Kernel-Size Lower Bounds: the Evidence from Complexity Theory

Lecture 6 (11.04.2013) Author: Mateusz Baranowski Kernel-Size Lower Bounds: The Evidence from Complexity Theory.

During the previous lecture we presented the main idea of Cross-Composition, a framework for proving that a problem does not have a polynomial kernel. We also presented a number of examples showing the use of this framework. In this lecture we will focus on the tools which were used for inventing Cross-Compositions.

1 Conjecture statements and their consequences

Let L ⊆ {0, 1}∗.

OR=(L) 1 t j Input: A list hx , . . . , x i of binary strings and a parameter k := maxj |x | Question: Is some xj ∈ L?

AND=(L) 1 t j Input: A list hx , . . . , x i of binary strings and a parameter k := maxj |x | Question: Is every xj ∈ L? i j We will usually assume that |x | = |x |, ∀i,j.

Claim 1 (OR-conjecture, [1]). The problem OR=(L) does not have a polynomial kernel for suﬃ- ciently hard languages L, in particular for NP -hard languages.

Assuming the OR-conjecture the following problems do not have a polynomial kernel:

• k-Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour,

• k-Graph Minor Order Test, k-Bounded Treewidth Subgraph Test,

• k-Planar Graph Subgraph Test, k-Planar Graph Induced Subgraph Test,

• (k, σ)-Short Nondeterministic Turing Machine Computation,

• w-Independent Set, w-Clique and w-Dominating Set

• ...

Claim 2 (AND-conjecture, [1]). The problem AND=(L) does not have a polynomial kernel for suﬃciently hard languages L, in particular for NP -hard languages.

Similarly to the OR-conjecture, assuming the AND-conjecture we can prove that the following do not have a polynomial kernel:

• k-Cutwidth, k-Modified Cutwidth, k-Search Number,

• k-Pathwidth, k-Treewidth, k-Branchwidth, • k-Gate Matrix Layout, k-Front Size,

• w-3-Coloring and w-3-Domatic Number. Theorem 3. Let L be NP -complete, P a parametrized problem, and suppose there is a polynomial time reduction R from an instance x¯ of OR=(L) (AND=(L)) to an equivalent instance of P , with

k(R(¯x)) ≤ poly(k(¯x)).

Then, if P has some poly(k)-kernelization A, so does OR=(L) (AND=(L)) (and the OR/AND- conjecture fails). Example 1: k-Treewidth Input: Graph G and a positive integer k. Question: Does G have treewidth ≤ k? Note that computing the exact treewidth is NP -hard. S Let tw(G) be the treewidth of G. Let G1,...,Gt be graphs. It is known that tw( i Gi) = maxi(tw(Gi)), thus we get [ ^ tw( Gi) ≤ k ⇔ [tw(Gi) ≤ k] i i |V (G)| Let us take the NP -complete problem L := {hGi | tw(G) ≤ 2 }. If all the input graphs Gi have the same number of vertices, then R : AND=(L) → k-Treewidth may be deﬁned by: [ |G1| R(hG ,...,G i) = G , 1 t i 2 i We see that |G | 1 = k(R(¯x)) ≤ poly(k(¯x)) 2 Thus using the AND-conjecture k-Treewidth cannot have a kernel of polynomial size. (The as- sumption of equal number of vertices can be circumvented using some padding, or using polynomial- time equivalence relation and sorting Gis according to equivalence classes of the relation.) By carefully analyzing [3] we can deduce the following theorem.

Theorem 4 (implicit, [3]). Assume NP * coNP/poly. If L is NP -complete and t(k) ≤ poly(k), t(·) no deterministic poly-time reduction R from OR=(L) to any other problem can achieve output size |R(¯x)| ≤ O(t log t), where t = t(k), k = k(¯x). As we can see this theorem can be used not only to prove that polynomial kernels are not possible, but also show tight lower bounds for polynomial kernels. Example 2: 100 1 t(k) t(·) 100 Let t(k) := k , then input (x , . . . , x ) to OR=(L) , of size k · k , cannot be reduced to a kernel of size ≤ k100. All previous results focused on deterministic algorithms, but we can extend our framework to work for randomized algorithms using the following theorem: Theorem 5 (Drucker, [2]). Assume NP * coNP/poly. If L is NP -complete, t(k) ≤ poly(k). No t(·) t(·) probabilistic poly-time reduction R from OR=(L) , AND=(L) to any problem, with P[success ≥ 0.99], can achieve |R(¯x)| ≤ t.

2 Background from complexity theory

We will show the idea behind the NP * coNP/poly conjecture. We will use the ordinary model of Boolean circuits with ∧, ∨, ¬ gates and bounded fan-in.

