# The Constant Inapproximability of the Parameterized Dominating Set Problem

aa r X i v : . [ c s . CC ] N ov The Constant Inapproximability of the ParameterizedDominating Set Problem

Yijia Chen

School of Computer ScienceFudan [email protected]

Bingkai Lin

Department of Computer ScienceUniversity of [email protected]

Abstract

We prove that there is no fpt-algorithm that can approximate the dominating set problem withany constant ratio, unless FPT = W [1] . Our hardness reduction is built on the second author’srecent W [1] -hardness proof of the biclique problem [22]. This yields, among other things, a proofwithout the PCP machinery that the classical dominating set problem has no polynomial timeconstant approximation under the exponential time hypothesis.

1. Introduction

The dominating set problem, or equivalently the set cover problem, was among the ﬁrst problemsproved to be NP-hard [20]. Moreover, it has been long known that the greedy algorithm achieves anapproximation ratio ≈ ln n [19, 30, 23, 8, 29]. And after a sequence of papers (e.g. [24, 28, 16, 1, 10]),this is proved to be best possible. In particular, Raz and Safra [28] showed that the dominating setproblem cannot be approximated with ratio c · log n for some constance c ∈ N unless P = NP [28].Under a stronger assumption NP DTIME (cid:0) n O ( log log n ) (cid:1) Feige proved that no approximation within (1 − ε ) ln n is feasible [16]. Finally Dinur and Steuer established the same lower bound assumingonly P = NP [10]. However, it is important to note that the approximation ratio ln n is measured interms of the size of an input graph G , instead of γ ( G ) , i.e., the size of its minimum dominating set. Asa matter of fact, the standard examples for showing the Θ( log n ) greedy lower bound have constant-size dominating sets. Thus, the size of the greedy solutions cannot be bounded by any function of γ ( G ) . So the question arises whether there is an approximation algorithm A that always outputs adominating set whose size can be bounded by ρ ( γ ( G )) · γ ( G ) , where the function ρ : N → N isknown as the approximation ratio of A . The constructions in [16, 1] indeed show that we can rule out ρ ( x ) ≤ ln x . To the best of our knowledge, it is not known whether this bound is tight. For instance,it is still conceivable that there is a polynomial time algorithm that always outputs a dominating set ofsize at most γ ( G ) .Other than looking for approximate solutions, parameterized complexity [12, 17, 27, 13, 9] ap-proaches the dominating set problem from a different perspective. With the expectation that in practicewe are mostly interested in graphs with relatively small dominating sets, algorithms of running time γ ( G ) · | G | O (1) can still be considered efﬁcient. Unfortunately, it turns out that the parameterizeddominating set problem is complete for the second level of the so-called W-hierarchy [11], and thusﬁxed-parameter intractable unless FPT = W [2] . So one natural follow-up question is whether theproblem can be approximated in fpt-time. More precisely, we aim for an algorithm with runningtime f ( γ ( G )) · | G | O (1) which always outputs a dominating set of size at most ρ ( γ ( G )) · γ ( G ) . Here,1 : N → N is an arbitrary computable function. The study of parameterized approximability wasinitiated in [4, 6, 14]. Compared to the classical polynomial time approximation, the area is still in itsvery early stage with few known positive and even less negative results. Our results.

We prove that any constant-approximation of the parameterized dominating set problemis W [1] -hard.

Theorem 1.1.

For any constant c ∈ N there is no fpt -algorithm A such that on every input graph G the algorithm A outputs a dominating set of size at most c · γ ( G ) , unless FPT = W [1] (which impliesthat the exponential time hypothesis ( ETH ) fails).

In the above statement, clearly we can replace “fpt-algorithm” by “polynomial time algorithm,”thereby obtaining the classical constant-inapproximability of the dominating set problem. But let usmention that our result is not comparable to the classical version, even if we restrict ourselves topolynomial time tractability. The assumption FPT = W [1] or ETH is apparently much stronger thanP = NP, and in fact ETH implies NP DTIME (cid:0) n O ( log log n ) (cid:1) used in aforementioned Feige’s result.But on the other hand, our lower bound applies even in case that we know in advance that a givengraph has no large dominating set. Corollary 1.2.

Let β : N → N be a nondecreasing and unbounded computable function. Considerthe following promise problem. M IN -D OMINATING -S ET β Instance:

A graph G = ( V, E ) with γ ( G ) ≤ β ( | V | ) . Solution:

A dominating set D of G . Cost: | D | . Goal: min.

Then there is no polynomial time constant approximation algorithm for M IN -D OMINATING -S ET β ,unless FPT = W [1] . The proof of Theorem 1.1 is crucially built on a recent result of the second author [22] whichshows that the parameterized biclique problem is W [1] -hard. We exploit the gap created in its hardnessreduction (see Section 2.1 for more details). In the known proofs of the classical inapproximabilityof the dominating set problem, one always needs the PCP theorem in order to have such a gap, whichmakes those proofs highly non-elementary. More importantly, it can be veriﬁed that reductions basedon the PCP theorem produce instances with optimal solutions of relatively large size, e.g., a graph G = ( V, E ) with γ ( G ) ≥ | V | Θ(1) . This is inevitable, since otherwise we might be able to solveevery NP-hard problem in subexponential time. As an example, if it is possible to reduce an NP-hard problem to the approximation of M IN -D OMINATING -S ET β for β ( n ) = log log log n , thenby brute-force searching for a minimum dominating set, we are able to solve the problem in time n O ( log log log n ) . It implies NP ⊆ DTIME (cid:0) n O ( log log log n ) (cid:1) . Because of this, Corollary 1.2, and hencealso Theorem 1.1, is unlikely provable following the traditional approach.Using a result of Chen et.al. [5] the lower bound in Theorem 1.1 can be further sharpened. Theorem 1.3.

