Weighted Circuit Satisfiability 18

Independent Setcan be reduced to Weighted Circuit Satisfiability:

x₁ x₂ x₃ x₄ x₆ x₇

Dominating Setcan be reduced toWeighted Circuit Satisfiability:

x₁ x₂ x₃ x₄ x₆ x₇

To expressDominating Set, we need more complicated circuits.

Weighted Circuit Satisfiability

¹⁸

Thedepth of a circuit is the maximum length of a path from an input to the output.

A gate islargeif it has more than2inputs. Theweft of a circuit is the maximum number of large gates on a path from an input to the output.

Independent Set: weft 1, depth3

x₂ x₃ x₄ x₆ x₇ x₁

Dominating Set: weft 2, depth 2

x₁ x₂ x₃ x₄ x₆ x₇

Depth and weft

19

LetC[t,d]be the set of all circuits having weft at most t and depth at mostd.

Definition

A problemP is in the class W[t]if there is a constant d and a parameterized reduction from P toWeighted Circuit SatisfiabilityofC[t,d].

We have seen thatIndependent Set is inW[1]and Dominating Setis inW[2].

Fact: Independent Setis W[1]-complete.

Fact: Dominating SetisW[2]-complete.

If anyW[1]-complete problem is FPT, then FPT=W[1]andevery problem inW[1] is FPT.

If anyW[2]-complete problem is inW[1], thenW[1]=W[2].

⇒If there is a parameterized reduction from Dominating Setto Independent Set, thenW[1]=W[2].

The W-hierarchy

20

LetC[t,d]be the set of all circuits having weft at most t and depth at mostd.

Definition

A problemP is in the class W[t]if there is a constant d and a parameterized reduction from P toWeighted Circuit SatisfiabilityofC[t,d].

We have seen thatIndependent Set is inW[1]and Dominating Setis inW[2].

Fact: Independent Setis W[1]-complete.

Fact: Dominating SetisW[2]-complete.

If anyW[1]-complete problem is FPT, thenFPT=W[1]andevery problem inW[1] is FPT.

If anyW[2]-complete problem is inW[1], thenW[1]=W[2].

⇒If there is a parameterized reduction from Dominating Setto Independent Set, thenW[1]=W[2].

The W-hierarchy

20

Weftis a term related to weaving cloth: it is the thread that runs from side to side in the fabric.

Weft

21

TypicalNP-hardness proofs: reduction from e.g.,Clique or 3SAT, representing each vertex/edge/variable/clause with a gadget.

v1 v2 v3 v4 v5 v6

C1 C2 C3 C4

Usually does not work for parameterized reductions: cannot afford the parameter increase.

Types of parameterized reductions:

Reductions keeping the structure of the graph. Clique⇒Independent Set

Independent Seton regular graphs ⇒Partial Vertex Cover

Reductions with vertex representations.

Multicolored Independent Set⇒Dominating Set Reductions with vertex and edge representations.

Parameterized reductions

22

TypicalNP-hardness proofs: reduction from e.g.,Clique or 3SAT, representing each vertex/edge/variable/clause with a gadget.

v1 v2 v3 v4 v5 v6

C1 C2 C3 C4

Usually does not work for parameterized reductions: cannot afford the parameter increase.

Types of parameterized reductions:

Reductions keeping the structure of the graph.

Clique⇒Independent Set

Independent Seton regular graphs ⇒Partial Vertex Cover

Reductions with vertex representations.

Multicolored Independent Set⇒Dominating Set Reductions with vertex and edge representations.

Parameterized reductions

22

Balanced Vertex Separator: Given a graphG and an inte-gerk, find a setS of at mostkvertices such that every component ofG −S has at most|V(G)|/2vertices.

Theorem

Balanced Vertex Separatorparameterized byk is W[1]-hard.

Balanced Vertex Separator

²³

Theorem

Balanced Vertex Separatorparameterized byk is W[1]-hard.

Proof: By reduction from Clique.

G G⁰

K`

|V(G)|=11,|E(G)|=22, k=4, `=3

We formG⁰ by

Subdividing every edge ofG.

Making the original vertices of G a clique.

Adding an `-clique for `=|V(G)|+|E(G)| −2(k+ ^k₂ ) (assuming the graph is sufficiently large, we have`≥1).

We have |V(G⁰)|= 2|V(G)|+2|E(G)| −2(k + ^k₂

) and the “big component” ofG⁰ has size|V(G)|+|E(G)|.

Balanced Vertex Separator

²³

Theorem

Balanced Vertex Separatorparameterized byk is W[1]-hard.

Proof: By reduction from Clique.

G G⁰

vertices, reducing the size of the big component to|V(G)|+|E(G)| −(k+ ^k₂

) =|V(G⁰)|/2.

Balanced Vertex Separator

²³

Theorem

Balanced Vertex Separatorparameterized byk is W[1]-hard.

Proof: By reduction from Clique.

G G⁰

⇐: We need to reduce the size of the large component of G⁰ by k+ ^k₂

by removingk vertices. This is only possible if thek vertices cut away ^k₂

isolated vertices, i.e., the k-vertices form a k-clique in G.

