Subexponential and FPT-time Inapproximability of Independent Set and Related Problems
aa r X i v : . [ c s . CC ] N ov Subexponential and FPT-time Inapproximabilityof Independent Set and Related Problems ⋆ Bruno Escoffier, Eun Jung Kim, and Vangelis Th. Paschos ⋆⋆ PSL Research University, Université Paris-Dauphine, LAMSADE, CNRS UMR 7243 {escoffier,eun-jung.kim,paschos}@lamsade.dauphine.fr
Abstract.
Fixed-parameter algorithms, approximation algorithms andmoderately exponential algorithms are three major approaches to algo-rithms design. While each of them being very active in its own, there isan increasing attention to the connection between different approaches.In particular, whether
Independent Set would be better approximableonce endowed with subexponential-time or FPT-time is a central ques-tion. In this paper, we present a strong link between the linear PCPconjecture and the inapproximability, thus partially answering this ques-tion.
In this paper we look into three approaches to algorithms design: Fixed-para-meter algorithms, approximation algorithms and moderately exponential algo-rithms. These three areas, each of them being very active in its own, have beenconsidered as foreign to each other until recently. Polynomial-time approxima-tion algorithm produces a solution whose quality is guaranteed to lie within acertain range from the optimum. One illustrative problem indicating the develop-ment of this area is
Independent Set . The approximability of
IndependentSet within constant ratios has remained as the most important open problemsfor a long time in the field. It was only after the novel characterization of the NP given by the PCP theorem [1, 2] that impossibility of such approximabilityhas been proven assuming P = NP . Subsequent improvements of the origi-nal PCP theorem, leading to corresponding refinements of the characterizationof NP have also led to the actual very strong inapproximability result for Inde-pendent Set , namely, that it is inapproximable within ratios Ω ( n ε − ) for any ε > , unless P = NP [31].Moderately exponential algorithm is to allow exponential running time forthe sake of optimality. In this case, the endeavor lies in limiting the growth of ⋆ Research supported by the French Agency for Research under the DEFIS programTODO, ANR-09-EMER-010. A preliminary version of this work appeared in [16]. ⋆⋆ Also, Institut Universitaire de France The approximation ratio of an algorithm computing a feasible solution for someproblem is the ratio of the value of the solution computed over the optimal value forthe problem. unning time function as slow as possible. Parameterized complexity providesan alternative framework to analyze the running time in a more refined way[14, 18]. The aim is to get an O ( f ( k ) · n c ) -time algorithm for some constant c (independent of k ). As these two research programs offer a generous running timecompared to polynomial-time approximation algorithms, a growing amount ofattention is paid to them as a way to cope with hardness in approximability.The first one deals with moderately exponential approximation . The goal of thisprogram is to explore approximability of highly inapproximable (in polynomialtime) problems in superpolynomial or moderately exponential time. Roughlyspeaking, if a given problem is solvable in time say O ∗ ( γ n ) but it is NP-hardto approximate within some ratio r , we seek r -approximation algorithms withcomplexity - significantly - lower than O ∗ ( γ n ) . This issue has been consideredfor several problems such as Set Cover [11, 5],
Coloring [3, 4],
IndependentSet and
Vertex Cover [6],
Bandwidth [12, 19].The second research program handles approximation by fixed parameter al-gorithms. In this approximation framework, we say that a parameterized (withparameter k ) problem Π is r -approximable if there exists an algorithm takingas inputs an instance I of Π and k and either computes a solution smaller orgreater than (depending on whether Π is, respectively, a minimization, or amaximization problem) rk , or returns “no”, asserting in this case that there isno solution of value at most or at least k . This line of research was initiated bythree independent works [15, 8, 10]. As an excellent overview in this direction,see [26].Several natural questions can be asked dealing with these two programs. Inparticular, the following ones have been asked several times (see for instance [26,15, 19, 6]) and of great interest: Q1 can a highly inapproximable in polynomial time problem be well-approxi-mated in subexponential time? Q2 does a highly inapproximable in polynomial time problem become well-approximable in parameterized time?Few answers have been obtained until now. Regarding Q1 , negative results canbe directly obtained by gap-reductions for certain problems. For instance, Col-oring is not approximable within ratio / − ǫ , since this would allow to deter-mine whether a graph is 3-colorable or not in subexponential time. This contra-dicts a widely-acknowledge computational assumption [23]: Exponential Time Hypothesis ( ETH ): There exists an ǫ > such that noalgorithm solves in time ǫn , where n is the number of variables.Regarding Q2 , [15] shows that assuming FPT = W[2], for any r the Indepen-dent Dominating Set problem is not r -approximable (in FPT time).Among interesting problems for which Q1 and Q2 are worth being asked are Independent Set , Coloring and