Assume

ETH holds. Then there is no fpt -algorithm which on every input graph G outputs a dominating set of size at most ε p log ( γ ( G )) · γ ( G ) for every < ε < . elated work. The existing literature on the dominating set problem is vast. The most relevant toour work is the classical approximation upper and lower bounds as explained in the beginning. But asfar as the parameterized setting is concerned, what was known is rather limited.Downey et. al proved that there is no additive approximation of the the parameterized dominatingset problem [15]. In the same paper, they also showed that the independent dominating set problemhas no fpt approximation with any approximation ratio. Recall that an independent dominating set isa dominating set which is an independent set at the same time. With this additional requirement, theproblem is no longer monotone , i.e., a superset of a solution is not necessarily a solution. Thus it isunclear how to reduce the independent dominating set problem to the dominating set problem by anapproximation-preserving reduction.In [7, 18] it is proved under ETH that there is no c p log γ ( G ) -approximation algorithm for thedominating set problem with running time O ( γ ( G ) ( log γ ( G )) d ) | G | O (1) , where c and d are some appro-priate constants. With the additional Projection Game Conjecture due to [26] and some of its furtherstrengthening, the authors of [7, 18] are able to even rule out γ ( G ) c -approximation algorithms withrunning time almost doubly exponential in terms of γ ( G ) . Clearly, these lower bounds are against farbetter approximation ratio than those of Theorem 1.1 and Theorem 1.3, while the drawback is that thedependence of the running time on γ ( G ) is not an arbitrary computable function.The dominating set problem can be understood as a special case of the weighted satisﬁabilityproblem of CNF-formulas, in which all literals are positive. The weighted satisﬁability problemsfor various fragments of propositional logic formulas, or more generally circuits, play very importantroles in parameterized complexity. In particular, they are complete for the W-classes. In [6] it is shownthat they have no fpt approximation of any possible ratio, again by using the non-monotoncity of theproblems. Marx strengthened this result signiﬁcantly in [25] by proving that the weighted satisﬁabilityproblem is not fpt approximable for circuits of depth 4 without negation gates, unless FPT = W [2] .Our result can be viewed as an attempt to improve Marx’s result to depth-2 circuits, although at themoment we are only able to rule out fpt approximations with constant ratio. Organization of the paper.

We ﬁx our notations in Section 2. In the same section we also explainthe result in [22] key to our proof. To help readability, we ﬁrst prove that the dominating set problem isnot fpt approximable with ratio smaller than / in Section 3. In the case of the clique problem, oncewe have inapproximability for a particular constant ratio, it can be easily improved to any constant bygap-ampliﬁcation via graph products. But dominating sets for general graph products are notoriouslyhard to understand (see e.g. [21]). So to prove Theorem 1.1, Section 4 presents a modiﬁed reductionwhich contains a tailor-made graph product. Section 5 discusses some consequences of our results.We conclude in Section 6.

2. Preliminaries

We assume familiarity with basic combinatorial optimizations and parameterized complexity, so weonly introduce those notions and notations central to our purpose. The reader is referred to the standardtextbooks (e.g., [3] and [12, 17]) for further background. N and N + denote the sets of natural numbers (that is, nonnegative integers) and positive integers,respectively. For every n ∈ N we let [ n ] := { , . . . , n } . R is the set of real numbers, and R ≥ := (cid:8) r ∈ R (cid:12)(cid:12) r ≥ (cid:9) . For a function f : A → B we can extend it to sets and vectors by deﬁning f ( S ) := The papers actually address the set cover problem, which is equivalent to the dominating set problem as mentioned inthe beginning. f ( x ) | x ∈ S } and f ( v ) := (cid:0) f ( v ) , f ( v ) , · · · , f ( v k ) (cid:1) , where S ⊆ A and v = ( v , v , · · · , v k ) ∈ A k for some k ∈ N + .Graphs G = ( V, E ) are always simple, i.e., undirected and without loops and multiple edges.Here, V is the vertex set and E the edge set, respectively. The size of G is | G | := | V | + | E | . Asubset D ⊆ V is a dominating set of G , if for every v ∈ V either v ∈ D or there exists a u ∈ D with { u, v } ∈ E . In the second case, we might say that v is dominated by u , and this can be easilygeneralized to v dominated by a set of vertices. The domination number γ ( G ) of G is the size of asmallest dominating set. The classical minimum dominating set problem is to ﬁnd such a dominatingset: M IN -D OMINATING -S ET Instance:

A graph G = ( V, E ) . Solution:

A dominating set D of G . Cost: | D | . Goal: min.The decision version of M IN -D OMINATING -S ET has an additional input k ∈ N . Thereby, we ask fora dominating set of size at most k instead of γ ( G ) . But it is well known that two versions can bereduced to each other in polynomial time. In parameterized complexity, we view the input k as theparameter and thus obtain the standard parameterization of M IN -D OMINATING -S ET : p -D OMINATING -S ET Instance:

A graph G and k ∈ N . Parameter: k . Problem:

Decide whether G has a dominating set of size at most k .As mentioned in the Introduction, p -D OMINATING -S ET is complete for the parameterized complex-ity class W [2] , the second level of the W-hierarchy. We will need another important parameterizedproblem, the parameterized clique problem p -C LIQUE

Instance:

A graph G and k ∈ N . Parameter: k . Problem:

Decide whether G has a clique of size at most k .which is complete for W [1] . Recall that a subset S ⊆ V is a clique in G = ( V, E ) , if for every u, v ∈ S we have either u = v or { u, v } ∈ E .Those W-classes are deﬁned by weighted satisﬁability problems for propositional formulas andcircuits. As they will be used only in Section 5, we postpone their deﬁnition until then. Parameterized approximability.

We follow the general framework of [6]. However, to lessen thenotational burden we restrict our attention to the approximation of the dominating set problem.

Deﬁnition 2.1.

Let ρ : N → R ≥ . An algorithm A is a parameterized approximation algorithm for p -D OMINATING -S ET with approximation ratio ρ if for every graph G and k ∈ N with γ ( G ) ≤ k thealgorithm A computes a dominating set D of G such that | D | ≤ ρ ( k ) · k.

