Polynomial Kernelizations for MIN F^+Pi_1 and MAX NP
aa r X i v : . [ c s . CC ] D ec Polynomial Kernelizations for MIN F + Π and MAX NP Stefan KratschMay 28, 2018
Abstract
It has been observed in many places that constant-factor approximable problems often ad-mit polynomial or even linear problem kernels for their decision versions, e.g.,
Vertex Cover , Feedback Vertex Set , and
Triangle Packing . While there exist examples like
Bin Pack-ing , which does not admit any kernel unless P = NP, there apparently is a strong relationbetween these two polynomial-time techniques. We add to this picture by showing that the nat-ural decision versions of all problems in two prominent classes of constant-factor approximableproblems, namely MIN F + Π and MAX NP, admit polynomial problem kernels. Problems inMAX SNP, a subclass of MAX NP, are shown to admit kernels with a linear base set, e.g.,the set of vertices of a graph. This extends results of Cai and Chen (JCSS 1997), stating thatthe standard parameterizations of problems in MAX SNP and MIN F + Π are fixed-parametertractable, and complements recent research on problems that do not admit polynomial kernel-izations (Bodlaender et al. JCSS 2009). Approximation and kernelization are two major ways of coping with NP-hardness in polynomialtime. The former relaxes the exactness requirement to that of finding good approximate solutions.The latter, as a formulation of preprocessing, shrinks the instance to a guaranteed size in termsof some difficulty parameter. For approximate solutions to a problem it is quite desirable to getsolutions within a constant-factor of the optimum, or even arbitrarily good approximations inpolynomial time through polynomial-time approximation schemes. In the world of preprocessing, polynomial kernelizations with a guaranteed size polynomial in the parameter are often the firstgoal, later aiming for stronger and stronger bounds down to linear kernels. Considering these twopolynomial-time techniques it is only natural to study the relation between them.This paper seeks to further the understanding of the relation between constant-factor approx-imation and polynomial kernelizations. This is motivated by the large number of problems thatboth techniques were successfully applied to so far, e.g.,
Vertex Cover , Max Sat , FeedbackVertex Set , and
Triangle Packing ; see Table 1 for approximability and kernelization resultsfor some well-known problems. Let us point out that there do exist examples that rule out a generalequivalence of these two notions, e.g.,
Connected Vertex Cover or Bin Packing . Both prob-lems have constant-factor approximation algorithms but none of them admits a polynomial kernel:The former admits a O (2 . k n c )-time algorithm [32], and hence a kernel of size O (2 . k ) , but ithas no kernel of polynomial size unless the polynomial hierarchy collapses [13]. The latter does not By a folklore result: run the algorithm for O ( n c +1 ) steps, it will either provide the correct answer (and we returna yes- or no-instance of constant size) or if it does not finish then it follows that n < . k and we have the kernel. Vertex Cover O ( k ) [11] Connected Vertex Cover
Feedback Vertex Set O ( k ) [39] Bin Packing . Minimum Fill-In O (opt) [33] O ( k ) [33] Treewidth O ( √ log opt) [17] not polynomial [5]Table 1: Approximation ratio and size of problem kernels for some optimization problems.even admit a polynomial-time algorithm for k = 2 unless P = NP, by an immediate reduction from Partition [22]. Consider also the
Minimum Fill-In problem, which has a polynomial kernel butthe best known ratio is O (opt) [33]. Since a general result is ruled out we take the natural approachof considering subclasses of the class of all constant-factor approximable problems (APX), namely MIN F + Π and MAX NP. Our work.
We prove that the standard parameterizations of problems in
MIN F + Π and MAX NP admit polynomial kernelizations. This extends results of Cai and Chen [8] who showed that the stan-dard parameterizations of all problems in
MIN F + Π and MAX SNP (a subclass of
MAX NP )are fixed-parameter tractable; or equivalently admit some (possibly exponential) kernelization. In-terestingly perhaps, both our results rely on the Sunflower Lemma due to Erd˝os and Rado [15].
Related work.
Kernelization has received significant interest over the last fifteen years, maturingfrom a technique to prove fixed-parameter tractability into its own field of research. In the literaturethere exist a significant number of positive results; we will only highlight a few from recent years,namely a kernel with O ( k ) vertices for Vertex Cover by Chen et al. [11], a O ( k ) vertices kernelfor Feedback Vertex Set by Thomass´e, and a O ( k d − ) vertices kernel for d - Hitting Set byAbu-Khzam [1]. Recently Bodlaender et al. [5] presented the first negative results concerning theexistence of polynomial kernelizations for some natural fixed-parameter tractable problems. Usingthe notion of a distillation algorithm and results due to Fortnow and Santhanam [21], they were ableto show that the existence of polynomial kernelizations for so-called compositional parameterizedproblems implies a collapse of the polynomial hierarchy to the third level. These seminal resultsled to an increased interest in polynomial lower bounds for kernelization as well as in polynomialkernelizations as a good way of understanding efficient preprocessing (and possibly ruling it outby means of polynomial lower bounds). A follow-up paper by Bodlaender et al. [7] proposedthe application of polynomial-time transformations, that allow only a polynomial increase in theparameter, to transfer lower and upper bounds between problems. A number of papers alreadyapply the framework of Bodlaender et al. [5, 7] to obtain polynomial lower bounds for a variety ofproblems. e.g., [13, 18, 29]. An important contribution to kernelization lower bounds was madeby Dell and van Melkebeek [12], who showed, amongst others, that
Feedback Vertex Set doesnot admit a kernelization to size O ( k − ǫ ) unless the polynomial hierarchy collapses, i.e., theremay be a kernelization with O ( k ) vertices but the number of O ( k ) edges is essentially optimal. Treewidth does not admit a polynomial kernelization unless there is an and-distillation algorithm for all NPcomplete problems [5]. Though unlikely, this is not known to imply a collapse of the polynomial hierarchy. H -minor-free graphs, given certain additional properties like finite integer index or quasi-compactness.Furthermore, two recent papers obtain complete classifications of three parameterized constraintsatisfaction problems into admitting or not admitting polynomial kernels depending on the languageof permitted constraints [30, 28]. For more background on kernelization we refer to the recentsurveys on kernelization given by Guo and Niedermeier [23] as well as by Bodlaender [4]. In anearlier paper, Mahajan et al. [31] studied MAX SNP problems and observe that kernelizationsfollow from the fact that NP-hard problems in
MAX SNP have guaranteed lower bounds for theoptimum value, motivating them to study these problems parameterized above such lower bounds.Cai and Huang [9] showed that all problems in
MAX SNP admit fixed-parameter approximationschemes.
