Enumerating maximal consistent closed sets in closure systems
EEnumerating maximal consistent closed sets inclosure systems
Lhouari Nourine and Simon Vilmin (cid:63)
LIMOS, Université Clermont Auvergne, Aubière, France [email protected] , [email protected] Abstract.
Given an implicational base, a well-known representation fora closure system, an inconsistency binary relation over a finite set, we areinterested in the problem of enumerating all maximal consistent closedsets (denoted by
MCCEnum for short). We show that
MCCEnum can-not be solved in output-polynomial time unless P = NP , even for lowerbounded lattices. We give an incremental-polynomial time algorithm tosolve MCCEnum for closure systems with constant Carathéodory num-ber. Finally we prove that in biatomic atomistic closure systems
MC-CEnum can be solved in output-quasipolynomial time if minimal gener-ators obey an independence condition, which holds in atomistic modularlattices. For closure systems closed under union (i.e., distributive),
MC-CEnum is solved by a polynomial delay algorithm [22, 25].
Keywords:
Closure systems, implicational base, inconsistency relation,enumeration algorithm
In this paper, we consider binary inconsistency relations (i.e., graphs) over im-plicational bases, a well-known representation for closure systems [7, 32]. Moreprecisely, we seek to enumerate maximal closed sets of a closure system givenby an implicational base that are consistent with respect to an inconsistencyrelation. We call this problem
Maximal Consistent Closed Sets Enumer-ation , or
MCCEnum for short.This problem finds applications for instance in minimization of sub-modularfunctions [22] or argumentation frameworks [12]. It is moreover a particular caseof dualization in closure systems given by an implicational bases, ubiquitous incomputer science [7, 11, 15]. This latter problem however cannot be solved inoutput-polynomial time unless P = NP [3] even when the input implicationalbase has premises of size at most two [10]. When restricted to graphs and impli-cational bases with premises of size one, or posets equivalently, the problem canbe solved in polynomial delay [22, 25].More generally, inconsistency relations combined with posets appear alsoin event structures [28], representations of median-semilattices [4] or cubical (cid:63) The second author is funded by the CNRS, France, ProFan project. a r X i v : . [ c s . CC ] F e b L. Nourine, S. Vilmin complexes [2] in which the term “inconsistency” is used. Recently in [21, 22],the authors derive a representation for modular semi-lattices based on incon-sistency and projective ordered spaces [20]. Furthermore, they characterize thecases where given an implicational base and an inconsistency relation, maximalconsistent closed sets coincide with maximal independent sets of the inconsis-tency relation, seen as a graph.In our contribution, we show first that enumerating maximal consistent closedsets cannot be solved in output-polynomial time unless P = NP , a surprisingresult which further emphasizes the hardness of dualization in lattices givenby implicational bases [3, 10]. In fact, we show that this problem is alreadyintractable for the well-known class of lower bounded lattices [1, 9, 16]. On thepositive side, we show that when the maximal size of minimal generators isbounded by a constant, the problem can be solved in incremental-polynomialtime. As a direct corollary, we obtain that MCCEnum can be solved efficientlyin a several classes of convex geometries where this parameter, also known as theCarathéodory number, is constant [26]. Finally, we focus on biatomic atomisticclosure systems [5,8]. We show that under an independence condition, the size ofa minimal generator is logarithmic in the size of the groundset. As a consequence,we get a quasi-polynomial time algorithm for enumerating maximal consistentclosed sets which can be applied to the well-known class of atomistic modularlattices [18, 20, 29, 31].The rest of the paper is organized as follows. Section 2 gives necessary defini-tions about closure systems and implicational bases. In Section 3 we show that
MCCEnum cannot be solved in output-polynomial time, in particular for lowerbounded closure systems. In Section 4, we show that if the size of a minimal gen-erator is bounded by a constant,
MCCEnum can be solved efficiently. Section 5is devoted to the class of biatomic atomistic closure systems. We conclude withopen questions and problems in 6.