4f the running time of A is bounded by f ( k ) · | G | O (1) where f : N → N is computable, then A is anfpt approximation algorithm.One might also deﬁne parameterized approximation directly for M IN -D OMINATING -S ET by tak-ing γ ( G ) as the parameter. The next result shows that essentially this leads to the same notion. Proposition 2.2 ([6, Proposition 5]) . Let ρ : N → R ≥ be a function such that ρ ( k ) · k is nondecreas-ing. Then the following are equivalent.(1) p -D OMINATING -S ET has an fpt approximation algorithm with approximation ratio ρ .(2) There exists a computable function g : N → N and an algorithm A that on every graph G computes a dominating set D of G with | D | ≤ ρ ( γ ( G )) · γ ( G ) in time g ( γ ( G )) · | G | O (1) . The Color-Coding.Lemma 2.3 ([2]) . For every n, k ∈ N there is a family Λ n,k of polynomial time computable functionsfrom [ n ] to [ k ] such that for every k -element subset X of [ n ] , there is an h ∈ Λ n,k such that h isinjective on X . Moreover, Λ n,k can be computed in time O ( k ) · n O (1) . [1] -hardness reduction of the parameterized biclique problem. Our starting point isthe following theorem proved in [22] which states that, on input a bipartite graph, it is W[1]-hard todistinguish whether there exist k vertices with large number of common neighbors or every k -vertexset has small number of common neighbors. Theorem 2.4 ([22, Theorem 1.3]) . There is a polynomial time algorithm A such that for every graph G with n vertices and k ∈ N with l n k +6 m > ( k + 6)! and | k + 1 the algorithm A constructs abipartite graph H = ( A ˙ ∪ B, E ) satisfying:(1) if G contains a clique of size k , i.e., K k ⊆ G , then there are s vertices in A with at least l n k +1 m common neighbors in B ;(2) otherwise K k * G , every s vertices in A have at most ( k + 1)! common neighbors in B ,where s = (cid:0) k (cid:1) . In our reductions from p -C LIQUE to p -D OMINATING -S ET , we use the following procedure toensure that the instance ( G, k ) of p -C LIQUE satisﬁes | k + 1 . Preprocessing.

On input a graph G and k ∈ N + , if does not divide k + 1 , let k ′ be the minimuminteger such that k ′ ≥ k and | k ′ + 1 . We construct a new graph G ′ by adding a clique with k ′ − k vertices into G and making every vertex of this clique adjacent to other vertices in G . It is easy to seethat k ′ ≤ k + 5 , and G contains a k -clique if and only if G ′ contains a k ′ -clique. Then we proceedwith G ← G ′ and k ← k ′ . 5 . The Case ρ < / As the ﬁrst illustration of how to use the gap created in Theorem 2.4, we show in this section that p -D OMINATING -S ET cannot be fpt approximated within ratio < / . This serves as a stepping stoneto the general constant-inapproximability of the problem. Theorem 3.1.

Let ρ < / . Then there is no fpt approximation of the parameterized dominating setproblem achieving ratio ρ unless FPT = W [1] .Proof : We ﬁx some ε, δ ∈ R with < ε < , < δ < / , and / − δ ε > ρ. (1)Let G be a graph with n vertices and k ∈ N a parameter. We set s := (cid:0) k (cid:1) , d := l sε m s , and t := (cid:24)(cid:18) − δ (cid:19) · d − / s (cid:25) . As a consequence, when k and n are sufﬁciently large, we have st < εd, (cid:18) − δ (cid:19) · dt ≤ s √ d, ( k + 1)! < δ √ d − , and d ≤ ⌈ n k +1 ⌉ . (2)By Theorem 2.4 (and the preprocessing) we can compute in fpt-time a bipartite graph H =( A ˙ ∪ B , E ) such that:- if K k ⊆ G , then there are s vertices in A with d common neighbors in B ;- if K k * G , then every s vertices in A have at most ( k + 1)! common neighbors in B .Then using the color-coding in Lemma 2.3, again in fpt-time, we construct two function families Λ A := Λ | A | ,s and Λ B := Λ | B | ,d such that- for every s -element subset X ⊆ A there is an h ∈ Λ A with h ( X ) = [ s ] ;- for every d -element subset Y ⊆ B there is an h ∈ Λ B with h ( Y ) = [ d ] .Deﬁne the bipartite graph H = (cid:0) A ( H ) ˙ ∪ B ( H ) , E ( H ) (cid:1) by A ( H ) := A × Λ A × Λ B , B ( H ) := B × Λ A × Λ B E ( H ) := n(cid:8) ( u, h , h ) , ( v, h , h ) (cid:9) (cid:12)(cid:12)(cid:12) u ∈ A , v ∈ B , h ∈ Λ A , h ∈ Λ B , and { u, v } ∈ E o . Moreover, deﬁne two colorings α : A ( H ) → [ s ] and β : B ( H ) → [ d ] by α ( u, h , h ) := h ( u ) and β ( v, h , h ) := h ( v ) . It is straightforward to verify that(H1) if K k ⊆ G , then there are s vertices of distinct α -colors in A ( H ) with d common neighbors ofdistinct β -colors in B ( H ) ;(H2) if K k * G , then every s vertices in A ( H ) have at most ( k + 1)! common neighbors in B ( H ) .6ow from H , α , and β we construct a new graph G ′ = (cid:0) V ( G ′ ) , E ( G ′ ) (cid:1) as follows. First, itsvertex set is deﬁned by V ( G ′ ) := B ( H ) ˙ ∪ (cid:8) x i , y i (cid:12)(cid:12) i ∈ [ d ] (cid:9) ˙ ∪ C ˙ ∪ W, where C := A ( H ) × [ t ] and W := n w b,j,i (cid:12)(cid:12)(cid:12) b ∈ B ( H ) , i ∈ [ t ] , j ∈ [ s ] o . Moreover, G ′ contains the following types of edges.(E1) { b, b ′ } ∈ E ( G ′ ) with b, b ′ ∈ B ( H ) , b = b ′ , and β ( b ) = β ( b ′ ) (cid:0) i.e., all vertices in B ( H ) withthe same color under β form a clique in G ′ (cid:1) .(E2) Let b ∈ B ( H ) and c := β ( b ) . Then { x c , b } , { y c , b } ∈ E ( G ′ ) .(E3) Let b, b ′ ∈ B ( H ) with β ( b ) = β ( b ′ ) and b = b ′ . Then (cid:8) w b,j,i , b ′ (cid:9) ∈ E ( G ′ ) for every i ∈ [ t ] and j ∈ [ s ] .(E4) (cid:8) ( a, i ) , w b,j,i (cid:9) ∈ E ( G ′ ) for every { a, b } ∈ E ( H ) , j = α ( a ) and i ∈ [ t ] .(E5) Let a, a ′ ∈ A ( H ) with a = a ′ and i ∈ [ t ] . Then (cid:8) ( a, i ) , ( a ′ , i ) (cid:9) ∈ E ( G ′ ) .To ease presentation, for every c ∈ [ d ] we set B c := (cid:8) b ∈ B ( H ) (cid:12)(cid:12) β ( b ) = c (cid:9) ∪ { x c , y c } . Claim 1. If D is a dominating set of G ′ , then D ∩ B c = ∅ for every c ∈ [ d ] . Proof of the claim.