Balanced Vertex Separator

²³

List Coloringis a generalization of ordinary vertex coloring:

given a graph G,

a set of colors C, and

a list L(v)⊆C for each vertex v,

the task is to find a coloringc where c(v)∈L(v) for every v. Theorem

Vertex Coloringis FPT parameterized by treewidth.

However, list coloring is more difficult:

Theorem

List Coloringis W[1]-hard parameterized by treewidth.

List Coloring

²⁴

Theorem

List Coloringis W[1]-hard parameterized by treewidth.

Proof: By reduction from Multicolored Independent Set. Let G be a graph with color classes V₁,. . .,V_k.

Theorem

List Coloringis W[1]-hard parameterized by treewidth.

Proof: By reduction from Multicolored Independent Set. Let G be a graph with color classes V₁,. . .,V_k.

Key idea

Represent the k vertices of the solution withk gadgets.

Connect the gadgets in a way that ensures that the represented values arecompatible.

Vertex representation

26

Odd Set: Given a set systemF over a universeU and an integer k, find a setS of at mostk elements such that|S∩F|is odd for everyF ∈ F.

Theorem

Odd Setis W[1]-hard parameterized byk.

First try: Reduction fromMulticolored Independent Set. LetU =V₁∪. . .V_k and introduce each set V_i intoF.

⇒The solution has to contain exactly one element from each V_i.

Ifxy ∈E(G), how can we express thatx ∈V_i andy ∈V_j cannot be selected simultaneously? Seems difficult:

introducing {x,y}into F forces thatexactly oneof x andy appears in the solution,

introducing {x} ∪(V_j \ {y}) intoF forces that either bothx andy or noneofx andy appear in the solution.

Odd Set

²⁷

Theorem

Odd Setis W[1]-hard parameterized byk.

First try: Reduction fromMulticolored Independent Set. LetU =V₁∪. . .V_k and introduce each set V_i intoF.

⇒The solution has to contain exactly one element from each V_i.

V1 V2 V3 V4 V5

Ifxy ∈E(G), how can we express thatx ∈V_i andy ∈V_j cannot be selected simultaneously?

Seems difficult:

introducing {x,y}into F forces thatexactly oneof x andy appears in the solution,

introducing {x} ∪(V_j \ {y}) intoF forces that either bothx andy or noneofx andy appear in the solution.

Odd Set

²⁷

Theorem

Odd Setis W[1]-hard parameterized byk.

First try: Reduction fromMulticolored Independent Set. LetU =V₁∪. . .V_k and introduce each set V_i intoF.

⇒The solution has to contain exactly one element from each V_i.

V1 V2 V3 V4 V5

x y

Ifxy ∈E(G), how can we express thatx ∈V_i andy ∈V_j cannot be selected simultaneously? Seems difficult:

introducing {x,y}into F forces thatexactly oneof x andy appears in the solution,

introducing {x} ∪(V_j \ {y}) intoF forces that either bothx andy or noneofx andy appear in the solution.

Odd Set

²⁷

Theorem

Odd Setis W[1]-hard parameterized byk.

First try: Reduction fromMulticolored Independent Set. LetU =V₁∪. . .V_k and introduce each set V_i intoF.

⇒The solution has to contain exactly one element from each V_i.

V1 V2 V3 V4 V5

x y

Ifxy ∈E(G), how can we express thatx ∈V_i andy ∈V_j cannot be selected simultaneously? Seems difficult:

introducing {x,y}into F forces thatexactly oneof x andy appears in the solution,

introducing {x} ∪(V_j \ {y}) intoF forces that either bothx andy or noneofx andy appear in the solution.

Odd Set

²⁷

Theorem

Odd Setis W[1]-hard parameterized byk.

First try: Reduction fromMulticolored Independent Set. LetU =V₁∪. . .V_k and introduce each set V_i intoF.

⇒The solution has to contain exactly one element from each V_i.

V1 V2 V3 V4 V5

x y

Ifxy ∈E(G), how can we express thatx ∈V_i andy ∈V_j cannot be selected simultaneously? Seems difficult:

introducing {x,y}into F forces thatexactly oneof x andy appears in the solution,

introducing {x} ∪(V_j \ {y}) intoF forces that either bothx andy or noneofx andy appear in the solution.

Odd Set

²⁷

Theorem

Odd Setis W[1]-hard parameterized byk.

First try: Reduction fromMulticolored Independent Set. LetU =V₁∪. . .V_k and introduce each set V_i intoF.

⇒The solution has to contain exactly one element from each V_i.

V1 V2 V3 V4 V5

x y

Ifxy ∈E(G), how can we express thatx ∈V_i andy ∈V_j cannot be selected simultaneously? Seems difficult:

introducing {x,y}into F forces thatexactly oneof x andy appears in the solution,

introducing {x} ∪(V_j \ {y}) intoF forces that either bothx andy or noneofx andy appear in the solution.

Odd Set

²⁷

Reduction fromMulticolored Clique. U :=Sk

i=1V_i ∪S

1≤i<j≤kE_i,j. k⁰:=k+ ^k₂

.

