INHERENT LIMITATIONS OF COMPUTER PROGRAMS

CSci 4011 INHERENT LIMITATIONS OF COMPUTER PROGRAMS

HARDEST PROBLEMS IN NP Theorem: A language B is NP-complete if: Definition: A language B is NP-complete if: 1. B  NP 2. A is NP-complete and A ≤PB 2. Every A in NP is poly-time reducible to B (i.e. B is NP-hard) If B is NP-Complete and P  NP, then There is no fast algorithm for B.

REDUCTION STRATEGIES A reduction by restriction shows that the source problem is a special case of the target problem. For example, 3SAT ≤P CNF-SAT because every satisfiable 3CNF is also a satisfiable CNF. Example. Prove 3SAT ≤P 4SAT by restriction. A 3CNF can be converted to an equivalent 4CNF by repeating one literal in each clause.

REDUCTION STRATEGIES A reduction from A to B by local replacement shows how to “translate” between “units” of A and “units” of B. Example. vertex cover “units” are vertices and edges; set cover “units” are elements and sets. Example. CIRCUIT-SAT units are gates, CNFSAT units are clauses, 3SAT units are 3-literal clauses.

GRADUATION A transcript is a set of course numbers a student has taken A major consists of: Pairs: exactly one of which must be taken Lists: at least one course of which must be taken GRADUATION = {〈T,M〉 | a subset of T satisfies M} For example: T = {1901A, 1902B, 1902A, 2011, 4041A, 4061, 4211} M = [1901A,1901B], [1902A,1902B] (4011,4041A,4041B), (4211,4707), (4061)

GRADUATION ∈NP: The subset is a proof that (T,M) ∈GRADUATION. 3SAT ≤PGRADUATION: T = {101, 102, 201, 202, 301, 302} (x1⋁ x2⋁ ¬x3) ∧ (¬x1 ⋁ x2 ⋁ x2) ∧ (¬x2⋁ x3⋁ x1) M = [101, 102], [201, 202], [301, 302] (101,201,302), (101,201,201), (101,202,301)

GRADUATION ∈NP: The subset is a proof that (T,M) ∈GRADUATION. 3SAT ≤PGRADUATION: Let  = C1∧ C2∧ … ∧Cm have variables x1…xk For each xi: add classes i01 and i02 to T. add pair (i01,i02) to M. For each Cj, we add a triple to M: if xi is a literal in Cj, the triple includes i01. if ¬xiis a literal in Cj, the triple includes i02.

UHAMPATH No HAM PATH Undirected HAM PATH B A B A C D C D E F E F UHAMPATH = {〈G,s,t〉 | G is an undirected graph with a Hamiltonian path from s to t}

HAMPATH≤PUHAMPATH Amid Aout B A Ain C D E F

HAMPATH ≤PUHAMPATH B A C D E F

3SAT P SUBSET-SUM We transform a 3-cnf formula  into 〈y1…yn, t〉:   3SAT  〈y1…yn,t〉  SUBSET-SUM The transformation can be done in time polynomial in the length of 

Each variable and each clause result in two yi’s. Each yi will have a digit for each clause and variable. x2 x1 C3 C2 C1 x1 (x1⋁ x2 ⋁ x2) ∧ (¬x1⋁ x2⋁ x2) ∧ (x1 ⋁ ¬x2⋁ x2) x2 C3 C2 C1

REDUCTION STRATEGIES A reduction by component design constructs gadgets that simulate variables and clauses. In particular a reduction must provide: a gadget to force the solution to the target instance to choose either x or ¬x a gadget to force the solution to the target instance to choose one literal from each clause a gadget to force the solution to the target instance to satisfy every clause

e f b a g d c K-CLIQUES

3SAT P CLIQUE We transform a 3-cnf formula  into 〈G,k〉 such that   3SAT  〈G,k〉  CLIQUE The transformation can be done in time polynomial in the length of 

x1 x2 x2 x1 x1 x1 x2 x2 x2 (x1  x1  x2)  (x1  x2  x2) (x1  x2  x2) k = #clauses

x1 x1 x2 x2 x2 x2 x1 x1 x1 x2 x2 x1 (x1  x1  x1)  (x1  x1  x2) (x2  x2  x2) (x2  x2  x1)

3SAT P CLIQUE We transform a 3-cnf formula  into 〈G,k〉 such that   3SAT  〈G,k〉  CLIQUE If  has k clauses, we create a graph with k clusters of 3 nodes each. Each cluster corresponds to a clause. Each node in a cluster is labeled with a literal from the clause. We do not connect any nodes in the same cluster We connect nodes in different clusters whenever they are not contradictory

3SAT P CLIQUE We transform a 3-cnf formula  into 〈G,k〉 such that   3SAT  〈G,k〉  CLIQUE  is satisfiable iff there is an assignment that makes at least one literal true in each clause. The nodes corresponding to these k true literals do not contradict each other, so they form a k- clique in G. A k-clique in G must have one node from each cluster. Setting the corresponding literals to true will satisfy 

e e b b a a d d c c VERTEX COVER Theorem: VERTEX-COVER is NP-Complete (1) VERTEX-COVER  NP (2) CLIQUE ≤P IND-SET ≤P VERTEX-COVER

e f b a g d i c h HAMILTONIAN PATHS

HAMPATH = { 〈G,s,t〉 | G is a graph with a Hamiltonian path from s to t } Theorem: HAMPATH is NP-Complete (1) HAMPATH  NP (2) 3SAT P HAMPATH

3SAT P HAMPATH We transform a 3-cnf formula  into 〈G,s,t〉 so that   3SAT  〈G,s,t〉  HAMPATH The transformation can be done in time polynomial in the length of 

x1 x1 x2 x2 s t (x1Ç x2Ç x2) Æ (:x1Ç x2Ç x2)

x11 x12 x21 x22 x1 x1 x2 x2 c1 s t (x1Ç x2Ç x2) Æ (:x1Ç x2Ç x2)

x11 x15 x14 x22 x21 x12 x24 x25 c1 c2 x1 x2 x2 x1 s t (x1Ç x2Ç x2) Æ (:x1Ç x2Ç x2)

c e e b b b b a a a a d d d d c c 3-COLORING

3SAT P 3COLOR We transform a 3-cnf formula  into a graph G so   3SAT  G  3COLOR The transformation can be done in time polynomial in the length of 

x1 x1 ? T F x2 x2 (x1Ç x2Ç x2)

? T F (x1Ç x2Ç x2) x1 x1 x2 x2 c13 c11 c12 c16 c14 c15

? T F (x1Ç x2Ç x2) Æ (x2 Ç :x1 Ç:x1) x1 x1 x2 x2 c13 c13 c11 c12 c11 c12 c16 c14 c16 c15 c14 c15

MOREGADGETS http://web.mat.bham.ac.uk/R.W.Kaye/minesw/

MOREGADGETS http://arxiv.org/abs/1203.1895

INHERENT LIMITATIONS OF COMPUTER PROGRAMS