We observe that every x c is only adjacent to vertices in B c . ⊣ Claim 2. If G contains a k -clique, then γ ( G ′ ) < (1 + ε ) d . Proof of the claim.

By (H1) the bipartite graph H has a K s,d biclique K with α ( A ( H ) ∩ K ) = [ s ] and β ( B ( H ) ∩ K ) = [ d ] . It is then easy to verify that (cid:0) B ( H ) ∩ K (cid:1) ˙ ∪ (cid:0) ( A ( H ) ∩ K ) × [ t ] (cid:1) is a dominating set of G ′ , whose size is d + s · t < (1 + ε ) d by (2). ⊣ Claim 3. If G contains no k -clique, then every s -vertex set of A ( H ) has at most ( k + 1)! < δ √ d − common neighbors in B ( H ) . Claim 4. If G contains no k -clique, then γ ( G ′ ) > (cid:18) − δ (cid:19) · d. Proof of the claim.

Let D be a dominating set of G ′ . By Claim 1 we have D ∩ B c = ∅ for every c ∈ [ d ] . Deﬁne e := (cid:12)(cid:12)(cid:12)(cid:8) c ∈ [ d ] (cid:12)(cid:12) | D ∩ B c | ≥ (cid:9)(cid:12)(cid:12)(cid:12) . e > (1 / − δ ) · d then | D | > d + e > (3 / − δ ) · d and we are done.So let us consider e ≤ (1 / − δ ) · d and without loss of generality | D ∩ B c | = 1 for every c ≤ (1 / δ ) · d . Fix such a c and assume D ∩ B c = { b c } . Recall x c , y c ∈ V ( G ′ ) are not adjacentto any vertex outside B c , and there is no edge between them, thus b c ∈ B c \ { x c , y c } = (cid:8) b ∈ B ( H ) (cid:12)(cid:12) α ( b ) = c (cid:9) . Let W := n w b c ,j,i (cid:12)(cid:12)(cid:12) i ∈ [ t ] , j ∈ [ s ] , and c ≤ (1 / δ ) · d o ⊆ W. (E3) implies that every w b c ,j,i ∈ W is not dominated by any vertex in D ∩ S c ∈ [ d ] B c . Therefore, ithas to be dominated by or included in D ∩ ( C ∪ W ) .If | D ∩ W | > (1 / − δ ) · d , then again we are done. So suppose | D ∩ W | ≤ (1 / − δ ) · d .Without loss of generality let W := n w b c ,j,i (cid:12)(cid:12)(cid:12) i ∈ [ t ] , j ∈ [ s ] , and c ≤ δd o ⊆ W and assume W ∩ D = ∅ . Thus W has to be dominated by D ∩ C . For later purpose, let Y := (cid:8) b c (cid:12)(cid:12) c ≤ δd (cid:9) . Obviously, | Y | ≥ δd − .Again we only need to consider the case | D ∩ C | ≤ (1 / − δ ) · d . Recall C = A ( H ) × [ t ] . Thusthere is an i ∈ [ t ] such that (cid:12)(cid:12)(cid:12) D ∩ (cid:0) A ( H ) × { i } (cid:1)(cid:12)(cid:12)(cid:12) ≤ (cid:18) − δ (cid:19) · dt . Let X := (cid:8) a ∈ A ( H ) (cid:12)(cid:12) ( a, i ) ∈ D (cid:9) , and in particular, | X | ≤ (1 / − δ ) · d/t . Since W is dominatedby D ∩ C , we have for all b ∈ Y and j ∈ [ s ] there exists a ∈ X such that (cid:8) ( a, i ) , w b,j,i (cid:9) ∈ E ( G ′ ) ,which means that { a, b } ∈ E ( H ) and α ( a ) = j . It follows that in the graph H every vertex of Y hasat least s neighbors in X . Recall that (1 / − δ ) · d/t ≤ s √ d by (2). There are at most √ d differenttypes of s -vertex sets in X , i.e., (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Xs (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) (1 / − δ ) · d/ts (cid:19) ≤ (cid:16) s √ d (cid:17) s = √ d. By the pigeonhole principle, there exists an s -vertex set of X ⊆ A ( H ) having at least | Y | / √ d ≥ δ √ d − common neighbors in Y ⊆ B ( H ) , which contradicts Claim 3. ⊣ .Claim 2 and Claim 4 indeed imply that there is an fpt-reduction from the clique problem to thedominating set problem which creates a gap great than / − δ ε . So if there is a ρ -approximation of the dominating set problem, by (1) we can decide the cliqueproblem in fpt time. ✷ . The Constant-Inapproximbility of p -D OMINATING -S ET Theorem 1.1 is a fairly direct consequence of the following theorem.