Dominating Set . They fit in the frame of Actually, the result is even stronger: it is impossible to obtain a ratio r = g ( k ) forany function g . Q1 and Q2 above: they are hard to approximate in polynomial time whiletheir approximability in subexponential or in parameterized time is still open.Note that Independent Set and
Dominating Set are moderately exponentialapproximable within any ratio − ε , for any ε > [5, 6], while Coloring isapproximable within ratio (1 + 1 /χ ( G )) , where χ ( G )) denotes the chromaticnumber of a graph G in moderately exponential time [3, 4].Our contribution in this paper is to establish a link between a major con-jecture in PCP theorem and inapproximability in subexponential-time and inFPT-time, assuming ETH. We first state the conjecture while the definition ofPCP is deferred to the next section. Linear PCP Conjecture ( LPC ): ∈ PCP , / [log | φ | + D, E ] , where | φ | is the size of the instance (sum of lengths of clauses), D and E are constant.Unlike ETH which is arguably recognized as a valid statement,
LPC is a wideopen question. In the emphasized statement given just below, we claim that if
LPC turns out to hold, it immediately implies that one of the most interestingquestions in subexponential and parameterized approximation is negatively an-swered. In particular, as shown in the sequel, assuming
ETH the followings holdfor
Independent Set on n vertices, for any constant < r < :(i) There is no r -approximation algorithm in time O (2 n − δ ) for any δ > .(ii) There is no r -approximation algorithm in time O (2 o ( n ) ) if LPC holds.(iii) There is no r -approximation algorithm in time O ( f ( k ) n O (1) ) if LPC holds.Remark that (i) is not conditional upon
LPC . In fact, this is an immediateconsequence of near-linear PCP construction achieved in [13]. Note that similarinapproximability results under
ETH for
Max-3Sat and
Max-3Lin for somesubexponential running time have been obtained in [28].In the following, Section 2 reviews some known consequences of near-linearPCP. In Section 3, we show how a combination of two classic reductions yields pa-rameterized inapproximabiliy bounds for
Independent Set provided that L PCand
ETH hold (point (iii) above); we also provide a parameterized approxima-tion preserving reduction that allows to transfer parameterized inapproximabil-ity results to
Dominating Set . In Section 4, we analyze known reductions inthe view of inapproximability in subexponential running time and present someresults similar to (i) and (ii).
Max-3Sat
A problem is in
PCP α,β [ q, p ] if there exists a PCP verifier which uses q randombits, reads at most p bits in the proof and is such that: – if the instance is positive, then there exists a proof such that V(erifier)accepts with probability at least α ;3 if the instance is negative, then for any proof V accepts with probability atmost β .Based upon the above definition, the following theorem is proved in [13] (seealso Theorem 7 in [28]), presenting a further refinement of the characterizationof NP. Theorem 1. [13] For every ǫ > , ∈ PCP ,ǫ [(1 + o (1)) log n + O (log(1 /ǫ )) , O (log(1 /ǫ ))] A recent improvement [28] of Theorem 1 (a PCP Theorem with two-query pro-jection tests, sub-constant error and almost-linear size) has some importantcorollaries in polynomial approximation. Among those, the following two areof particular interest in what follows.
Corollary 1. [28] Under
ETH , for every ǫ > , and δ > , it is impossibleto distinguish between instances of max 3-lin with m equations where at least (1 − ǫ ) m are satisfiable from instances where at most (1 / ǫ ) m are satisfiable,in time O (2 m − δ ) . Corollary 2. [28] Under
ETH , for every ǫ > , and δ > , it is impossibleto distinguish between instances of Max-3Sat with m clauses where at least (1 − ǫ ) m are satisfiable from instances where at most (7 / ǫ ) m are satisfiable,in time O (2 m − δ ) . The following is a stronger version of Corollary 2: it holds if L PC holds. Thiswill be our working hypothesis.
Hypothesis 1
Under
ETH , there exists r < such that: for every ǫ > it is impossible to distinguish between instances of Max-3Sat with m clauseswhere at least (1 − ǫ ) m are satisfiable from instances where at most ( r + ǫ ) m aresatisfiable, in time o ( m ) . Using the well known sparsification lemma (Lemma 1), which intuitivelyallows to work with 3-SAT formula with linear lengths (the sum of the lengths ofclauses is linearly bounded in the number of variables), a very standard argumentgives the validity of Hypothesis 1 under L PC, see Lemma 2.