MIN F + Π and MAX NP. Two decades ago Papadimitriou and Yannakakis [36] initiated thesyntactic study of optimization problems to extend the understanding of approximability. Theyintroduced the classes
MAX NP and
MAX SNP as natural variants of NP based on Fagin’s [16]syntactic characterization of NP. Essentially problems are in
MAX NP or MAX SNP if theiroptimum value can be expressed as the maximum number of tuples for which some existential, re-spectively quantifier-free, first-order formula holds. They showed that every problem in these twoclasses is approximable to within a constant factor of the optimum. Arora et al. complemented thisby proving that no
MAX SNP -complete problem has a polynomial-time approximation scheme,unless P=NP [2]. Contained in
MAX SNP there are some well-known maximization problems,such as
Max Cut , Max q -Sat , and Independent Set on graphs of bounded degree. Its super-class
MAX NP also contains
Max Sat amongst others.Kolaitis and Thakur generalized the approach of examining the logical definability of opti-mization problems and defined further classes of minimization and maximization problems [26, 27].Amongst others they introduced the class
MIN F + Π of problems whose optimum can be expressedas the minimum weight of an assignment (i.e., number of ones) that satisfies a certain universalfirst-order formula. They proved that every problem in MIN F + Π is approximable to within aconstant factor of the optimum. In MIN F + Π there are problems like Vertex Cover , d - HittingSet , and
Triangle Edge Deletion . Organization of the paper.
Section 2 covers the definitions of the classes
MIN F + Π and MAX NP , as well as the necessary details from parameterized complexity. In Sections 3 and 4 wepresent polynomial kernelizations for the standard parameterizations of problems in
MIN F + Π and MAX NP respectively. Section 5 summarizes our results and poses some open problems.
Logic and complexity classes.
A (relational) vocabulary is a set σ of relation symbols, eachhaving some fixed integer as its arity. Atomic formulas over σ are of the form R ( z , . . . , z t ) where R is a t -ary relation symbol from σ and the z i are variables. A literal is an atomic formula orthe negation of an atomic formula. The set of quantifier-free (relational) formulas over σ is theclosure of the set of all atomic formulas under negation, conjunction, and disjunction. A formula3n conjunctive normal form is a conjunction of disjunctions of literals, called clauses . A formula in disjunctive normal form is a disjunction of conjunctions of literals, called disjuncts . Definition 1 ( MIN F + Π , MAX NP ) . A tuple A = ( A, R , . . . , R t ) where A is a finite set andeach R i is an r i -ary relation over A is called a finite structure of type ( r , . . . , r t ).Let Q be an optimization problem on finite structures of type ( r , . . . , r t ).(a) The problem Q is contained in MIN F + Π if its optimum on finite structures A = ( A, R , . . . , R t )of type ( r , . . . , r t ) can be expressed asopt Q ( A ) = min S {| S | : ( A , S ) | = ( ∀ x ∈ A c x ) : ψ ( x , S ) } , where S is a single relation symbol and ψ ( x , S ) is a quantifier-free formula in conjunctive nor-mal form over the vocabulary { R , . . . , R t , S } on variables { x , . . . , x c x } . Furthermore, ψ ( x , S ) ispositive in S , i.e., S does not occur negated in ψ ( x , S ).(b) The problem Q is contained in MAX NP if its optimum on finite structures A = ( A, R , . . . , R t )of type ( r , . . . , r t ) can be expressed asopt Q ( A ) = max S |{ x ∈ A c x : ( A , S ) | = ( ∃ y ∈ A c y ) : ψ ( x , y , S ) }| , where S = ( S , . . . , S u ) is a tuple of s i -ary relation symbols S i and ψ ( x , y , S ) is a quantifier-free formula in disjunctive normal form over the vocabulary { R , . . . , R t , S , . . . , S u } on variables { x , . . . , x c x , y , . . . , y c y } .The definition of MAX SNP is similar to that of
MAX NP but without the existential quan-tification of y , i.e., opt Q ( A ) = max S |{ x : ( A , S ) | = ψ ( x , S ) }| . Remark 1.
Since the formulas ψ depend only on the problem Q they are of constant length withrespect to inputs A . Thus there is no strict need to require normal forms, but the chosen ones fitthe quantification nicely, e.g., we can view ( ∀ x ) : ψ ( x , S ) as a large conjunctive normal form. Example 1 ( Minimum Vertex Cover ) . Let G = ( V, E ) be a finite structure of type (2) thatrepresents a graph by a set V of vertices and a binary relation E over V as its edges. The optimumof Minimum Vertex Cover on structures G can be expressed as:opt V C ( G ) = min S ⊆ V {| S | : ( G, S ) | = ( ∀ ( u, v ) ∈ V ) : ( ¬ E ( u, v ) ∨ S ( u ) ∨ S ( v )) } . This implies that
Minimum Vertex Cover is contained in
MIN F + Π . Example 2 ( Maximum Satisfiability ) . Formulas in conjunctive normal form can be representedby finite structures F = ( F, P, N ) of type (2 , F be the set of all clauses and variables, andlet P and N be binary relations over F . Let P ( x, c ) be true if and only if x is a literal of theclause c and let N ( x, c ) be true if and only if ¬ x is a literal of the clause c . The optimum of MaxSat on structures F can be expressed as:opt MS ( F ) = max T ⊆ F |{ c ∈ F : ( F , T ) | = ( ∃ x ∈ F ) : ( P ( x, c ) ∧ T ( x )) ∨ ( N ( x, c ) ∧ ¬ T ( x )) }| . Thus
Max Sat is contained in
MAX NP .For a detailed introduction to
MIN F + Π , MAX NP , and
MAX SNP we refer the readerto [36, 26, 27]. An introduction to logic and complexity can be found in [35].4 arameterized complexity.
Parameterized complexity provides a multivariate analysis of com-binatorially hard problems, considering at least one additional parameter of input instances apartfrom their size. This allows a more fine-grained analysis of the required runtimes than the merestatement of NP-hardness could provide. In the following we give the necessary formal definitions,namely fixed-parameter tractability, standard parameterizations, and kernelization.