Theorem 4.1 (Main) . There is an algorithm A such that on input a graph G , k ≥ , and c ∈ N thealgorithm A computes a graph G c such that(i) if K k ⊆ G , then γ ( G c ) < . · d c ;(ii) if K k * G , then γ ( G c ) > c · d c / ,where d = (cid:0) · c · ( k + 1) (cid:1) · k +3 c . Moreover the running time of A is bounded by f ( k, c ) · | G | O ( c ) for a computable function f : N × N → N .Proof of Theorem 1.3: Suppose for some ε > there is an fpt-algorithm A ( G ) which outputs adominating set for G of size at most ε p log ( γ ( G )) · γ ( G ) . Of course we can further assume that ε < . Then on input a graph G and k ∈ N , let c := l k − ε/ m = o ( k ) and d := (cid:0) · c · ( k + 1) (cid:1) · k +3 c . We have ε p log (1 . · d c ) = O (cid:16) ε p c · k · log k (cid:17) = o (cid:16) k ε (cid:17) = o ( c ) . By Theorem 4.1, we can construct a graph G c with properties (i) and (ii) in time f ( k, c ) · | G | O ( c ) = h ( k ) · | G | o ( k ) for an appropriate computable function h : N → N . Thus, G contains a clique of size k if and only if A ( G c ) returns a dominating set of size at most . · d c · ε p log (1 . · d c ) = o ( c · d c ) < c · d c , where the inequality holds for sufﬁciently large k (cid:0) and hence sufﬁciently large c · d c (cid:1) .Therefore we can determine whether G contains a k -clique in time g ( k ) · | G | o ( k ) for some com-putable g : N → N . This contradicts a result in Chen et.al. [5, Theorem 4.4] under ETH. ✷ We start by showing a variant of Theorem 2.4.

Theorem 4.2.

Let ∆ ∈ N + be a constant and d : N + → N + a computable function. Then there is an fpt -algorithm that on input a graph G and a parameter k ∈ N with | k + 1 constructs a bipartitegraph H = (cid:0) A ( H ) ˙ ∪ B ( H ) , E ( H ) (cid:1) together with two colorings α : A ( H ) → [∆ s ] and β : B ( H ) → [ d ( k )] such that:(H1) if K k ⊆ G , then there are ∆ s vertices of distinct α -colors in A ( H ) with d ( k ) common neighborsof distinct β -colors in B ( H ) ;(H2) if K k * G , then every ∆( s −

1) + 1 vertices in A ( H ) have at most ( k + 1)! common neighborsin B ( H ) , here s = (cid:0) k (cid:1) .Proof : Let G be a graph with n vertices and k ∈ N . Assume without loss of generality l n k +6 m > ( k + 6)! and l n k +1 m ≥ d ( k ) . By Theorem 2.4 we can construct in polynomial time a bipartite graph H = ( A ˙ ∪ B , E ) suchthat for s := (cid:0) k (cid:1) :– if K k ⊆ G , then there are s vertices in A with at least d ( k ) common neighbors in B ;– if K k * G , then every s vertices in A have at most ( k + 1)! common neighbors in B .Deﬁne A := A × [∆] , B := B , and E := (cid:8) { ( u, i ) , v } (cid:12)(cid:12) ( u, i ) ∈ A × [∆] , v ∈ B , and { u, v } ∈ E (cid:9) . It is easy to verify that in the bipartite graph ( A ˙ ∪ B , E ) – if K k ⊆ G , then there are ∆ s vertices in A with at least d ( k ) common neighbors in B ;– if K k * G , then every ∆( s −

1) + 1 vertices in A have at most ( k + 1)! common neighbors in B .Applying Lemma 2.3 on (cid:0) n ← | A | , k ← ∆ s (cid:1) and (cid:0) n ← | B | , k ← d ( k ) (cid:1) we obtain two function families Λ A := Λ | A | , ∆ s and Λ B := Λ | B | ,d ( k ) with the stated properties.Finally the desired bipartite graph H is deﬁned by (cid:16) ( A × Λ A × Λ B ) ˙ ∪ ( B × Λ A × Λ B ) , E ) (cid:17) with E := n(cid:8) ( u, h , h ) , ( v, h , h ) (cid:9) (cid:12)(cid:12)(cid:12) u ∈ A , v ∈ B , h ∈ Λ A , h ∈ Λ B , and { u, v } ∈ E o and the colorings α ( u, h , h ) := h ( u ) and β ( v, h , h ) := h ( v ) . ✷ Setting the parameters.

Let ∆ := 2 . Recall that k ≥ , s = (cid:0) k (cid:1) ≥ , and c ∈ N + . We ﬁrst deﬁne d := d ( k ) := (cid:0) · c · ( k + 1) (cid:1) · k +3 c . It is easy to check that:(i) d − s > c · s c (cid:16) = c · (cid:0) k (cid:1) c (cid:17) .(ii) d > (cid:0) k + 1)! (cid:1) s .(iii) d > (cid:0) s · c (cid:1) s . 10hen let t := c · d c − s . (3)From (ii), (iii), and (3) we conclude ∆ sct < . · d c , c · d c t ≤ s √ d, and ( k + 1)! < s √ d . (4)Moreover by (i) and ∆ = 2 we have c · d c + c ∆ c s c d c − + s < c d c . (5) Construction of G c . We invoke Theorem 4.2 to obtain H = ( A ˙ ∪ B, E ) , α , and β . Then weconstruct a new graph G c = (cid:0) V ( G c ) , E ( G c ) (cid:1) as follows. First, the vertex set of G c is given by V ( G c ) := [ i ∈ [ d ] c V i ˙ ∪ C ˙ ∪ W, where V i := (cid:8) v ∈ B c (cid:12)(cid:12) β ( v ) = i (cid:9) for every i ∈ [ d ] c ,C := A × [ c ] × [ t ] , and W := n w v , j ,i (cid:12)(cid:12)(cid:12) v ∈ V i for some i ∈ [ d ] c , j ∈ [∆ s ] c and i ∈ [ t ] o . Moreover, G c contains the following types of edges.(E1) For each i ∈ [ d ] c , V i forms a clique.(E2) Let i ∈ [ d ] c and v , v ′ ∈ V i . If for all ℓ ∈ [ c ] we have v ( ℓ ) = v ′ ( ℓ ) then { w v , j ,i , v ′ } ∈ E ( G c ) forevery i ∈ [ t ] and j ∈ [∆ s ] c .(E3) Let i ∈ [ t ] . Then (cid:8) ( u, ℓ, i ) , w v , j ,i (cid:9) ∈ E ( G c ) if { u, v ( ℓ ) } ∈ E and j ( ℓ ) = α ( u ) .(E4) Let u, u ′ ∈ A ( H ) with u = u ′ , ℓ ∈ [ c ] , and i ∈ [ t ] . Then (cid:8) ( u, ℓ, i ) , ( u ′ , ℓ, i ) (cid:9) ∈ E ( G c ) .Theorem 4.1 then follows from the completeness and the soundness of this reduction. Lemma 4.3 (Completeness) . If G contains k -clique, then γ ( G c ) < . d c . Lemma 4.4 (Soundness) . If G contains no k -clique then γ ( G c ) > c · d c / . We ﬁrst show the easier completeness.