Lemma 1. [23] For all ǫ > , a 3-SAT formula φ on n variables can be writtenas the disjunction of at most ǫn φ i on (at most) n variablessuch that φ i contains each variable in at most c ǫ clauses for some function c ǫ .Moreover, this reduction takes at most p ( n )2 ǫn time. Lemma 2. If L PC holds, then Hypothesis 1 also holds. Note that L PC as expressed in this article implies that Hypothesis 1 holds event withreplacing (1 − ǫ ) m by m . However, we define Hypothesis 1 with this lighter statement (1 − ǫ ) m in order, in particular, to emphasize the fact that perfect completeness isnot required in the LPC conjecture. roof. Suppose that ∈ PCP , / [log | φ | + D, E ] , where | φ | is the sum ofthe lengths of clauses in the instance, D and E are constants.Given an ǫ > , let ǫ ′ such that < ǫ ′ < ǫ . Given an instance φ of on n variables, we apply the sparsification lemma (with ǫ ′ ) to get ǫ ′ n instances φ i on at most n variables. Since each variable appears at most c ǫ ′ times in φ i , theglobal size of φ i is | φ i | c ǫ ′ n .Then for each formula φ i we use the previous PCP assumption. The size ofthe proof is at most E | R | = c ′ | φ i | cn for some constants c ′ , c that depend on ǫ ′ (where | R | = log n + D is the number of random bits) since E | R | is the totalnumber of bits that we read in the proof. Take one variable for each bit in theproof: x , · · · , x cn . For each random string R : take all the E possibilities forthe E variables read, and write a CNF formula which is satisfied if and only ifthe verifier accepts. This can be done with a formula with a constant number ofclauses, say C , each clause having a constant number of variables, say C ( C and C depends on E ).If we consider the CNF formed by all theses CNF for all the random clauses,we get a CNF with C | R | clauses on variables x , · · · , x cn . The clauses are on C variables but by adding a constant number of variables we can replace a clauseon C variables by an equivalent set of clauses on 3 variables. This way we get a3-CNF formula and multiply the number of variables and the number of clausesby a constant, so they are still linear in n . For each R you have a set of say C ′ clauses.Suppose that we start from a satisfiable formula φ i . Then there exists a prooffor which the verifier always accepts. By taking the corresponding values for thevariables x i , and extending it properly to the new variables y, all the clauses aresatisfied.Suppose that we start from a non satisfiable formula φ i . Then for any proof(i.e. any truth values of variables), the verifier rejects for at least half of therandom strings. If the verifier rejects for a random string R , then in the set ofclauses corresponding to this variable at least one clause is not satisfied. It meansthat among the C ′ | R | clauses (total number of clauses), at least / · | R | arenot satisfied, ie a fraction / (2 C ′ ) of the clauses.Then either m = C ′ | R | = O ( n ) clauses are satisfiable, or at least m/ (2 C ′ ) clauses are not satisfied by each assignment. Distinguishing between these sets intime o ( m ) would determine whether φ i is satisfiable or not in o ( n ) . Doing thisfor each φ i would solve in time p ( n )2 ǫ ′ n + 2 ǫ ′ n O (2 o ( n ) ) = O (2 ǫn ) (where p is a polynomial). This is valid for any ǫ > so it would contradicting ETH . ⊓⊔ Dealing with
Independent Set , it is easy to see that, for any increasing andunbounded function r ( n ) , the problem is approximable within ratio /r ( n ) insubexponential time (recall that ratios n ǫ − are are very unlikely to be achievedin polynomial time). Indeed, simply consider all the subsets of V of size atmost n/r ( n ) and return the largest independent set among these sets. If a maxi-mum independent set has size at most n/r ( n ) then the algorithm finds it, other-wise the algorithm outputs a solution of size n/r ( n ) , while the size of an optimum5olution is at most n . The running time of the algorithm is O ∗ ( (cid:0) nn/r ( n ) (cid:1) ) that issubexponential in n .Let us note that Independent Set has the so called self-improvement prop-erty [21] claiming, roughly speaking, that either it is polynomially approximableby a polynomial time approximation schema, or no polynomial algorithm existsthat guarantees some constant approximation ratio, unless P = NP .With a similar proof, the above self-improvement property can be proved for Independent Set also in the case of parameterized approximation.
Lemma 3. [17] The following statements are equivalent for
Independent Set : – there exists r ∈ (0 , such that there exists an r -approximation parameterizedalgorithm; – for any r ∈ (0 , there exists an r -approximation parameterized algorithm. A graph G is a ( n, d, α ) -expander graph if (i) G has n vertices, ( ii ) G is d-regular, ( iii ) all the eigenvalues λ of G but the largest one is suchthat | λ | αd . Fact 1.
For any k ∈ N ∗ and any α > there exists d and a ( k , d, α ) -expandergraph. Moreover, d depends only on α , and this graph can be computed in poly-nomial time for every fixed α . This fact follows from the following lemmas.
Lemma 4 ([20], or Th. 8.1 in [22]).
For every positive integer k , there existsa ( k , , √ / -expander graph, computable in polynomial time. If G is a graph with adjacency matrix M , let us denote G k the graph withadjacency matrix M k . Lemma 5 (Fact 1.2 in [29]). If G is a ( n, d, α ) -expander graph, then G k is a ( n, d k , α k ) -expander graph.Proof. G k is obviously d k regular, and the eigenvalues of G k are the eigenvaluesof G to the power of k . ⊓⊔ Proof of Fact 1.
Take α > and let p be the smallest integer such that (5 √ / p α . G p is as required. The proof of Fact 1 is completed. ⊓⊔ Let G be a graph on n vertices and H be a ( n, d, α ) -expander graph. Let t be a positive integer. We build the graph G ′ t on N = nd t − vertices: each vertexcorresponds to a ( t − -random walk x = ( x , · · · , x t ) on H (meaning that x is chosen at random, and x i +1 is chosen randomly in the set of neighbors of x i ),and two vertices x = ( x , · · · , x t ) and y = ( y , · · · , y t ) in G ′ t are adjacent iff { x , · · · , x t , y , · · · , y t } is a clique in G . Theorem 2 (claims 3.15 and 3.16 in [22]).