Definition 2 (Parameterized problem, Fixed-parameter tractability) . A parameterized problem p - Q over the alphabet Σ is a subset of Σ ∗ × N ; the second component is called the parameter .A parameterized problem p - Q is fixed-parameter tractable if there exists an algorithm A , apolynomial p , and a computable function f : N → N such that A decides ( x, k ) ∈ p - Q in time f ( k ) · p ( | x | ). FPT is the class of all fixed-parameter tractable problems. Definition 3 (Standard parameterization) . Let Q be a maximization (minimization) problem.Its standard parameterization is defined as p - Q := { ( A , k ) | opt Q ( A ) ≥ k } (respectively p - Q := { ( A , k ) | opt Q ( A ) ≤ k } for minimization problems).Basically, the standard parameterization of an optimization problem is its decision version,asking whether the optimum is at least k (respectively at most k ), parameterized by k . Definition 4 (Kernelization) . Let p - Q ⊆ Σ ∗ × N be a parameterized problem over Σ. A polynomial-time computable function K : Σ ∗ × N → Σ ∗ × N is a kernelization of p - Q if there is a computablefunction h : N → N such that for all ( x, k ) ∈ Σ ∗ × N , and letting ( x ′ , k ′ ) := K (( x, k )), we have1. ( x, k ) ∈ p - Q ⇔ ( x ′ , k ′ ) ∈ p - Q as well as2. | x ′ | ≤ h ( k ) and k ′ ≤ h ( k ).We call h the size of the problem kernel ( x ′ , k ′ ). The kernelization K is polynomial if h is apolynomial. We say that p - Q admits a (polynomial) kernelization if there exists a (polynomial)kernelization of p - Q .Essentially, a kernelization is a polynomial-time data reduction that comes with a guaranteedupper bound on the size of the resulting instance in terms of the parameter.For an introduction to parameterized complexity we refer the reader to [14, 19, 34]. Hypergraphs and sunflowers. A hypergraph is a tuple H = ( V, E ) consisting of a finite set V ,its vertices, and a family E of subsets of V , its edges. A hypergraph has dimension d if each edgehas cardinality at most d . A hypergraph is d - uniform if each edge has cardinality exactly d . Definition 5 (Sunflower) . Let H be a hypergraph. A sunflower of cardinality r is a set F = { f , . . . , f r } of edges of H such that every pair has the same intersection C , i.e., for all 1 ≤ i < j ≤ r : f i ∩ f j = C . The set C is called the core of the sunflower, the disjoint sets f i \ C are called petals .The following lemma is the beautiful Sunflower Lemma due to Erd˝os and Rado [15]. Lemma 1 (Sunflower Lemma) . Let k, d ∈ N and let H be a d -uniform hypergraph with morethan k d · d ! edges. Then there is a sunflower of cardinality k + 1 in H . For every fixed d there isan algorithm that computes such a sunflower in time polynomial in | E ( H ) | .
5e give a short sketch of its algorithmic proof. The idea is to greedily select disjoint sets. If atleast k + 1 sets are found then they form a sunflower with core C = ∅ . Otherwise all other sets mustintersect the at most dk elements of the selected sets. Then the search continues among those setsthat contain the most frequent element, i.e., occurring in at least | E ( H ) | /dk sets. This terminatesafter d − d − d -dimensional hypergraphs. Corollary 1.
The same holds for d -dimensional hypergraphs with more than k d · d ! · d edges.Proof. For some d ′ ∈ { , . . . , d } , H has more than k d · d ! ≥ k d ′ · d ′ ! edges of cardinality d ′ . Let H d ′ be the d ′ -uniform subgraph induced by the edges of cardinality d ′ . We apply the Sunflower Lemmaon H d ′ and obtain a sunflower F of cardinality k + 1 in time polynomial in | E ( H d ′ ) | ≤ | E ( H ) | .Clearly F is also a sunflower of H . + Π The class
MIN F + Π was introduced by Kolaitis and Thakur in a framework of syntactically de-fined classes of optimization problems [26]. In a follow-up paper they showed that every problemin MIN F + Π is constant-factor approximable [27]. We will prove that the standard parameteri-zation of any problem in MIN F + Π admits a polynomial kernelization.Let us fix some optimization problem Q from MIN F + Π that takes as input finite structuresof type ( r , . . . , r t ). Accordingly let R , . . . , R t be relation symbols of arity r , . . . , r t . Since Q ∈
MIN F + Π there is a c S -ary relation symbol S and a quantifier-free formula ψ ( x , S ) in conjunctivenormal form such that:1. the formula ψ ( x , S ) is positive in S , i.e., there are no literals ¬ S ( x , . . . , x c S ) and2. the optimum value of Q on input A of type ( r , . . . , r t ) can be expressed asopt Q ( A ) = min S ⊆ A cS {| S | : ( A , S ) | = ( ∀ x ∈ A c x ) : ψ ( x , S ) } . We denote by s the maximum number of occurrences of S in any clause of ψ ( x , S ). This valueplays a crucial role in our kernelization bound. For the polynomial kernelization we consider thestandard parameterization of Q , denoted by p - Q : Input:
A finite structure A of type ( r , . . . , r t ) and an integer k . Parameter: k . Task:
Decide whether opt Q ( A ) ≤ k .We will see that, given an instance ( A , k ), deciding whether opt Q ( A ) ≤ k is equivalent todeciding an instance of s - Hitting Set , defined as follows:
Input:
A hypergraph H = ( V, E ) of dimension s and an integer k . Parameter: k . Task:
Decide whether H has a hitting set of size at most k , i.e., S ⊆ V , | S | ≤ k , suchthat S has a nonempty intersection with every edge of H .6he following definition formalizes the procedure of plugging in a specific tuple x ∈ A c x into ψ ( x , S ). That way all occurrences of relation symbols R i can be evaluated, as they are part ofthe input ( A , k ), leaving only literals S ( · ). Definition 6.