Let F containV_i (1≤i ≤k) and E_i,j (1≤i <j ≤k).

For every v ∈V_i andx 6=i, we introduce the sets: (V_i \ {v})∪ {every edge from E_i,x with endpoint v} (V_i \ {v})∪ {every edge from E_x,i with endpoint v}

E1,2 E1,3 E1,4 E2,3 E2,4 E3,4

V1 V2 V3 V4

Odd Set

²⁸

Reduction fromMulticolored Clique. U :=Sk

i=1V_i ∪S

1≤i<j≤kE_i,j. k⁰:=k+ ^k₂

.

Let F containV_i (1≤i ≤k) and E_i,j (1≤i <j ≤k).

For every v ∈V_i andx 6=i, we introduce the sets:

(V_i \ {v})∪ {every edge from E_i,x with endpoint v} (V_i \ {v})∪ {every edge from E_x,i with endpoint v}

E1,2 E1,3 E1,4 E2,3 E2,4 E3,4

V1 V2 V3 V4

Odd Set

²⁸

Reduction fromMulticolored Clique. U :=Sk

i=1V_i ∪S

1≤i<j≤kE_i,j. k⁰:=k+ ^k₂

.

Let F containV_i (1≤i ≤k) and E_i,j (1≤i <j ≤k).

For every v ∈V_i andx 6=i, we introduce the sets:

(V_i \ {v})∪ {every edge from E_i,x with endpoint v} (V_i \ {v})∪ {every edge from E_x,i with endpoint v}

E1,2 E1,3 E1,4 E2,3 E2,4 E3,4

V1 V2 V3 V4

Odd Set

²⁸

Reduction fromMulticolored Clique. U :=Sk

i=1V_i ∪S

1≤i<j≤kE_i,j. k⁰:=k+ ^k₂

.

Let F containV_i (1≤i ≤k) and E_i,j (1≤i <j ≤k).

For every v ∈V_i andx 6=i, we introduce the sets:

(V_i \ {v})∪ {every edge from E_i,x with endpoint v} (V_i \ {v})∪ {every edge from E_x,i with endpoint v}

E1,2 E1,3 E1,4 E2,3 E2,4 E3,4

V1 V2 V3 V4

Odd Set

²⁸

Reduction fromMulticolored Clique.

For every v ∈V_i andx 6=i, we introduce the sets:

(V_i \ {v})∪ {every edge from E_i,x with endpoint v} (V_i \ {v})∪ {every edge from E_x,i with endpoint v} v ∈V_i selected ⇐⇒ edges with endpointv are selected

fromE_i,x andE_x,i

v_i ∈V_i selected

v_j ∈V_j selected ⇐⇒ edgev_iv_j is selected inE_i,x

E1,2 E1,3 E1,4 E2,3 E2,4 E3,4

V1 V2 V3 V4

Odd Set

²⁸

Reduction fromMulticolored Clique.

For every v ∈V_i andx 6=i, we introduce the sets:

(V_i \ {v})∪ {every edge from E_i,x with endpoint v} (V_i \ {v})∪ {every edge from E_x,i with endpoint v} v ∈V_i selected ⇐⇒ edges with endpointv are selected

fromE_i,x andE_x,i v_i ∈V_i selected

v_j ∈V_j selected ⇐⇒ edgev_iv_j is selected inE_i,x

E1,2 E1,3 E1,4 E2,3 E2,4 E3,4

V1 V2 V3 V4

Odd Set

²⁸

Key idea

Represent the vertices of the clique by k gadgets.

Represent the edges of the clique by ^k₂

gadgets.

Connect edge gadgetE_i,j to vertex gadgetsV_i andV_j such that if E_i,j represents the edge betweenx ∈V_i andy ∈V_j, then it forcesV_i to x andV_j toy.

Vertex and edge representation

29

The following problems areW[1]-hard:

Odd Set

Exact Odd Set(find a set of size exactly k . . . ) Exact Even Set

Unique Hitting Set

(at mostk elements that hit each set exactly once) Exact Unique Hitting Set

(exactly k elements that hit each set exactly once)

Open question:

?

^{Even Set}: Given a set systemF and an integerk, find a nonempty setS of at mostk elements such|F∩S|is even for every F ∈ F.

Variants of Odd Set

³⁰

The following problems areW[1]-hard:

Odd Set

Exact Odd Set(find a set of size exactly k . . . ) Exact Even Set

Unique Hitting Set

(at mostk elements that hit each set exactly once) Exact Unique Hitting Set

(exactly k elements that hit each set exactly once) Open question:

?

^{Even Set}: Given a set systemF and an integerk, find a nonempty setS of at mostk elements such|F∩S|is even for every F ∈ F.

Variants of Odd Set

³⁰

By parameterized reductions, we can show that lots of

parameterized problems are at least as hard asClique, hence unlikely to be fixed-parameter tractable.

Connection with Turing machines gives some supporting evidence for hardness (only of theoretical interest).

TheW-hierarchy classifies the problems according to hardness (only of theoretical interest).

Important trick inW[1]-hardness proofs: vertex and edge representations.

Summary

31

In document Lower bounds (Pldal 33-64)