Let G be a graph on n verticesand H be a ( n, d, α ) -expander graph. If b > α , then: If ω ( G ) bn then ω ( G ′ t ) ( b + 2 α ) t N ; – If ω ( G ) > bn then ω ( G ′ t ) > ( b − α ) t N . We are now able to prove the gap amplification with linear size amplification.
Theorem 3.
Let G be a graph on n vertices (for a sufficiently large n ) and a > b be two positive real numbers. Then for any real r > one can build inpolynomial time a graph G r such that: – G r has N Cn vertices for C independent of G ( C may depend on r ); – If ω ( G ) bn then ω ( G r ) b r N ; – If ω ( G ) > an then ω ( G r ) > a r N ; – b r /a r r .Proof. Let k = ⌈√ n ⌉ . We modify G by adding k − n dummy (isolated) vertices.Let G ′ be the new graph. It has n ′ = k vertices. Note that n ′ ( √ n + 1) = n + 2 √ n + 1 = n + o ( n ) . Let n be such that − ǫ n/n ′ for a small ǫ .Thanks to Fact 1, we consider a ( k , d, α ) -expander graph H for a sufficientlysmall α (the value of which will be fixed later). According to Theorem 2 (appliedon G ′ ) we build in polynomial time a graph G ′ t on N = n ′ d t vertices such that(choosing α < b/ ): – If ω ( G ) bn then ω ( G ′ ) = ω ( G ) bn ′ , hence ω ( G ′ t ) ( b + 2 α ) t N ; – If ω ( G ) > an then ω ( G ′ ) = ω ( G ) an ′ (1 − ǫ ) , hence ω ( G ′ t ) > ( a (1 − ǫ ) − α ) t N .We choose ǫ and α such that a (1 − ǫ ) − α > b + 2 α , and then t such that ( a (1 − ǫ ) − α ) t / ( b + 2 α ) t r . The number of vertices of G ′ t is clearly linear in n (first point of the theorem). b r = ( b + 2 α ) t and a r = ( a (1 − ǫ ) − α ) t fulfillsitems 2, 3 and 4. ⊓⊔ It is shown in [9] that, under
ETH , for any function f no algorithm running intime f ( k ) n o ( k ) can determine whether there exists an independent set of size k ,or not (in a graph with n vertices). A challenging question is to obtain a similarresult for approximation algorithms for Independent Set . In the sequel, wepropose a reduction from
Max-3Sat to Independent Set that, based uponthe negative result of Corollary 2, only gives a negative result for some function f (because Corollary 2 only avoids some subexponential running time). However,this reduction gives the desired inapproximability result if Hypothesis 1, whichis an enforcement of Corollary 2, is used.Based upon Hypothesis 1, the following theorem on parameterized inapprox-imability bound can be proved. Its proof essentially combines the parameterizedreduction in [9] and a classic gap-creating reduction. Theorem 4.
Under Hypothesis 1 and
ETH , for every ǫ > , no parameterizedapproximation algorithm for Independent Set running in time f ( k ) N o ( k ) canachieve approximation ratio r + ǫ in graphs of order N . roof. Suppose that such an algorithm exists for some ǫ > . W.l.o.g., we canassume that f is increasing, and that f ( k ) > k . Take an instance I of Max-3Sat , let K be an integer that will be fixed later, and do the following: Partitionthe m clauses into K groups H , · · · , H K each of them containing, roughly, m/K clauses each. Each group H i involves a number s i m/K of variables. For allpossible values of these variables, add a vertex in the graph G I if these valuessatisfy at least λm/K clauses in H i (the value of λ will also be fixed later).Finally, add an edge between two vertices if they have one contradicting variable.In particular the vertices corresponding to the same group of clauses form aclique. It is easy to see that the so-constructed graph contains N K m/K vertices.The following easy claim holds. Claim.
If a variable assignment satisfies at least λm/K clauses in at most s groups, then it satisfies at most λm + s (1 − λ ) m/K clauses. Proof of claim.