Let A = ( A, R , . . . , R t ) be a finite structure of type ( r , . . . , r t ) and let x ∈ A c x .We define ψ x ( S ) to be the formula obtained in the following way:1. Replace all variables x , . . . , x c x by the chosen elements of A .2. Replace all literals R i ( z ) and ¬ R i ( z ) by 1 (true) or 0 (false) depending on whether z iscontained in R i (note that z is a concrete tuple from A r i by Step 1).3. Delete all clauses that contain a 1 and delete all occurrences of 0.Observe that application of Definition 6 yields an equivalent formula in the sense that ( A , S ) | = ψ ( x , S ) if and only if ( A , S ) | = ψ x ( S ), since we only replace literals according to the input. It is easyto see that ψ x ( S ) is a formula in conjunctive normal form on literals S ( z ) for some z ∈ A c S ; there areat most s literals per clause. A formula ψ x ( S ) can have empty clauses when all literals R i ( · ), ¬ R i ( · )in a clause are evaluated to 0 and there are no literals S ( · ). In that case, no assignment to S cansatisfy the formula ψ x ( S ), or equivalently ψ ( x , S ). Thus ( A , k ) is a no-instance and we mayreject it or return a dummy no-instance of constant size. Note that clauses of ψ x ( S ) cannotcontain contradicting literals since ψ ( x , S ) is positive in S . Henceforth we assume all clauses offormulas ψ x ( S ) to be nonempty.We continue by defining a mapping Φ from finite structures A to hypergraphs H . Then weshow that ( A , k ) is a yes-instance for p - Q if and only if (Φ( A ) , k ) is a yes-instance for s - HittingSet
Definition 7.
Let A be an instance of Q . We define Φ( A ) := H with H = ( V, E ). We let E bethe family of all sets e = { z , . . . , z p } such that ( S ( z ) ∨ · · · ∨ S ( z p )) is a clause of a ψ x ( S ) forsome x ∈ A c x . We let V be the union of all sets e ∈ E .The hypergraphs H obtained from the mapping Φ have dimension s since each ψ x ( S ) has atmost s literals per clause. The following lemma establishes the equivalence of ( A , k ) and ( H , k ) =(Φ( A ) , k ). Lemma 2.
Let A = ( A, R , . . . , R t ) be a finite structure of type ( r , . . . , r t ) and let k be an integer.Then ( A , k ) is a yes-instance of p - Q if and only if (Φ( A ) , k ) is a yes-instance of s - Hitting Set .Proof.
It suffices to show that for all S ⊆ A c S :( A , S ) | = ( ∀ x ∈ A c x ) : ψ ( x , S ) if and only if S is a hitting set for Φ( A ) . Let H = Φ( A ) = ( V, E ) and let S ⊆ A c S :( A , S ) | = ( ∀ x ∈ A c x ) : ψ ( x , S ) ⇔ ( A , S ) | = ( ∀ x ∈ A c x ) : ψ x ( S ) ⇔ ( ∀ x ∈ A c x ) : each clause of ψ x ( S ) has a literal S ( z ) for which z ∈ S ⇔ S has a nonempty intersection with every set e ∈ E ⇔ S is a hitting set for H = ( V, E ) . Since the number of ones in the assignment to S , i.e., the number of tuples z ∈ A c S with S ( z ) = 1,translates directly to the cardinality of the hitting set and vice versa, the lemma follows.7ur kernelization will consist of the following three steps:1. Map the given instance ( A , k ) for p - Q to an equivalent instance ( H , k ) = (Φ( A ) , k ) for s - Hitting Set according to Definition 7 and Lemma 2.2. Use a polynomial kernelization for s - Hitting Set on ( H , k ) to obtain an equivalent in-stance ( H ′ , k ) with size polynomial in k .3. Use ( H ′ , k ) to derive an equivalent instance ( A ′ , k ) of p - Q . That way we will be able toconclude that ( A ′ , k ) is equivalent to ( H , k ) and hence also to ( A , k ).There are two kernelizations for s - Hitting Set : one by Flum and Grohe [19] based on theSunflower Lemma due to Erd˝os and Rado [15] and a recent one by Abu-Khzam [1] based on crowndecompositions. For our purpose of deriving an equivalent instance for p - Q , these kernelizationshave the drawback of shrinking sets during the reduction, since we need to find an equivalentinstance of p - Q afterwards. To shrink edges we would need to shrink clauses of the formula ψ ( x , S ),but we may only change the instance ( A , k ). Fortunately we are able to modify Flum and Grohe’skernelization to use only edge deletions. Remark 2.
Crown decompositions frequently produce the strongest kernelization results by virtueof proving certain decisions to be optimal, usually independent of the solution size k . Kernelizationbased on sunflowers makes use of the solution size, showing that certain decisions are forced.The sunflower-based kernelization for s - Hitting Set uses the fact that a sunflower of cardi-nality greater than k forces an element of its core to be selected; recall that the petals are pairwisedisjoint. Thus such a sunflower may be replaced by its core. In our case the idea is to shrinksunflowers from size at least k + 2 down to size k + 1. This way the selection of an element from thecore is still forced, but we are able to reduce the size of our instance without shrinking of edges. Theorem 1.
There exists a polynomial kernelization of s - Hitting Set that, given an instance ( H , k ) , computes an instance ( H ∗ , k ) such that E ( H ∗ ) ⊆ E ( H ) , H ∗ has O ( k s ) edges, and the sizeof ( H ∗ , k ) is O ( k s ) as well.Proof. Let ( H , k ) be an instance of s - Hitting Set , with H = ( V, E ). If H contains a sunflower F = { f , . . . , f k +1 } of cardinality k + 1 then every hitting set of H must have a nonempty intersectionwith the core C of F or with the k + 1 disjoint sets f \ C, . . . , f k +1 \ C . Thus every hitting set ofat most k elements must have a nonempty intersection with C .Now consider a sunflower F = { f , . . . , f k +1 , f k +2 } of cardinality k + 2 in H and let H ′ =( V, E \ { f k +2 } ). We show that the instances ( H , k ) and ( H ′ , k ) are equivalent. Clearly every hittingset for H is also a hitting set for H ′ since E ( H ′ ) ⊆ E ( H ). Let S ⊆ V be a hitting set of size atmost k for H ′ . Since F \ { f k +2 } is a sunflower of cardinality k + 1 in H ′ , it follows that S has anonempty intersection with its core C . Hence S has a nonempty intersection with f k +2 ⊇ C too.Thus S is a hitting set of size at most k for H , implying that ( H , k ) and ( H ′ , k ) are equivalent.We turn this fact into a kernelization, by starting with H ∗ = H and by repeating the followingstep while H ∗ has more than ( k + 1) s · s ! · s edges. By Corollary 1 we obtain a sunflower ofcardinality k + 2 in H ∗ in time polynomial in | E ( H ∗ ) | . We delete an edge of the detected sunflowerfrom the edge set of H ∗ , thereby reducing the cardinality of the sunflower to k + 1. Thus, bythe argument from the previous paragraph, we maintain that ( H , k ) and ( H ∗ , k ) are equivalent.Furthermore E ( H ∗ ) ⊆ E ( H ) and H ∗ has no more than ( k + 1) s · s ! · s ∈ O ( k s ) edges. Since we delete8n edge of H ∗ in each step, there are O ( | E ( H ) | ) steps, and the total time is polynomial in | E ( H ) | .Deleting all isolated vertices from H ∗ yields a size of O ( s · k s ) = O ( k s ) since each edge contains atmost s vertices.The following lemma proves that every s - Hitting Set instance that is “sandwiched” betweentwo equivalent instances must be equivalent to both.
