When can an FPT decision algorithm be used to count? January 2016 Kitty Meeks
Deciding, counting and enumerating DECISION Is there a witness? 2/19
Deciding, counting and enumerating DECISION Is there a witness? APPROX COUNTING Approximately how many witnesses? 2/19
Deciding, counting and enumerating DECISION Is there a witness? APPROX COUNTING Approximately how many witnesses? EXACT COUNTING Exactly how many witnesses? 2/19
Deciding, counting and enumerating EXTRACTION DECISION Identify a single Is there a witness? witness APPROX COUNTING Approximately how many witnesses? EXACT COUNTING Exactly how many witnesses? 2/19
Deciding, counting and enumerating EXTRACTION DECISION Identify a single Is there a witness? witness APPROX COUNTING UNIFORM SAMPLING Approximately how Pick a single witness many witnesses? uniformly at random EXACT COUNTING Exactly how many witnesses? 2/19
Deciding, counting and enumerating EXTRACTION DECISION Identify a single Is there a witness? witness APPROX COUNTING UNIFORM SAMPLING Approximately how Pick a single witness many witnesses? uniformly at random EXACT COUNTING ENUMERATION Exactly how many List all witnesses witnesses? 2/19
If we can decide, we can find a witness 3/19
If we can decide, we can find a witness 3/19
If we can decide, we can find a witness 3/19
If we can decide, we can find a witness 3/19
If we can decide, we can find a witness 3/19
If we can decide, we can find a witness 3/19
If we can decide, we can find a witness 3/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can decide, we can find a witness Theorem (Björklund, Kaski and Kowalik, 2014) There exists an algorithm that extracts a witness using at most n � � 2 k log 2 k + 2 queries to a deterministic inclusion oracle. 4/19
If we can count approximately, we can decide 5/19
If we can count approximately, we can decide ... at least with high probability. An FPRAS for a counting problem Π is a randomised approximation scheme that takes an instance I of Π (with | I | = n ), and numbers ǫ > 0 and 0 < δ < 1, and in time poly ( n , 1 /ǫ, log ( 1 /δ )) outputs a rational number z such that P [( 1 − ǫ )Π( I ) ≤ z ≤ ( 1 + ǫ )Π( I )] ≥ 1 − δ. 5/19
If we can count approximately, we can decide ... at least with high probability. An FPRAS for a counting problem Π is a randomised approximation scheme that takes an instance I of Π (with | I | = n ), and numbers ǫ > 0 and 0 < δ < 1, and in time poly ( n , 1 /ǫ, log ( 1 /δ )) outputs a rational number z such that P [( 1 − ǫ )Π( I ) ≤ z ≤ ( 1 + ǫ )Π( I )] ≥ 1 − δ. Set ǫ < 1 2 , and we will distinguish between 0 and at least 1 with probability at least 1 − δ . 5/19
Uniform sampling is harder than finding a witness G EN C YCLE Input: A directed graph G. Output: A cycle selected uniformly, at random, from the set of all directed cycles of G . Theorem (Jerrum, Valiant, Vazirani, 1986) Suppose there exists a polynomial time bounded Probabilistic Turing Machine which solves the problem G EN C YCLE . Then NP = RP. 6/19
Self-reducibility A relation R ⊆ Σ ∗ × Σ ∗ is self-reducible if and only if: there exists a polynomial time computable function g ∈ Σ ∗ → N such that xRy = ⇒ | y | = g ( x ) ; there exist polynomial time computable functions ψ ∈ Σ ∗ × Σ ∗ → Σ ∗ and σ ∈ Σ ∗ → N satisfying: σ ( x ) = O ( log | x | ) g ( x ) > 0 = ⇒ σ ( x ) > 0 ∀ x ∈ Σ ∗ | ψ ( x , w ) | ≤ | x | ∀ x , w ∈ Σ ∗ , and such that, for all x ∈ Σ ∗ , y = y 1 . . . y n ∈ Σ ∗ , � x , y 1 . . . y n � ∈ R ⇐ ⇒ � ψ ( x , y 1 . . . y σ ( x ) ) , y σ ( x )+ 1 . . . y n � ∈ R . 7/19
Self-reducibility A relation R ⊆ Σ ∗ × Σ ∗ is self-reducible if and only if: there exists a polynomial time computable function g ∈ Σ ∗ → N such that xRy = ⇒ | y | = g ( x ) ; there exist polynomial time computable functions ψ ∈ Σ ∗ × Σ ∗ → Σ ∗ and σ ∈ Σ ∗ → N satisfying: σ ( x ) = O ( log | x | ) g ( x ) > 0 = ⇒ σ ( x ) > 0 ∀ x ∈ Σ ∗ | ψ ( x , w ) | ≤ | x | ∀ x , w ∈ Σ ∗ , and such that, for all x ∈ Σ ∗ , y = y 1 . . . y n ∈ Σ ∗ , � x , y 1 . . . y n � ∈ R ⇐ ⇒ � ψ ( x , y 1 . . . y σ ( x ) ) , y σ ( x )+ 1 . . . y n � ∈ R . Theorem (Jerrum, Valiant, Vazirani, 1986) For self-reducible problems, approximate counting and almost-uniform sampling are polynomial-time inter-reducible. 7/19
Parameterised subgraph problems Let Φ be a family ( φ 1 , φ 2 , . . . ) of functions, such that φ k is a mapping from labelled graphs on k -vertices to { 0 , 1 } . p- I NDUCED S UBGRAPH W ITH P ROPERTY ( Φ ) ( p- ISWP (Φ) ) Input: A graph G = ( V , E ) and an integer k . Parameter: k . Question: Is there a tuple ( v 1 , . . . , v k ) ∈ V k such that v 1 , . . . , v k are all distinct and φ k ( G [ v 1 , . . . , v k ]) = 1? 8/19
Parameterised subgraph problems Let Φ be a family ( φ 1 , φ 2 , . . . ) of functions, such that φ k is a mapping from labelled graphs on k -vertices to { 0 , 1 } . p- I NDUCED S UBGRAPH W ITH P ROPERTY ( Φ ) ( p- ISWP (Φ) ) Input: A graph G = ( V , E ) and an integer k . Parameter: k . Question: Is there a tuple ( v 1 , . . . , v k ) ∈ V k such that v 1 , . . . , v k are all distinct and φ k ( G [ v 1 , . . . , v k ]) = 1? p- MISWP (Φ) Input: A graph G = ( V , E ) , an integer k and a colouring f : V → { 1 , . . . , k } . Parameter: k . Question: Is there a tuple ( v 1 , . . . , v k ) ∈ V k such that { f ( v 1 ) , . . . , f ( v k ) } = { 1 , . . . , k } and φ k ( G [ v 1 , . . . , v k ]) = 1? 8/19
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