Consider an assignment as the one claimed in claim’s statement.This assignment satisfies at most m/K clauses in at most s groups, and atmost λm/K in the other K − s groups, so in total at most sm/K + ( K − s ) λm/K = λm + s (1 − λ ) m/K , that completes the proof of the claim. ✸ Now, let us go back to the proof of the theorem. Assume an independent setof size at least t in G I . Then one can achieve a partial solution that satisfiesat least λm/K clauses in at least t groups. So, at least tλm/K clauses aresatisfiable. In other words, if at most ( r + ǫ ′ ) m clauses are satisfiable, then amaximum independent set in G I has size at most K r + ǫ ′ λ . Suppose that at least (1 − ǫ ′ ) m clauses are satisfiable. Then, using Lemma 3, there exists a solutionsatisfying at least λm/K clauses in at least − ǫ ′ − λ − λ K groups; otherwise, it shouldbe λm + s (1 − λ ) m/K < (1 − ǫ ′ ) m . Then, there exists an independent set of size − ǫ ′ − λ − λ K in G I .Now, set K = ⌈ φ ( m ) / (1 − ǫ ) ⌉ where φ is the inverse function of f (i.e., φ = f − ). Set also λ = 1 − ǫ , and ǫ ′ = ǫ . Run the assumed ( r + ǫ ) -approximationparameterized algorithm for Independent Set in G I with parameter k =(1 − ǫ ) K . Then, if at least (1 − ǫ ′ ) m equations are satisfiable, there exists anindependent set of size at least − ǫ ′ − λ − λ K = (1 − ǫ /ǫ ) K = (1 − ǫ ) K = k ; so, thealgorithm must output an independent set of size at least ( r + ǫ ) k . Otherwise,if at most ( r + ǫ ′ ) equations are satisfiable, the size of an independent set is atmost K r + ǫ ′ λ = K r + ǫ − ǫ = k r + ǫ (1 − ǫ )(1 − ǫ ) = k ( r + rǫ + o ( ǫ )) .So, for ǫ sufficiently small, the algorithm allows to distinguish between thetwo cases of Max-3Sat (for ǫ ′ ).The running time of the yielded algorithm is f ( k ) N o ( k ) , but f ( k ) = f ((1 − ǫ ) K ) = m , and N o ( k ) = N k/ψ ( k ) for some increasing and unbounded function ψ ,and N o ( k ) = ( K m/K ) k/ψ ( k ) = 2 o ( m ) . ⊓⊔ Using Lemma 3 together with Theorem 4, the following result can be easilyderived. 8 orollary 3.
Under Hypothesis 1 and
ETH , for any r ∈ (0 , there is no r -ap-proximation parameterized algorithm for Independent Set (i.e., an algorithmthat runs in time f ( k ) p ( n ) for some function f and some polynomial p ). Let us now deal with
Dominating Set that is known to be W[2]-hard [14].Existence of FPT-approximation algorithms for this problem is an open ques-tion [15]. Here, we present an approximation preserving reduction (fitting theparameterized framework) that works with the special set of instances producedin the proof of Theorem 4. This reduction will allow us to obtain a lower bound(based on the same hypothesis) for the approximation of min dominating set from Theorem 4.Consider a graph G ( V, E ) on n vertices where V is a set of K cliques C , · · · , C K . We build a graph G ′ ( V ′ , E ′ ) such that G has an independent set ofsize α if and only if G ′ has a dominating set of size K − α . The graph G ′ is builtas follows. For each clique C i in G , add a clique C ′ i of the same size in G ′ . Addalso: an independent set S i of size K , each vertex in S i being adjacent to allvertices in C ′ i and a special vertex t i adjacent to all the vertices in C ′ i . For eachedge e = ( u, v ) with u and v not in the same clique in G , add an independentset W e of size K . Suppose that u ∈ C i and v ∈ C j . Then, each vertex in W e islinked to t i and to all vertices in C ′ i but u (and t j and all vertices in C ′ j but v ).Informally, the reduction works as follows. The set S i ensures that we haveto take at least one vertex in each C ′ i , the fact that | W e | = 3 K ensures that it isnever interesting to take a vertex in W e . If we take vertex t i in a dominating set,this will mean that we do not take any vertex in the set C i in the correspondingindependent set in G . If we take one vertex in C ′ i (but not t i ), this vertex will bein the independent set in G . Let us state this property in the following lemma. Lemma 6. G has an independent set of size α if and only if G ′ has a dominatingset of size K − α .Proof. Suppose that G has an independent set S of size α . Then, S has one vertexin α sets C i , and no vertex in the other K − α sets. We build a dominating set T in G ′ as follows: for each vertex in S we take its copy in G ′ . For each clique C i without vertices in S , we take t i and one (anyone) vertex in C ′ i . The dominatingset T has size α + 2( K − α ) = 2 K − α . For each C ′ i there exists a vertex in T ;so, vertices in C ′ i , t i and vertices in S i are dominated. Now take a vertex in W e with e = ( u, v ) , u ∈ C i and v ∈ C j . If C i ∩ S = ∅ (or C j ∩ S = ∅ ), then t i ∈ T (or t j ∈ T ) and, by construction, t i is adjacent to all vertices in W e . Otherwise,there exist w ∈ S ∩ C i and x ∈ S ∩ C j . Since S is an independent set, either w = u or x = v . If w = u , by construction w (its copy in C ′ i ) is adjacent to allvertices in W e and, similarly, for x if x = v . So, T is a dominating set.Conversely, suppose that T is a dominating set of size K − α . Since S i is anindependent set of size K , we can assume that T ∩ S i = ∅ and the same occurswith W e . In particular, there exists at least one vertex in T in each C i . Now,suppose that T has two different vertices u and v in the same C i . Then we canreplace v by t i getting a dominating set (vertices in S i are still dominated by u ,and any vertex in some W e which is adjacent to v is adjacent to t i ). So, we can9ssume that T has the following form: exactly one vertex in each C i , and K − α vertices t i . Hence, there are α C ′ i cliques where t i is not in T . We consider in G the set S constituted by the α vertices in T in these α sets. Take two vertices u, v in S with, say, u ∈ C ′ i and v ∈ C ′ j (with t i T and t j T ). If there werean edge e = ( u, v ) in G , neither u nor v would have dominated a vertex in W e (by construction). Since neither t i nor t j is in T , this set would not have been adominating set, a contradiction. So S is an independent set. ⊓⊔ Theorem 5.