Lemma 3.
Let ( H , k ) be an instance of s - Hitting Set and let ( H ∗ , k ) be an equivalent instancewith E ( H ∗ ) ⊆ E ( H ) . Then for any H ′ with E ( H ∗ ) ⊆ E ( H ′ ) ⊆ E ( H ) the instance ( H ′ , k ) isequivalent to ( H , k ) and ( H ∗ , k ) .Proof. Observe that hitting sets for H can be projected to hitting sets for H ′ (i.e., restricted tothe vertex set of H ′ ) since E ( H ′ ) ⊆ E ( H ). Thus if ( H , k ) is a yes-instance then ( H ′ , k ) is a yes-instance too. The same argument holds for ( H ′ , k ) and ( H ∗ , k ). Together with the fact that ( H , k )and ( H ∗ , k ) are equivalent, this proves the lemma.Now we are well equipped to prove that p - Q admits a polynomial kernelization. The mainremaining difficulty lies in finding an instance of p - Q that is equivalent to the kernelized s - HittingSet instance that we already know how to obtain. It is in fact easier to find an instance of p - Q that is equivalent to a sandwiched instance. Theorem 2.
Let
Q ∈
MIN F + Π . The standard parameterization p - Q of Q admits a polynomialkernelization.Proof. Let ( A , k ) be an instance of p - Q . By Lemma 2 we have that ( A , k ) is a yes-instance of p - Q ifand only if ( H , k ) = (Φ( A ) , k )) is a yes-instance of s - Hitting Set . We apply the kernelization fromTheorem 1 to ( H , k ) and obtain an equivalent s - Hitting Set instance ( H ∗ , k ) such that E ( H ∗ ) ⊆ E ( H ) and H ∗ has O ( k s ) edges.Recall that every edge of H , say { z , . . . , z p } , corresponds to a clause ( S ( z ) ∨ · · · ∨ S ( z p ))of ψ x ( S ) for some x ∈ A c x . Thus for each edge e ∈ E ( H ∗ ) ⊆ E ( H ) we can select a tuple x e such that e corresponds to a clause of ψ x e ( S ). Let X be the set of the selected tuples x e for alledges e ∈ E ( H ∗ ). Let A ′ ⊆ A be the set of all components of tuples x e ∈ X , ensuring that X ⊆ A ′ c x .Let R ′ i be the restriction of R i to A ′ and let A ′ = ( A ′ , R ′ , . . . , R ′ t ).Let ( H ′ , k ) = (Φ( A ′ ) , k ). By definition of Φ and by construction of H ′ we know that E ( H ∗ ) ⊆ E ( H ′ ) ⊆ E ( H ) since X ⊆ A ′ c x ⊆ A c x . Thus, by Lemma 3, we have that ( H ′ , k ) is equivalentto ( H , k ). Furthermore, by Lemma 2, ( H ′ , k ) is a yes-instance of s - Hitting Set if and onlyif ( A ′ , k ) is a yes-instance of p - Q . Thus ( A ′ , k ) and ( A , k ) are equivalent instances of p - Q .We conclude the proof by giving an upper bound on the size of ( A ′ , k ) that is polynomial in k .The set X contains at most | E ( H ∗ ) | ∈ O ( k s ) tuples. These tuples have no more than c x · | E ( H ∗ ) | different components. Hence the size of A ′ is O ( c x · k s ) = O ( k s ). Thus the size of ( A ′ , k ) is O ( k sm ),where m is the largest arity of a relation R i , i.e., m = max { r , . . . , r t } . Thus ( A ′ , k ) is an instanceequivalent to ( A , k ) with size polynomial in k , since c x , s , and m are constants independent of theinput. Papadimitriou and Yannakakis introduced
MAX SNP as well as its superclass
MAX NP andshowed that every problem from these classes is constant-factor approximable [36]. We show thatthe standard parameterization of any
MAX NP problem admits a polynomial kernelization.9gain let us fix some problem
Q ∈
MAX NP . Let ( r , . . . , r t ) be the type of input structuresfor Q and let R , . . . , R t be matching relation symbols. By definition of MAX NP there is a tupleof relation symbols S = ( S , . . . , S u ) of arity s , . . . , s u and a formula ψ ( x , y , S ) in disjunctivenormal form over the vocabulary { R , . . . , R t , S , . . . , S u } such that for all finite structures A oftype ( r , . . . , r t ) the optimum value of Q on input A can be expressed asopt Q ( A ) = max S |{ x ∈ A c x : ( A , S ) | = ( ∃ y ∈ A c y ) : ψ ( x , y , S ) }| . Let s be the maximum number of occurrences of relations S , . . . , S u in any disjunct of ψ ( x , y , S ).The standard parameterization p - Q of Q is the following problem: Input:
A finite structure A of type ( r , . . . , r t ) and an integer k . Parameter: k . Task:
Decide whether opt Q ( A ) ≥ k .We define formulas ψ x , y ( S ) similarly to Definition 6 in Section 3. Definition 8.
Let A = ( A, R , . . . , R t ) be a finite structure of type ( r , . . . , r t ), let x ∈ A c x , andlet y ∈ A c y . We define ψ x , y ( S ) to be the formula obtained by the following steps:1. Replace all variables x , . . . , x c x , y , . . . , y c y by the chosen elements of A .2. Replace all literals R i ( z ) and ¬ R i ( z ), for some z ∈ A r i , by 1 (true) or 0 (false) depending onwhether z is contained in R i .3. Delete all disjuncts that contain a 0 and delete all occurrences of 1; note the difference toDefinition 6 through using a different normal form.4. Delete all disjuncts that contain contradicting literals S j ( z ) , ¬ S j ( z ) since they cannot besatisfied.We explicitly allow empty disjuncts that are satisfied by definition for the sake of simplicity (theyoccur when all literals in a disjunct are evaluated to 1).It is easy to see that ψ ( x , y , S ) and ψ x , y ( S ) are equivalent for any choice of x , y , and S ,i.e., ( A , S ) | = ψ ( x , y , S ) iff ( A , S ) | = ψ x , y ( S ). Moreover, we can compute all formulas ψ x , y ( S )for x ∈ A c x , y ∈ A c y in polynomial time, since c x , c y , and the length of ψ ( x , y , S ) are constantsindependent of A . Definition 9.
