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 ⊆ f0; 1g∗. OR=(L) 1 t j Input: A list hx ; : : : ; x i of binary strings and a parameter k := maxj jx j Question: Is some xj 2 L? AND=(L) 1 t j Input: A list hx ; : : : ; x i of binary strings and a parameter k := maxj jx j Question: Is every xj 2 L? i j We will usually assume that jx j = jx j, 8i;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 jV (G)j Let us take the NP -complete problem L := fhGi j tw(G) ≤ 2 g. If all the input graphs Gi have the same number of vertices, then R : AND=(L) ! k-Treewidth may be defined by: [ jG1j R(hG ;:::;G i) = G ; 1 t i 2 i We see that jG j 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 jR(¯x)j ≤ 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 jR(¯x)j ≤ 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 2 P=poly, if n 9fCn : f0; 1g ! f0; 1ggn>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 2 P=poly, also BP P ⊂ 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 2 L , 9yA(x; y) = 1 Definition 8. We say that L is in NP=poly if there exists a sequence of polynomial sized circuit fCn(x; y)gn on n + poly(n) input bits such that: x 2 Ln , 9yCn(x; y) = 1 Recall that coNP = fL j L¯ 2 NP g, thus coNP=poly = fL j L¯ 2 NP=polyg. 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(9) 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 poly- nomial time to a d-Round Game(9). 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 8d>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 8d>2Σd = Σ2 Theorem 15 (Yap '83). Assuming NP ⊆ coNP=poly, we have p p 8d>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) 2 L0 , [xj 2 L]: j Furthermore, suppose that we have the output-size bound jR(¯x)j ≤ 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¯ 2 NP=poly As L 2 coNP=poly , L¯ 2 NP=poly. We want to use R to build a non-uniform proof system witnessing membership in L¯. Let r = jR(x1; : : : ; xt(n))j. n 1 t(n) j Definition 17. Let x 2 f0; 1g . We say thatx ¯ = (x ; : : : ; x ) contains x if 9jx = x . Definition 18. We define the shadow of x 2 f0; 1gn by: shadow(x) := fz j z = R(¯x) ^ x¯ contains xg ⊆ f0; 1gr Note that if z2 = L0 is the shadow of x, then x2 = L. ¯0 Claim 19 ([3]). There exists a set Z ⊆ Lr, with jZj ≤ poly(n); such that for every x 2 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 2 Z and check whether R(¯x) = z if so, then we know that x 2 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..