Under Hypothesis 1 and
ETH , for every ǫ > , no approximationalgorithm running in time f ( k ) N o ( k ) can achieve approximation ratio smallerthan − r − ǫ for Dominating Set in graphs of order N .Proof. In the proof of Theorem 4, we produce a graph G I which is made of K cliques and such that: if at least (1 − ǫ ) m clauses are satisfiable in I , then thereexists an independent set of size (1 − O ( ǫ )) K ; otherwise (at most ( r + ǫ ) m clauses are satisfiable in I ), the maximum independent set has size at most ( r + O ( ǫ )) K . The previous reduction transforms G I in a graph G ′ I such that,applying Lemma 6, in the first case there exists a dominating set of size atmost K − (1 − O ( ǫ )) K = K (1 + O ( ǫ )) while, in the second case, the size ofa dominating set is at least K − ( r + O ( ǫ )) K = K (2 − r − O ( ǫ )) . Thus, weget a gap with parameter k ′ = K (1 + O ( ǫ )) . Note that the number of verticesin G ′ I is N ′ = N + K + 3 K + 3 K | E I | = O ( N ) (where E I is the set of edgesin G I ). If we were able to distinguish between these two sets of instances in time f ( k ′ ) N ′ o ( k ′ ) , this would allow to distinguish the corresponding independent setinstances in time f ( k ′ ) N ′ o ( k ′ ) = g ( k ) N o ( k ) since k ′ = K (1 + O ( ǫ )) = k (1 + O ( ǫ )) ( k = K (1 − ǫ ) being the parameter chosen for the graph G I ). ⊓⊔ Such a lower bound immediately transfers to
Set Cover since a graph on n vertices for Dominating Set can be easily transformed into an equivalentinstance of
Set Cover with ground set and set system both of size n . Corollary 4.
Under Hypothesis 1 and
ETH , for every ǫ > , no approximationalgorithm running in time f ( k ) m o ( k ) can achieve approximation ratio smallerthan − r − ǫ for Set Cover in instances with m sets. Independent Set andrelated problems in subexponential time
As mentioned in Section 2, an almost-linear size
PCP construction [28] for allows to get the negative results stated in Corollaries 1 and 2. In this section, wepresent further consequences of Theorem 1, based upon a combination of knownreductions with (almost) linear size amplifications of the instance.First, Theorem 1 combined with the reduction in [1] showing inapproxima-bility results for
Independent Set in polynomial time, leads to the followingresult. 10 heorem 6.
Under
ETH , for any r > and any δ > , there is no r -approximation algorithm for Independent Set running in time O (2 N − δ ) ,where N is the size of the input graph for Independent Set .Proof.
Given an ǫ > , let ǫ ′ such that < ǫ ′ < ǫ . Given an instance φ of on n variables, we first apply the sparsification lemma (with ǫ ′ ) to get ǫ ′ n instances φ i on at most n variables. Since each variable appears at most c ǫ ′ timesin φ i , the global size of φ i is | φ i | c ǫ ′ n .Consider a particular φ i , r > and δ > . We use the fact that ∈ PCP ,r [(1 + o (1)) log | φ | + D r , E r ] (where D r and E r are constants that dependonly on r ), in order to build the following graph G φ i (see also [1]). For anyrandom string R , and any possible value of the E r bits read by V, add a vertexin the graph if V accepts. If two vertices are such that they have at least onecontradicting bit (they read the same bit which is 1 for one of them and 0 forthe other one), add an edge between them. In particular, the set of verticescorresponding to the same random string is a clique.Assume that φ i is satisfiable. Then there exists a proof for which the verifieraccepts for any random string R . Take for each random string R the vertexin G φ i corresponding to this proof. There is no conflict (no edge) between anyof these | R | vertices, hence α ( G φ i ) = 2 | R | (where, in a graph G , α ( G ) denotesthe size of a maximum independent set).If φ i is not satisfiable, then α ( G φ i ) r | R | . Indeed, suppose that there isan independent set of size α > r | R | . This independent set corresponds to a setof bits with no conflict, defining part of a proof that we can arbitrarily extendto a proof Π . The independent set has α vertices corresponding to α randomstrings (for which V accepts), meaning that the probability of acceptance for thisproof Π is at least α/ | R | > r , a contradiction with the property of the verifier.Furthermore, G φ i has N | R | E r C ′ | φ i | o (1) = Cn o (1) vertices (forsome constants C, C ′ that depend on ǫ ′ ) since | φ i | c ǫ ′ n . Then, one can see that,for any r ′ > r , an r ′ -approximation algorithm for Independent Set runningin time O (2 N − δ ) would allow to decide whether φ i is satisfiable or not in time O (2 n − δ ′ ) for some δ ′ < δ . Doing this for each of the formula φ i would allow todecide whether φ is satisfiable or not in time p ( n )2 ǫ ′ n + 2 ǫ ′ n O (2 n − δ ′ ) = O (2 ǫn ) (where p is a polynomial). This is valid for any ǫ > so it would contradicting ETH . ⊓⊔ Since (for k N ), N k − δ = O (2 N − δ ′ ) , for some δ ′ < δ , the following resultalso holds. Corollary 5.