Proof of Lemma 4.3:

By (H1) in Theorem 4.2, if G contains a subgraph isomorphic to K k , then thebipartite graph H has a K ∆ s,d -subgraph K such that α ( A ∩ K ) = [∆ s ] and β ( B ∩ K ) = [ d ] . Let D := ( B ∩ K ) c ˙ ∪ (cid:0) ( A ∩ K ) × [ c ] × [ t ] (cid:1) . Obviously, | D | = d c + ∆ sct < . · d c by (4). And (E1) and (E4) imply that D dominates everyvertex in C and every vertex in V i for all i ∈ [ d ] c .To see that D also dominates W , let w v , j ,i be a vertex in W . First consider the case where v ( ℓ ) / ∈ B ∩ K for all ℓ ∈ [ c ] . Since β (cid:0) ( B ∩ K ) c (cid:1) = [ d ] c , there exists a vertex v ′ ∈ ( B ∩ K ) c with β ( v ′ ) = β ( v ) and v ( ℓ ) = v ′ ( ℓ ) for all ℓ ∈ [ c ] . Then w v , j ,i is dominated by v ′ because of (E2).Otherwise assume v ( ℓ ) ∈ B ∩ K for some ℓ ∈ [ c ] , then A ∩ K ⊆ N H ( v ( ℓ )) = (cid:8) u ∈ A (cid:12)(cid:12) { u, v ( ℓ ) } ∈ E (cid:9) . There exists a vertex u ∈ A ∩ K such that α ( u ) = j ( ℓ ) and (cid:8) v ( ℓ ) , u (cid:9) ∈ E . By(E3), w v , j ,i is adjacent to ( u, ℓ, i ) . ✷ Here, we assume d c − s is an integer. Otherwise, let d ← d s which maintains (i)– (iii). .2. Soundness.Lemma 4.5. Suppose c, ∆ , t ∈ N + and ∆ < t . Let V ⊆ [ t ] c . If there exists a function θ : V → [ c ] such that for all i ∈ [ c ] we have (cid:12)(cid:12)(cid:12)(cid:8) v ( i ) (cid:12)(cid:12) v ∈ V and θ ( v ) = i (cid:9)(cid:12)(cid:12)(cid:12) ≤ t − ∆ , (6) then | V | ≤ t c − ∆ c .Proof : When c = 1 , we have | V | ≤ t − ∆ by (6). Suppose the lemma holds for c ≤ n and consider c = n + 1 . Given V ⊆ [ t ] n +1 and θ , let C n +1 := (cid:8) v ( n + 1) (cid:12)(cid:12) v ∈ V and θ ( v ) = n + 1 (cid:9) . By (6), | C n +1 | ≤ t − ∆ . If | C n +1 | < t − ∆ , we add (cid:0) t − ∆ − | C n +1 | (cid:1) arbitrary integers from [ t ] \ C n +1 to C n +1 . So we have | C n +1 | = t − ∆ . Let A := (cid:8) v ∈ V (cid:12)(cid:12) v ( n + 1) ∈ C n +1 (cid:9) and B := V \ A . Itfollows that | A | ≤ ( t − ∆) t c − , (7) (cid:12)(cid:12)(cid:12)(cid:8) v ( n + 1) (cid:12)(cid:12) v ∈ B (cid:9)(cid:12)(cid:12)(cid:12) ≤ ∆ , and θ ( v ) ∈ [ c − for v ∈ B . Let V ′ := (cid:8) ( v , v , · · · , v n ) (cid:12)(cid:12) ∃ v n +1 ∈ [ t ] , ( v , v , · · · , v n , v n +1 ) ∈ B (cid:9) . We deﬁne a function θ ′ : V ′ → [ c − as follows. For all v ′ ∈ V ′ , choose v ∈ B with the minimum v ( c ) such that for all i ∈ [ c − it holds v ′ ( i ) = v ( i ) . By the deﬁnition of V ′ , such a v must exist, andwe let θ ′ ( v ′ ) := θ ( v ) . By (6), (cid:12)(cid:12)(cid:12)(cid:8) v ′ ( i ) (cid:12)(cid:12) v ′ ∈ V ′ and θ ′ ( v ′ ) = i (cid:9)(cid:12)(cid:12)(cid:12) ≤ t − ∆ for all i ∈ [ c − . Applyingthe induction hypothesis, we get | V ′ | ≤ t c − − ∆ c − . Obviously, | B | ≤ ∆ | V ′ | ≤ ∆ t c − − ∆ c . (8)From (7) and (8), we deduce that | V | = | A | + | B | ≤ ( t − ∆) t c − + ∆ t c − − ∆ c ≤ t c − ∆ c . ✷ We are now ready to prove the soundness of our reduction.