Let A = ( A, R , . . . , R t ) be a finite structure of type ( r , . . . , r t ).(a) We define X A ⊆ A c x as the set of all tuples x such that ( ∃ y ) : ψ x , y ( S ) holds for some S : X A = { x : ( ∃S ) : ( A , S ) | = ( ∃ y ) : ψ x , y ( S ) } . (b) For x ∈ A c x we define Y A ( x ) as the set of all tuples y such that ψ x , y ( S ) holds for some S : Y A ( x ) = { y : ( ∃S ) : ( A , S ) | = ψ x , y ( S ) } . The sets X A and Y A ( x ) can be computed in polynomial time because the number of tuples x ∈ A c x respectively y ∈ A c y is polynomial in the size of A and the formula ψ ( x , y , S ) is of constantlength independent of A . 10 emma 4. Let ( A , k ) be an instance of p - Q . If | X A | ≥ k · s then opt Q ( A ) ≥ k , i.e., ( A , k ) is ayes-instance. Remark 3.
In the following proof we consider assignments to variables of the formulas ψ x , y ( S ). Wepoint out that assigning true or false to some variable S i ( z ) corresponds to including or excluding,respectively, the tuple z in S i . Note that there are P ui =1 | A | s i variables, one for each possible tupleof a relation S i of arity s i . Proof of Lemma . We follow Papadimitriou and Yannakakis’ [36] proof for the fact that all prob-lems in
MAX NP are constant-factor approximable. For each x ∈ X A we fix a tuple y ∈ Y A ( x )such that ψ x , y ( S ) is satisfiable. This yields m = | X A | formulas, say ψ , . . . , ψ m . Now, for eachformula ψ i let f i denote the fraction of all assignments to S (i.e., inclusion or exclusion of tuples z in the relations S j ) that satisfies ψ i .We will create an assignment that satisfies at least P f i formulas ψ i . Let y be a variablethat has not been assigned yet. We assume that ℓ variables are unassigned at that point andthat P f ′ i ≥ P f i , where the fractions f ′ i are with respect to assignments to these ℓ remainingvariables. For i ∈ { , . . . , m } , let p i and n i denote the fraction of assignments to the remainingvariables that satisfies ψ i in which y is set to true or false, respectively. Thus there are 2 ℓ ( p i + n i )assignments which satisfy ψ i . Assign true to y if P p i ≥ P n i ; else, assign false. We show that thesum of fractions f ′ i never decreases (always taking f ′ i to be with respect to the remaining unassignedvariables): If y is set to true, then 2 ℓ p i assignments to the other ℓ − ψ i , whichcorresponds to a fraction of 2 ℓ p i / ℓ − = 2 p i . Thus if P p i ≥ P n i then m X i =1 p i ≥ m X i =1 p i + m X i =1 n i ≥ m X i =1 f ′ i ≥ m X i =1 f i . Note that P mi =1 p i is the sum of fractions of satisfying assignments taken with respect to theremaining ℓ − P p i < P n i and y is assigned false. Thusthe sum of fractions never decreases.When all variables are assigned a value, f ′ i is equal to 1 if ψ i is satisfied and 0 else. Thus, thisassignment satisfies at least P f ′ i ≥ P f i formulas ψ i (recall that each satisfied formula contributesa tuple to the solution).It is easy to see that f i ≥ − s for each formula ψ i . Since ψ i is satisfiable there exists asatisfiable disjunct. To satisfy a disjunct of at most s literals, at most s variables need to beassigned accordingly. Since the assignment to all other variables can be arbitrary this impliesthat f i ≥ − s . Thus we have that P f i ≥ m · − s . Therefore | X A | = m ≥ k · s implies that theassignment satisfies at least k formulas, i.e., that opt Q ( A ) ≥ k .Henceforth we assume that | X A | < k · s . The remaining and more involved part is to boundand reduce the size of the sets Y A ( · ). Note the difference between X A and sets Y A ( · ): everytuple x ∈ X A can add to the solution value, whereas tuples y ∈ Y A ( x ) only provide different waysof satisfying ( ∃ y ∈ A c y ) : ψ x , y ( S ). Hence our goal is to shrink the sets Y A ( x ) without harmingsatisfiability. We consider ( ∃ y ∈ A c y ) : ψ x , y ( S ) on the level of single disjuncts. Definition 10.
Let ( A , k ) be an instance of p - Q with A = ( A, R , . . . , R t ). For x ∈ A c x wedefine D A ( x ) as the set of all disjuncts of ψ x , y ( S ) over all y ∈ Y A ( x ).11o reduce the size of sets D A ( x ), which will lead to a decreased number of tuples in Y A ( x ),we again make use of the Sunflower Lemma. We will see that large sunflowers among disjunctsin D A ( x ) represent redundant ways of satisfying ( ∃ y ∈ A c y ) : ψ x , y ( S ). The size of each D A ( x ) isbounded by the size of Y A ( x ) ⊆ A c y times the number of disjuncts of ψ ( x , y , S ) which is a constantindependent of A . Thus the size of each D A ( x ) is bounded by a polynomial in the input size.The following definition of intersection and sunflowers among disjuncts treats disjuncts like setsof literals. Definition 11.
We define the intersection of two disjuncts as the conjunction of all literals thatoccur in both disjuncts. A sunflower of a set of disjuncts is a subset such that each pair of disjunctsin the subset has the same intersection (modulo permutation of the literals).
Definition 12. A partial assignment is a set L of literals such that no literal is the negation ofanother literal in L . A formula is satisfiable under L if there exists an assignment that satisfies theformula and each literal in L , i.e., there is an extension of the partial assignment L that satisfies F (as well as, naturally, all literals in L ).The following lemma is the basis of our data reduction. It shows that satisfiability under smallpartial assignments can be maintained in a reduced set of disjuncts. Lemma 5.