Under
ETH , for any r > and any δ > , there is no r -approximation algorithm for Independent Set (parameterized by k ) runningin time O ( N k − δ ) , where N is the size of the input graph. The results of Theorem 6 and Corollary 5 can be immediately extended to prob-lems that are linked to
Independent Set by approximability preserving reduc-tions (that preserve at least constant ratios) and have linear amplifications ofthe sizes of the instances. 11or instance, this is the case of
Set Packing (preservation of constant ratiosand of ratios functions of the input size with amplification that is the identityfunction). This holds for the
Bipartite Subgraph problem where, given agraph G ( V, E ) , the goal is to find a maximum-size subset V ′ ⊆ V such that thegraph G [ V ′ ] is a bipartite graph. Proposition 1.
Under
ETH , for any r > and any δ > , there is no r -approximation algorithm for either Set Packing or Bipartite Subgraph running in time O (2 n − δ ) in a graph of order n .Proof. Consider the following reduction from
Independent Set to BipartiteSubgraph ([30]). Let G ( V, E ) be an instance of Independent Set of order n .Construct a graph G ′ ( V ′ , E ′ ) for Bipartite Subgraph by taking two distinctcopies of G (denote them by G and G , respectively) and adding the followingedges: a vertex v i of copy G is linked with a vertex v j of G , if and only ifeither i = j or ( v i , v j ) ∈ E . G ′ has n vertices. Let now S be an independentset of G . Then, obviously, taking the two copies of S in G and G induces abipartite graph of size | S | . Conversely, consider an induced bipartite graph in G ′ of size t , and take the largest among the two color classes. By construction itcorresponds to an independent set in G , whose size is at least t/ (note that itcannot contain 2 copies of the same vertex). So, any r -approximate solution for Bipartite Subgraph in G ′ can be transformed into an r -approximate solutionfor Independent Set in G . Observe finally that the size of G ′ is two times thesize of G . ⊓⊔ Dealing with minimization problems, Theorem 6 and Corollary 5 can beextended to
Coloring , thanks to the reduction given in [24].Given a graph G whose vertex set is partitioned into K cliques each of size S ,and given a prime number q > S , a graph H q having the following propertiescan be built in polynomial time: (i) the vertex set of H q is partitioned into q K cliques, each of size q ; (ii) α ( H q ) max { q α ( G ); q ( α ( G ) −
1) + K ; qK } ; (iii) if α ( G ) = K , then χ ( H q ) = q .Note that this reduction uses the particular structure of graphs producedin the inapproximability result in [1] (as in Theorem 6). Then, we deduce thefollowing result. Proposition 2.
Under
ETH , for any r > and any δ > , there is no r -approximation algorithm for Coloring running in time O (2 n − δ ) in a graph oforder n .Proof. Fix a ratio r > , and let r IS > be such that r IS + r IS /r . Start fromthe graph G φ i produced in the proof of Theorem 6 for ratio r IS . The vertex setof G φ i is partitioned into K = 2 | R | cliques, each of size at most E r . By addingdummy vertices (a linear number, since E r is a fixed constant), we can assumethat each clique has the same size S = 2 E r , so the number of vertices in G φ i is N = KS = 2 | R | E r .Let q > max { S, /r IS } be a prime number, and consider the graph H q pro-duced from G φ i by the reduction in [24] mentioned above. If φ i is satisfiable,12 ( G φ i ) = K and then by the third property of the graph H q , χ ( H q ) = q . Oth-erwise, by the second property α ( H q ) max { q α ( G φ ); q ( α ( G φ ) −
1) + K ; qK } .Formula φ i being not satisfiable, α ( G φ i ) r IS K . By the choice of q , qK q r IS K , so α ( H q ) q r IS K + K = ( q r IS + 1) K . Since the number of verticesin H q is Kq , we get that χ ( H q ) > q / ( q r IS + 1) . The gap created for thechromatic number in the two cases is then at least: q ( q r IS + 1) q = 1 r IS + 1 /q > r IS + r IS > r The result follows since H q has Kq vertices and q is a constant (that dependsonly on the ratio r and on the constant number of bits p read by V), so the sizeof H q is linear in the size of G φ i . ⊓⊔ We consider the approximability of
Vertex Cover and
Min-Sat in subexpo-nential time. The following statement provides a lower bound to such a possi-bility.
Proposition 3.