Proof of Lemma 4.4:

Let D be a dominating set of G c . Deﬁne a := (cid:12)(cid:12)(cid:12)(cid:8) i ∈ [ d ] c (cid:12)(cid:12) | D ∩ V i | ≥ c + 1 (cid:9)(cid:12)(cid:12)(cid:12) . If a > d c / , then | D | ≥ ( c + 1) a > c · d c / and we are done.So let us consider a ≤ d c / . Thus, the set I := (cid:8) i ∈ [ d ] c (cid:12)(cid:12) | D ∩ V i | ≤ c (cid:9) has size | I | ≥ d c / . Let i ∈ I and assume that D ∩ V i = (cid:8) v , v , . . . , v c ′ (cid:9) for some c ′ ≤ c . Wedeﬁne a v i ∈ V i as follows. If c ′ = 0 , we choose an arbitrary v i ∈ V i . Otherwise, let v i ( ℓ ) := ( v ℓ ( ℓ ) for all ℓ ∈ [ c ′ ]; v ( ℓ ) for all c ′ < ℓ ≤ c. Since the coloring β is obtained by the color-coding used in the proof of Theorem 4.2, for every b ∈ [ d ] it holds that { v ∈ B | β ( v ) = b } 6 = ∅ , hence V i = ∅ . β ( v i ) = i .(E2) implies that for every j ∈ [∆ s ] c and every i ∈ [ t ] , the vertex w v i , j ,i is not dominated by D ∩ V i . Observe that w v i , j ,i cannot be dominated by other D ∩ V i ′ with i ′ = i either, by (E2) and (E3).Therefore every vertex in the set W := (cid:8) w v i , j ,i (cid:12)(cid:12) i ∈ I , j ∈ [∆ s ] c , and i ∈ [ t ] (cid:9) is not dominated by D ∩ S i ∈ [ d ] c V i . As a consequence, W has to be dominated by or included in D ∩ ( C ∪ W ) .If | D ∩ W | > c · d c / , then again we are done. So suppose | D ∩ W | ≤ c · d c / and let W := W \ D . It follows that W has to be dominated by D ∩ C . Once again we only need toconsider the case | D ∩ C | ≤ c · d c / , and hence there is an i ′ ∈ [ t ] such that (cid:12)(cid:12)(cid:12) D ∩ (cid:0) A × [ c ] × { i ′ } (cid:1)(cid:12)(cid:12)(cid:12) ≤ c · d c t . (9)Then we deﬁne Z := (cid:8) w v , j ,i ∈ W (cid:12)(cid:12) i = i ′ (cid:9) = (cid:8) w v i , j ,i ′ (cid:12)(cid:12) i ∈ I , j ∈ [∆ s ] c , and w v i , j ,i ′ / ∈ D (cid:9) . So Z has to be dominated by D ∩ C , and in particular those vertices of the form ( u, ℓ, i ′ ) ∈ D ∩ C .Moreover, | Z | ≥ ∆ c s c | I | − | D ∩ W | ≥ ∆ c s c | I | − c · d c / . (10)Our next step is to upper bound | Z | . To that end, let X := (cid:8) u ∈ A (cid:12)(cid:12) ( u, ℓ, i ′ ) ∈ D for some ℓ ∈ [ c ] (cid:9) . Thus Z is dominated by those vertices ( u, ℓ, i ′ ) with u ∈ X . And by (9) | X | ≤ c · d c t . Set Y := n v ∈ B (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) N H ( v ) ∩ X (cid:12)(cid:12) > ∆( s − o . Recall that c · d c / (3 t ) ≤ s √ d by (4). Hence X has at most √ d different subsets of size ∆( s −

1) + 1 ,i.e., (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) X ∆( s −

1) + 1 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | X | ∆( s − ≤ | X | ∆ s ≤ √ d. We should have | Y | ≤ √ d · ( k + 1)! ≤ d + s , (11)where the second inequality is by (4). Otherwise, by the pigeonhole principle, there exists a (∆( s − -vertex set of X ⊆ A ( H ) having at least | Y | / √ d > ( k +1)! common neighbors in Y ⊆ B ( H ) .However, if G contains no k -clique, then by (H2) every (cid:0) ∆( s −

