, y3, y4 } , Z = { z1, z2, z3, z4 } For each triplet t= (xi, yj, zk ) T, create an integer wt with 3n digits that has a 1 in positions i, n+j, and 2n+k. use base m+1 Claim. 3D-matching iff some subset sums to W = 111,, 111. Triplet ti x1

x2 x3 x4 y1 y2 y3 y4 z1 z2 z3 z4 wi x1 y2 z3 1

0 0 0 0 1 0 0 0 0 1 0 100,001,000,010 x2 y4 z2

0 1 0 0 0 0 0 1 0 1 0 0 10,000,010,100 x1 y1 z1

1 0 0 0 1 0 0 0 1 0 0 0 100,010,001,000 x2 y2

z4 0 1 0 0 0 1 0 0 0 0 0 1 10,001,000,001 x4 y3

z4 0 0 0 1 0 0 1 0 0 0 0 1 100,100,001 x3

y1 z2 0 0 1 0 1 0 0 0 0 1 0 0 1,010,000,100 x3

y1 z3 0 0 1 0 1 0 0 0 0 0 1 0 1,010,000,010

x3 y1 z1 0 0 1 0 1 0 0 0 1 0 0 0 1,010,001,000

x4 y4 z4 0 0 0 1 0 0 0 1 0 0 0 1

100,010,001 111,111,111,111 43 Partition SUBSET-SUM. Given natural numbers w1, , wn and an integer W, is there a subset that adds up to exactly W? PARTITION. Given natural numbers v1, , vm , can they be partitioned into two subsets that add up to the same value? i vi Claim. SUBSET-SUM P PARTITION. Pf. Let W, w1, , wn be an instance of SUBSET-SUM. Create instance of PARTITION with m = n+2 elements. v1 = w1, v2 = w2, , vn = wn, vn+1 = 2 i wi - W, vn+2 = i wi +W There exists a subset that sums to W iff there exists a partition since two new elements cannot be in the same partition. vn+1 = 2 i wi - W vn+2 = i wi + W W i wi - W

subset A subset B 44 4 Color Theorem Planar 3-Colorability PLANAR-3-COLOR. Given a planar map, can it be colored using 3 colors so that no adjacent regions have the same color? YES instance. 46 Planar 3-Colorability PLANAR-3-COLOR. Given a planar map, can it be colored using 3 colors so that no adjacent regions have the same color? NO instance. 47 Planarity Def. A graph is planar if it can be embedded in the plane in such a way that no two edges cross. Applications: VLSI circuit design, computer graphics. planar K5: non-planar

K3,3: non-planar Kuratowski's Theorem. An undirected graph G is non-planar iff it contains a subgraph homeomorphic to K5 or K3,3. homeomorphic to K3,3 48 Planarity Testing Planarity testing. [Hopcroft-Tarjan 1974] O(n). simple planar graph can have at most 3n edges Remark. Many intractable graph problems can be solved in polytime if the graph is planar; many tractable graph problems can be solved faster if the graph is planar. 49 Planar Graph 3-Colorability Q. Is this planar graph 3-colorable? 50 Planar 3-Colorability and Graph 3-Colorability Claim. PLANAR-3-COLOR P PLANAR-GRAPH-3-COLOR. Pf sketch. Create a vertex for each region, and an edge between

regions that share a nontrivial border. 51 Planar Graph 3-Colorability Claim. W is a planar graph such that: In any 3-coloring of W, opposite corners have the same color. Any assignment of colors to the corners in which opposite corners have the same color extends to a 3-coloring of W. Pf. Only 3-colorings of W are shown below (or by permuting colors). 52 Planar Graph 3-Colorability Claim. 3-COLOR P PLANAR-GRAPH-3-COLOR. Pf. Given instance of 3-COLOR, draw graph in plane, letting edges cross. Replace each edge crossing with planar gadget W. In any 3-coloring of W, a a' and b b'. If a a' and b b' then can extend to a 3-coloring of W.

b b a' a a' a b' a crossing gadget W b' 53 Planar Graph 3-Colorability Claim. 3-COLOR P PLANAR-GRAPH-3-COLOR.

Pf. Given instance of 3-COLOR, draw graph in plane, letting edges cross. Replace each edge crossing with planar gadget W. In any 3-coloring of W, a a' and b b'. If a a' and b b' then can extend to a 3-coloring of W. a' a multiple crossings a W W W a' gadget W 54

Planar k-Colorability PLANAR-2-COLOR. Solvable in linear time. PLANAR-3-COLOR. NP-complete. PLANAR-4-COLOR. Solvable in O(1) time. Theorem. [Appel-Haken, 1976] Every planar map is 4-colorable. Resolved century-old open problem. Used 50 days of computer time to deal with many special cases. First major theorem to be proved using computer. False intuition. If PLANAR-3-COLOR is hard, then so is PLANAR-4COLOR and PLANAR-5-COLOR. 55 Polynomial-Time Detour Graph minor theorem. [Robertson-Seymour 1980s] Corollary. There exist an O(n3) algorithm to determine if a graph can be embedded in the torus in such a way that no two edges cross. Pf of theorem. Tour de force. 56 Polynomial-Time Detour

Graph minor theorem. [Robertson-Seymour 1980s] Corollary. There exist an O(n3) algorithm to determine if a graph can be embedded in the torus in such a way that no two edges cross. Mind boggling fact 1. The proof is highly non-constructive! Mind boggling fact 2. The constant of proportionality is enormous! Unfortunately, for any instance G = (V, E) that one could fit into the known universe, one would easily prefer n70 to even constant time, if that constant had to be one of Robertson and Seymour's. - David Johnson Theorem. There exists an explicit O(n) algorithm. Practice. LEDA implementation guarantees O(n3). 57

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