Under
ETH , for any r > and any δ > , there is no (7 / − ǫ ) -approximation algorithm for Vertex Cover running in time O (2 N − δ ) ingraphs of order N , nor for Min-Sat running in time m − δ in CNF formulæwith m clauses.Proof. We combine Corollary 1 with the following classical reduction. Consideran instance I of max 3-lin on m equations. Build the following graph G I : – for any equation and any of the eight possible values of the 3 variables in it,add a vertex in the graph if the equation is satisfied; – if two vertices are such that they have one contradicting variable (the samevariable has value 1 for one vertex and 0 for the other one), then add anedge between them.In particular, the set of vertices corresponding to the same equation is a clique.Note that each equation is satisfied by exactly 4 values of the variables in it.Then, the number of vertices in the graph is N = 4 m . Consider an independentset S in the graph G I . Since there is no conflict, it corresponds to a partialassignment that can be arbitrarily completed into an assignment τ for the wholesystem. Each vertex in S corresponds to an equation satisfied by τ (and S has atmost one vertex per equation), so τ satisfies (at least) | S | equations. Reciprocally,if an assignment τ satisfies α clauses, there is obviously an independent setof size α in G I . Hence, if (1 − ǫ ) m equations are satisfiable, there exists anindependent set of size at least (1 − ǫ ) m , i.e., a vertex cover of size at most N − (1 − ǫ ) m = N (3 / ǫ/ . If at most (1 / ǫ ) m equations are satisfiable,then each vertex cover has size at least N − (1 / ǫ ) m = N (7 / − ǫ/ .We now handle Min-Sat problem via the following reduction (see [25]). Givena graph G , build the following instance on Min-Sat . For each edge ( v i , v j ) adda variable x ij . For each vertex v i add a clause c i . Variable x ij appears positivelyin c i and negatively in c j . Then, take a vertex cover V ∗ of size k ; for any x ij
13x the variable to true if v i ∈ V ∗ , to false otherwise. Consider a clause c j with v j V ∗ . If x ij is in c j then v i is in V ∗ hence x ij is true; if x ji is in c j then, byconstruction, x ji is false. So c j is not satisfied, and the assignment satisfies atmost k clauses. Conversely, consider a truth assignment that satisfies k clauses c i , · · · , c i k . Consider the vertex set V ∗ = { v i , · · · , v i k } . For an edge ( v i , v j ) ,if x ij is set to true then c i is satisfied and v i is in V ∗ , otherwise c j is satisfiedand v j is in V ∗ , so V ∗ is a vertex cover of size k . Since the number of clausesin the reduction equals the number of vertices in the initial graph, the result isconcluded. ⊓⊔ All the results given in this section are valid under
ETH and rule out some ratioin subexponential time of the form n − δ . It is worth noticing that if L P C holds,then all these result would hold for any subexponential time.
Corollary 6. If L P C holds, under
ETH the negative results of Theorem 6 andPropositions 1, 2 and 3 hold for any time complexity o ( n ) .Proof. Using
LPC , the same proof as in Theorem 6 creates for each φ i a graph on N = O ( n ) variables with either an independent set of size αN (if φ i is satisfiable)or a maximum independent set of size at most α/ N (if φ i is not satisfiable).Then using expander graphs, Theorem 3 allows to amplify this gap from 1/2 toany constant r > while preserving the linear size of the instance. Results forthe other problems immediately follow from the same arguments as above. ⊓⊔ This paper presents conditional lower bounds of approximation ratio in FPT- andsubexponential-time. Assuming
ETH , we prove inapproximability in time n − δ for any δ > for the problems such as: Independent Set , Set Packing , Bipartite Subgraph , Coloring , Vertex Cover . If L inear PCP Conjectureturns out to hold, even in time o ( n ) we cannot approximate any better. Alsoassuming ETH , we proved that
Linear PCP Conjecture implies FPT-timeinapproximabilty of
Independent Set (for any ratio) and
Dominating Set (for some ratio).Our effort in this paper is only a first step and we wish to motivate furtherresearch. There remains a range of problems to be tackled, among which wepropose the followings. – Our inapproximability results, in particular those in FPT-time, are condi-tional upon L inear PCP Conjecture. Is it possible to relax the condition toa more plausible one? – Or, we dare ask whether (certain) inapproximability results in FPT-timeimply strong improvement in PCP theorem. For example, would the converseof Lemma 2 hold?Note that we have considered in this article constant approximation ratios. Inthis sense, Theorem 6 is “tight” with respect to approximation ratios since, as14entioned in Section 2, ratio /r ( n ) is achievable in subexponential time forany increasing and unbounded function r . However, dealing with parameterizedapproximation algorithms, achieving a non constant ratio is also an open ques-tion. More precisely, finding in FPT-time an independent set of size g ( k ) whenthere exists an independent set of size k is not known for any unbounded andincreasing function g .Finally, let us note that, in the same vein of our work, [27] in his recentpaper initiates a proof checking view of parameterized complexity, by proposing aparameterized PCP and by giving a parameterized PCP characterization of W[1].Possible links between these two approaches are worth being investigated infuture works. References
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