Let ( A , k ) be an instance of p - Q . For each x ∈ A c x there exists a set D ∗A ( x ) ⊆ D A ( x ) of cardinality O ( k s ) such that:1. For every partial assignment L of at most sk literals, D ∗A ( x ) contains a disjunct satisfiableunder L , if and only if D A ( x ) contains a disjunct satisfiable under L .2. D ∗A ( x ) can be computed in time polynomial in |A| .Proof. Let A = ( A, R , . . . , R t ) be a finite structure of type ( r , . . . , r t ), let x ∈ A c x , and let D A ( x )be a set of disjuncts according to Definition 10. We compute the set D ∗A ( x ) starting from D ∗A ( x ) = D A ( x ) and successively shrinking sunflowers while the cardinality of D ∗A ( x ) is greater than ( sk +1) s · s ! · s .We compute a sunflower of cardinality sk + 2, say F = { f , . . . , f sk +2 } , in time polynomialin | D ∗A ( x ) | using Corollary 1. We delete a disjunct of F , say f sk +2 , from D ∗A ( x ). Let O and P becopies of D ∗A before respectively after deleting f sk +2 . Observe that F ′ = F \ { f sk +2 } is a sunflowerof cardinality sk + 1 in P . Let L be a partial assignment of at most sk literals and assume that nodisjunct in P is satisfiable under L . This means that for each disjunct of P there is a literal in L that contradicts it, i.e., a literal that is the negation of a literal in the disjunct. We focus on thesunflower F ′ in P . There must be a literal in L , say ℓ , that contradicts the intersection of at leasttwo disjuncts of F ′ , say f and f ′ , since | F ′ | = sk + 1 and | L | ≤ sk . Therefore ℓ is the negation ofa literal in the intersection of f and f ′ , i.e., the core of F ′ . Thus ℓ contradicts also f sk +2 and weconclude that no disjunct in O = P ∪ { f sk +2 } is satisfiable under the partial assignment L . Thereverse argument holds since all disjuncts of P are contained in O . Thus each step maintains thedesired property (1).At the end D ∗A ( x ) contains no more than ( sk + 1) s · s ! · s ∈ O ( k s ) disjuncts. The computationtakes time polynomial in the size of A since the cardinality of D A ( x ) is bounded by a polynomialin the size of A and a disjunct is deleted in each step.12s in the previous section we are able to generate a kernelized instance of another problem,that is easier to handle. The sets D ∗A ( x ) describe a possibly different formula for each x , however,it is more convenient to view them as an image of the original instance on which it is easier to drawconclusions. Again, we will use the “sandwiching” trick. Lemma 6.
Let ( A , k ) be an instance of p - Q with A = ( A, R , . . . , R t ) and let x ∈ A c x . Let D ′A ( x ) be a subset of D A ( x ) such that D ∗A ( x ) ⊆ D ′A ( x ) ⊆ D A ( x ) . For any partial assignment L of atmost sk literals it holds that D A ( x ) contains a disjunct satisfiable under L if and only if D ′A ( x ) contains a disjunct satisfiable under L .Proof. Let L be a partial assignment of at most sk literals. If D A ( x ) contains a disjunct satis-fiable under L , then, by Lemma 5, this holds also for D ∗A ( x ). For D ∗A ( x ) and D ′A ( x ) this holdssince D ∗A ( x ) ⊆ D ′A ( x ). The same is true for D ′A ( x ) and D A ( x ). Theorem 3.
Let
Q ∈
MAX NP . The standard parameterization p - Q of Q admits a polynomialkernelization.Proof. The proof is organized in three parts. First, given an instance ( A , k ) of p - Q , we constructan instance ( A ′ , k ) of p - Q in time polynomial in the size of ( A , k ). In the second part, we provethat ( A , k ) and ( A ′ , k ) are equivalent. In the third part, we conclude the proof by showing thatthe size of ( A ′ , k ) is bounded by a polynomial in k . We recall the assumption that | X A | < k · s ,based on Lemma 4.(I.) Let ( A , k ) be an instance of p - Q . We use the sets D A ( x ) and D ∗A ( x ) according to Definition 10and Lemma 5. Recall that D A ( x ) is the set of all disjuncts of ψ x , y ( S ) for every y ∈ Y A ( x ). Thus, foreach disjunct d ∈ D ∗A ( x ) ⊆ D A ( x ), we can select a y d ∈ Y A ( x ) such that d is a disjunct of ψ x , y d ( S ).Let Y ′A ( x ) ⊆ Y A ( x ) be the set of these selected tuples y d . Let D ′A ( x ) be the set of all disjunctsof ψ x , y ( S ) for y ∈ Y ′A ( x ). Since D ∗A ( x ) contains some disjuncts of ψ x , y ( S ) for y ∈ Y ′A ( x ) and D A ( x )contains all disjuncts of ψ x , y ( S ) for y ∈ Y A ( x ) ⊇ Y ′A ( x ), we have that D ∗A ( x ) ⊆ D ′A ( x ) ⊆ D A ( x ).For each x this takes time O ( | D ∗A ( x ) |·| Y ∗A ( x ) | ) ⊆ O ( k s ·| A | c y ). Computing Y ′A ( x ) for all x ∈ A c x takes time O ( | A | c x · k s · | A | c y ), i.e., time polynomial in the size of ( A , k ) since k is never largerthan | A | c x . Let A ′ ⊆ A be the set of all components of x ∈ X A and y ∈ Y ′A ( x ) for all x ∈ X A . This ensuresthat X A ⊆ ( A ′ ) c x and Y ′A ( x ) ⊆ ( A ′ ) c y for all x ∈ X A . Let R ′ i be the restriction of R i to A ′ andlet A ′ = ( A ′ , R ′ , . . . , R ′ t ).