Deﬁnition 6. We say that a problem L has polynomial size circuits, and write L ∈ P/poly, if

n ∃{Cn : {0, 1} → {0, 1}}n>0, such that size(Cn) ≤ poly(n),Cn(x) ≡ L(x).

Notice that we do not assume anything about Cn (it can even be uncomputable). Thus, let L ⊆ 1∗, then L ∈ P/poly, also BPP ⊂ P/poly.

Deﬁnition 7. Recall that a decision problem L is NP if there exists a polynomial time algorithm A(x, y) on n + poly(n) input bits such that:

x ∈ L ⇔ ∃yA(x, y) = 1

Deﬁnition 8. We say that L is in NP/poly if there exists a sequence of polynomial sized circuit {Cn(x, y)}n on n + poly(n) input bits such that:

x ∈ Ln ⇔ ∃yCn(x, y) = 1

Recall that coNP = {L | L¯ ∈ NP }, thus coNP/poly = {L | L¯ ∈ NP/poly}.

Deﬁnition 9. Let C(y1, y2, . . . , yk) be a circuit with k input blocks. Consider a game where two players take turns setting y1, . . . , yk. The ﬁrst player (P1) wins when C(y1, y2, . . . , yk) = 1, otherwise the second player (P2) wins. We call this game a 2-player game.

d-Round Game(∃) Input: A d-block circuit C(y1, . . . , yd). Question: Can the ﬁrst player force a win (C = 1) in a 2-player game?

p Deﬁnition 10. We deﬁne the complexity class Σd as the set of languages Karp-reducible in polynomial time to a d-Round Game(∃).

We refer to this class as the “dth level of Polynomial Hierarchy”. Note the following facts from complexity theory.

Corollary 11. p NP = Σ1 Corollary 12. p p Σd ⊆ Σd+1 There is a common belief that there is no eﬃcient reduction of a (d + 1)-Round Game to an equivalent d-Round Game, which formally we state as: Claim 13. p p ∀d>0Σd 6= Σd+1 This theory leads us to the theorems about the “collapse of the Polynomial Hierarchy”.

Theorem 14 (Karp-Lipton ’82). Assuming NP ⊆ P/poly, we have

p p ∀d>2Σd = Σ2 Theorem 15 (Yap ’83). Assuming NP ⊆ coNP/poly, we have

p p ∀d>3Σd = Σ3 This fact is frequently referred to as the “collapse of the Polynomial Hierarchy to its third level”.

3 Proof of the main result

Theorem 16 ([3]). Let L be an NP -complete language, L0 another language, and t(n) ≤ poly(n). Suppose there is a polynomial time reduction

R(¯x) = R(x1, . . . , xt(n)) taking t(n) inputs of length n, and producing output such that _ R(¯x) ∈ L0 ⇔ [xj ∈ L]. j

Furthermore, suppose that we have the output-size bound

|R(¯x)| ≤ O(t(n) log t(n))

Then NP ⊆ coNP/poly.

Proof. Since L is NP -complete, to prove the theorem it is enough to show that

L¯ ∈ NP/poly

As L ∈ coNP/poly ⇔ L¯ ∈ NP/poly. We want to use R to build a non-uniform proof system witnessing membership in L¯. Let r = |R(x1, . . . , xt(n))|. n 1 t(n) j Definition 17. Let x ∈ {0, 1} . We say thatx ¯ = (x , . . . , x ) contains x if ∃jx = x . Definition 18. We define the shadow of x ∈ {0, 1}n by:

shadow(x) := {z | z = R(¯x) ∧ x¯ contains x} ⊆ {0, 1}r

Note that if z∈ / L0 is the shadow of x, then x∈ / L. ¯0 Claim 19 ([3]). There exists a set Z ⊆ Lr, with

|Z| ≤ poly(n), such that for every x ∈ L¯n, shadow(x) ∩ Z 6= ∅. Since R is a compression of L then some instances of L0 are in the image of many sequencesx ¯, hence is in many shadows. We can collect these ,,popular” instances to hit all shadows (of L¯n). We will use Z as the advice. The machine will nondeterministically selectx ¯ containing x and z ∈ Z and check whether R(¯x) = z if so, then we know that x ∈ L¯.

References

[1] Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. Technical report, Lect. Notes Comput. Sci, 2007.

[2] Andrew Drucker. Electronic colloquium on computational complexity, report no. 112 (2012) new limits to classical and quantum instance compression, 2012.

[3] Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct pcps for np. Electronic Colloquium on Computational Complexity (ECCC), 2007.