1) + 1 (cid:1) -vertex set of A ( H ) has atmost ( k + 1)! common neighbors in B ( H ) , and we obtain a contradiction.13et Z := (cid:8) w v , j ,i ′ ∈ Z (cid:12)(cid:12) there exists an ℓ ∈ [ c ] with v ( ℓ ) ∈ Y (cid:9) (cid:16) ⊆ Z (cid:17) = (cid:8) w v i , j ,i ′ (cid:12)(cid:12) i ∈ I , j ∈ [∆ s ] c , w v i , j ,i ′ / ∈ D , and there exists an ℓ ∈ [ c ] with v i ( ℓ ) ∈ Y (cid:9) and Z := Z \ Z = (cid:8) w v i , j ,i ′ (cid:12)(cid:12) i ∈ I , j ∈ [∆ s ] c , w v i , j ,i ′ / ∈ D , and v i ( ℓ ) / ∈ Y for all ℓ ∈ [ c ] (cid:9) . Moreover, let I := { i ∈ I | there exists a w v i , j ,i ′ ∈ Z } . From the deﬁnition, we can deduce thatfor all i ∈ I there exists an ℓ ∈ [ c ] such that i ( ℓ ) ∈ β ( Y ) . Then | I | ≤ c | Y | d c − and hence | Z | ≤ | I | ∆ c s c ≤ c | Y | d c − ∆ c s c . To estimate | Z | , let us ﬁx an i ∈ I and thus ﬁx the tuple v i ∈ B c , and consider the set J i := (cid:8) j ∈ [∆ s ] c (cid:12)(cid:12) w v i , j ,i ′ ∈ Z (cid:9) . Recall that Z is dominated by those vertices ( u, ℓ, i ′ ) with u ∈ X , so for every j ∈ J i the vertex w v i , j ,i ′ is adjacent to some ( u, ℓ, i ′ ) in the dominating set D with u ∈ X . Moreover, for every ℓ ∈ [ c ] , in theoriginal graph H the vertex v i ( ℓ ) ∈ B has at most ∆( s − neighbors in X , by the fact that v i ( ℓ ) / ∈ Y and our deﬁnition of the set Y .Deﬁne a function θ : J i → [∆ s ] such that for each j ∈ J i , if w v i , j ,i ′ is adjacent to a vertex ( u, ℓ, i ′ ) ∈ D with u ∈ X , then θ ( j ) = ℓ . As argued above, such a ( u, ℓ, i ′ ) must exist, and if thereare more than one such, choose an arbitrary one.Let j ∈ J i and ℓ := θ ( j ) . By (E3), in the graph H the vertex v i ( ℓ ) is adjacent to some vertex u ∈ X with α ( u ) = j ( ℓ ) . It follows that for each ℓ ∈ [ c ] we have (cid:12)(cid:12)(cid:12)(cid:8) j ( ℓ ) (cid:12)(cid:12) j ∈ J i and θ ( j ) = ℓ (cid:9)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:8) α ( u ) (cid:12)(cid:12) u ∈ X adjacent to v i ( ℓ ) (cid:9)(cid:12)(cid:12)(cid:12) ≤ ∆( s − . Applying Lemma 4.5, we obtain (cid:12)(cid:12) J i (cid:12)(cid:12) ≤ ∆ c s c − ∆ c . Then (cid:12)(cid:12) Z (cid:12)(cid:12) = X i ∈ I (cid:12)(cid:12) J i (cid:12)(cid:12) ≤ | I | (∆ c s c − ∆ c ) . By (10) and the deﬁnition of Z and Z , we should have ∆ c s c | I | − c · d c / ≤ | Z | = | Z | + | Z | ≤ c | Y | d c − ∆ c s c + | I | (∆ c s c − ∆ c ) . That is, c · d c / c | Y | d c − ∆ c s c ≥ ∆ c | I | ≥ c d c / . Combined with (11), we have c · d c + c ∆ c s c d c − + s ≥ c d c , which contradicts the equation (5). ✷ . Some Consequences Proof of Corollary 1.2:

Let c ∈ N + , and assume that A is a polynomial time algorithm which oninput a graph G = ( V, E ) with γ ( G ) ≤ β ( | V | ) outputs a dominating set D with | D | ≤ c · γ ( G ) .Without loss of generality, we further assume that given ≤ k ≤ n it can be tested in time n O (1) whether k > c · β ( n ) .Now let G be an arbitrary graph. We ﬁrst simulate A on G , and there are three possible outcomesof A .– A does not output a dominating set. Then we know γ ( G ) > β ( | V | ) . So in time O ( | V | ) ≤ O ( β − ( γ ( G ))) we can exhaustively search for a minimum dominating set D of G .– A outputs a dominating set D with | D | > c · β ( | V | ) . We claim that again γ ( G ) > β ( | V | ) .Otherwise, the algorithm A would have behaved correctly with | D | ≤ c · γ ( G ) ≤ c · β ( | V | ) . So we do the same brute-force search as above.– A outputs a dominating set D with | D | ≤ c · β ( | V | ) . If | D | > c · γ ( G ) , then c · β ( | V | ) ≥ | D | > c · γ ( G ) , i.e. , β ( | V | ) > γ ( G ) , which contradicts our assumption for A . Hence, | D | ≤ c · γ ( G ) and we can output D := D .To summarize, we can compute a dominating set D with | D | ≤ c · γ ( G ) in time f ( γ ( G )) · | G | O (1) forsome computable f : N → N . This is a contradiction to Theorem 1.1. ✷ Now we come to the approximability of the monotone circuit satisﬁability problem.M

ONOTONE -C IRCUIT -S ATISFIABILITY

Instance:

A monotone circuit C . Solution:

A satisfying assignment S of C . Cost:

The weight of | S | . Goal: min.Recall that a Boolean circuit C is monotone if it contains no negation gates; and the weight of anassignment is the number of inputs assigned to .As mentioned in the Introduction, Marx showed [25] that M ONOTONE -C IRCUIT -S ATISFIABILITY has no fpt approximation with any ratio ρ for circuits of depth 4, unless FPT = W [2] . Corollary 5.1.

Assume

FPT = W [1] . Then M ONOTONE -C IRCUIT -S ATISFIABILITY has no constantfpt approximation for circuits of depth 2.Proof :

This is an immediate consequence of Theorem 1.1 and the following well-known approximation-preserving reduction from M

ONOTONE -C IRCUIT -S ATISFIABILITY to M IN -D OMINATING -S ET . Let G = ( V, E ) be a graph. We deﬁne a circuit C ( G ) = ^ v ∈ V _ { u,v }∈ E X u . G of size k and a satisfying assignmentof C ( G ) of weight k . ✷ Remark 5.2.

Of course the constant ratio in Corollary 5.1 can be improved according to Theorem 1.3.

6. Conclusions

We have shown that p -D OMINATING -S ET has no fpt approximation with any constant ratio, and infact with a ratio slightly super-constant. The immediate question is whether the problem has fptapproximation with some ratio ρ : N → N , e.g., ρ ( k ) = 2 k . We tend to believe that it is not the case.Our proof does not rely on the deep PCP theorem, instead it exploits the gap created in the W [1] -hardness proof of the parameterized biclique problem in [22]. In the same paper, the second author hasalready proved some inapproximability result which was shown by the PCP theorem before. Exceptfor the derandomization using algebraic geometry in [22] the proofs are mostly elementary. Of coursewe are working under some stronger assumptions, i.e., ETH and FPT = W [1] . It remains to beseen whether we can take full advantage of such assumptions to prove lower bounds matching thoseclassical ones or even improve them as in Corollary 1.2. Acknowledgement.

We thank Edouard Bonnet for pointing out a mistake in an earlier version of thepaper.

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