(II.) We will now prove that opt Q ( A ) ≥ k if and only if opt Q ( A ′ ) ≥ k , i.e., that ( A , k ) and ( A ′ , k )are equivalent. Assume that opt Q ( A ) ≥ k and let S = ( S , . . . , S u ) such that |{ x : ( A , S ) | = ( ∃ y ) : ψ ( x , y , S ) }| ≥ k . This implies that there must exist tuples x , . . . , x k ∈ A c x and y , . . . , y k ∈ A c y such that S satisfies ψ x i , y i ( S ) for i = 1 , . . . , k . Thus S must satisfy at least one disjunct ineach ψ x i , y i ( S ) since these formulas are in disjunctive normal form. Accordingly let d , . . . , d k bedisjuncts such that S satisfies the disjunct d i in ψ x i , y i ( S ) for i = 1 , . . . , k . We show that thereexists S ′ such that: |{ x : ( A ′ , S ′ ) | = ( ∃ y ) : ψ ( x , y , S ′ ) }| ≥ k. For p = 1 , . . . , k we apply the following step: If y p ∈ Y ′A ( x p ) then do nothing. Otherwise considerthe partial assignment L consisting of the at most sk literals of the disjuncts d , . . . , d k . Theset D A ( x p ) contains a disjunct that is satisfiable under L , namely d p . By Lemma 6, it follows That is, ( A , k ) is a no-instance if k > | A | c x since k exceeds the number of tuples x ∈ A c x . D ′A ( x p ) also contains a disjunct satisfiable under L , say d ′ p . Let y ′ p ∈ Y ′A ( x p ) such that d ′ p is adisjunct of ψ x p , y ′ p ( S ). Such a y ′ p can be found by selection of D ′A ( x p ). Change S in the followingway to satisfy the disjunct d ′ p . For each literal of d ′ p of the form S i ( z ) add z to the relation S i .Similarly for each literal of the form ¬ S i ( z ) remove z from S i . This does not change the factthat S satisfies the disjunct d i in ψ x i , y i ( S ) for i = 1 , . . . , k since, by selection, d ′ p is satisfiableunder L . Then we replace y p by y ′ p and d p by d ′ p . Thus we maintain that S satisfies d i in ψ x i , y i ( S )for i = 1 , . . . , k .After these steps we obtain S as well as tuples x , . . . , x k , y , . . . , y k with y i ∈ Y ′A ( x i ), anddisjuncts d , . . . , d k such that S satisfies d i in ψ x i , y i ( S ) for i = 1 , . . . , k . Let S ′ be the restrictionof S to A ′ . Then we have that ( A ′ , S ′ ) | = ψ x i , y i ( S ′ ) for i = 1 , . . . , k since A ′ is defined to contain thecomponents of tuples x ∈ X A and of all tuples y ∈ Y ′A ( x ) for x ∈ X A . Hence x i ∈ { x : ( A ′ , S ′ ) | =( ∃ y ) : ψ ( x , y , S ′ ) } for i = 1 , . . . , k . Thus opt Q ( A ′ ) ≥ k .For the reverse direction assume that opt Q ( A ′ ) ≥ k . Since A ′ ⊆ A it follows that { x : ( A ′ , S ′ ) | = ( ∃ y ) : ψ ( x , y , S ′ ) } ⊆ { x : ( A , S ′ ) | = ( ∃ y ) : ψ ( x , y , S ′ ) } . Thus |{ x : ( A , S ′ ) | = ( ∃ y ) : ψ ( x , y , S ′ ) }| ≥ k , implying that opt Q ( A ) ≥ k . Therefore opt Q ( A ) ≥ k if and only if opt Q ( A ′ ) ≥ k . Hence ( A , k ) and ( A ′ , k ) are equivalent instances of p - Q .(III.) We conclude the proof by providing an upper bound on the size of ( A ′ , k ) that is polynomialin k . For the sets Y ′A ( x ) we selected one tuple y for each disjunct in D ∗A ( x ). Thus | Y ′A ( x ) | ≤| D ∗ ( x ) | ∈ O ( k s ) for all x ∈ X A . The set A ′ contains the components of tuples x ∈ X A and of alltuples y ∈ Y ′A ( x ) for x ∈ X A . Thus | A ′ | ≤ c x · | X A | + c y · X x ∈ X A | Y ′A ( x ) |≤ c x · | X A | + c y · | X A | · O ( k s ) < c x · k · s + c y · k · s · O ( k s ) = O ( k s +1 ) . For each relation R ′ i we have | R ′ i | ≤ | A ′ | r i ∈ O ( k ( s +1) r i ). Thus the size of ( A ′ , k ) is boundedby O ( k ( s +1) m ), where m is the largest arity of a relation R i .For MAX SNP there is a fairly immediate stronger kernelization that relies on Lemma 4.
Corollary 2.
Let
Q ∈
MAX SNP . The standard parameterization p - Q of Q admits a polynomialkernelization with a linear bound on the size of the base set of the obtained finite structure.Proof. Let
Q ∈
MAX SNP be an optimization problem on finite structures of type ( r , . . . , r t ).Let S = ( S , . . . , S u ) be a tuple of relation symbols of arity s , . . . , s u . Finally let ψ ( x , S ) be aformula in disjunctive normal form such that the optimum value of Q on a finite structure A oftype ( r , . . . , r t ) can be expressed asopt Q ( A ) = max S |{ x : ( A , S ) | = ψ ( x , S ) }| . Now, let ( A , k ) be an instance of p - Q , with A = ( A, R , . . . , R t ). Similarly to Definition 9, weconsider the set X A of all tuples x such that ψ x ( S ) holds for some S : X A = { x : ( ∃S ) : ( A , S ) | = ψ x ( S ) } .
14y Lemma 4, if | X A | ≥ k · s then opt Q ( A ) ≥ k and we may accept A as a yes-instance. Oth-erwise | X A | ∈ O ( k ) and by restricting A to those elements that occur in elements of X A weobtain A ′ with | A ′ | ∈ O ( k ). Also restricting the relations R i to A ′ we obtain an equivalent in-stance A ′ = ( A ′ , R ′ , . . . , R ′ t ) of total size O ( k m ) where k = max { r , . . . , r t } . We have constructively established that the standard parameterizations of problems in
MIN F + Π and MAX NP admit polynomial kernelizations. Thus a strong relation between constant-factorapproximability and polynomial kernelizability has been showed for two large classes of problems. Itremains an open problem to give a more general result that covers all known examples (e.g.,
Feed-back Vertex Set ). It might be profitable to consider closures of
MAX SNP under reductionsthat preserve constant-factor approximability. Khanna et al. [25] proved that APX and APX-PBare the closures of
MAX SNP under PTAS-preserving reductions and E-reductions, respectively.Since both classes contain
Bin Packing which does not admit a polynomial kernelization, this leadsto the question whether polynomial kernelizability or fixed-parameter tractability are maintainedunder restricted versions of these reductions.Furthermore, it would be interesting to see whether polynomial lower bounds similar to theresults of Dell and van Melkebeek [12] can be proven. It is easy, however, to construct artificialexamples with almost redundant relations of high arity being part of the finite structures, so thefocus may have to be on exhibiting meaningful families of problems in
MIN F + Π and MAX NP and showing lower bounds for them.
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