All the objects considered in this paper are finite. Let X be a set. We denoteby X its powerset. For any n ∈ (cid:78) , we write [ n ] for the set { , . . . , n } . We willsometimes use the notation x . . . x n as a shortcut for { x , . . . , x n } . The size ofa subset A of X is denoted by | A | . If H = ( X, E ) is a hypergraph, we denoteby IS ( H ) its independent sets (or stable sets). We write MIS ( H ) for its maximalindependent sets. Similarly, if G = ( X, E ) is a graph, its independent sets (resp.maximal independent sets) are written IS ( G ) (resp. MIS ( G ) ).We recall principal notions on lattices and closure systems [18]. A mapping φ : 2 X ! X is a closure operator if for any Y, Z ⊆ X , Y ⊆ φ ( Y ) (extensive), Y ⊆ Z implies φ ( Y ) ⊆ φ ( Z ) (isotone), and φ ( φ ( Y )) = φ ( Y ) (idempotent). Wecall φ ( Y ) the closure of Y . The family F = { φ ( Y ) | Y ⊆ X } ordered by set-inclusion forms a closure system or lattice . A closure system F ⊆ X is a setsystem such that X ∈ F and for any F , F ∈ F , F ∩ F also belongs to F .Elements of F are closed sets and we say that F is closed if F ∈ F . Each closure numerating maximal consistent closed sets 3 system F induces a unique closure operator φ such that φ ( Y ) = (cid:84) { F ∈ F | Y ⊆ F } , for any Y ⊆ X . Thus, there is a one-to-one correspondence between closuresystems and operators. Without loss of generality, we will assume that φ and F are standard : φ ( ∅ ) = ∅ and for any x ∈ X , φ ( x ) \ { x } is closed. Note that ∅ isthus the minimum element of F , called the bottom . Similarly, X is the top of F .Let φ be a closure operator with corresponding closure system F . Let F , F ∈F . We say that F and F are comparable if F ⊆ F or F ⊆ F . They are incomparable otherwise. A subset S of F is an antichain if its elements arepairwise incomparable. If for any F ∈ F , F ⊂ F ⊆ F implies F = F , we saythat F covers F , and denote it F ≺ F . An atom is a closed set covering thebottom ∅ of F . Dually, a co-atom is a closed set covered by the top X of F . Wedenote by C ( F ) the set of co-atoms of F . Let M ∈ F . We say that M is meet-irreducible in F if for any F , F ∈ F , M = F ∩ F entails either F = M or F = M . In this case, M has a unique cover M ∗ in F . The set of meet-irreducibleelements of F is denoted by M ( F ) . Dually, J ∈ F is a join-irreducible element of F if for any F , F ∈ F , J = φ ( F ∪ F ) implies J = F or J = F . Then, J coversa unique element J ∗ in F . We denote by J ( F ) the join-irreducible elements of F . When F and φ are standard, there is a one-to-one correspondence between X and J ( F ) given by J ( F ) = { φ ( x ) | x ∈ X } . Furthermore, x ∗ = φ ( x ) ∗ = φ ( x ) \ x .Consequently, we will identify X with J ( F ) .Let x ∈ X . A minimal generator of x is an inclusion-wise minimal subset A x of X such that x ∈ φ ( A x ) . We consider { x } as a trivial minimal generator of x .Following [26], the Carathéodory number c( F ) of F is the least integer k suchthat for any A ⊆ X and any x ∈ X , x ∈ φ ( A ) implies the existence of some A (cid:48) ⊆ A with | A (cid:48) | ≤ k such that x ∈ φ ( A (cid:48) ) . At first, this notion was used forconvex geometries, but its definition applies to any closure system. Moreover, theCarathéodory number of F is the maximal possible size of a minimal generator(see Proposition 4.1 in [26], which can be applied to any closure system). A key of F is a minimal subset K ⊆ X such that φ ( K ) = X . We denote by K theset of keys of F . The number of keys in K is denoted by | K | . It is well-known(see for instance [11]) that maximal independent sets MIS ( K ) of K , viewed asa hypergraph over X , are exactly co-atoms of F . We define arrow relationsfrom [17]. Let x ∈ X and M ∈ M ( F ) . We write x " M if x / ∈ M but x ∈ M ∗ .Dually, we write M x if x / ∈ M but x ∗ ⊆ M .We move to implicational bases [7, 32]. An implication si an expression ofthe form A ! B with A, B ⊆ X . We call A the premise and B the conclusion .A set Σ of implications over X is an implicational base over X . We denoteby | Σ | the number of implications in Σ . A subset F ⊆ X satisfies or models Σ if for any A ! B ∈ Σ , A ⊆ F implies B ⊆ F . The family F = { F ⊆ X | F satisfies Σ } is a closure system whose induced closure operator φ is the forward chaining algorithm . This procedure starts from any subset Y of X andconstructs a sequence Y = Y ⊆ · · · ⊆ Y k = φ ( Y ) of subsets of X such that forany i ∈ [ k ] , Y i = Y i − ∪ { B | ∃ A ! B ∈ Σ s.t. A ⊆ Y i − } . The algorithm stopswhen Y i − = Y i . L. Nourine, S. Vilmin ∅ Fig. 1: On the left, a consistency-graph G c over X = { , , , , } with in-consistent pairs , and . On the right, the closure system associated to Σ = { ! , ! , ! , ! } . Black and white dots stand for inconsistentand consistent closed sets respectively. We have maxCC ( Σ, G c ) = { , , } .We now introduce our main problem. Following [2, 21, 22] we call an incon-sistency relation any symmetric and irreflexive relation over X . Such a relationis sometimes called a site [4] or a conflict relation [28]. Usually, inconsistencyrelations need to satisfy more conditions in order to capture median or modular-semilattices [4, 21]. As we do not need further restrictions here, we can chooseto model inconsistency as a graph G c = ( X, E c ) , and call it a consistency-graph .An edge uv of E c represents an inconsistent pair of elements in X . A subset Y which does not contain any inconsistent pair (i.e., an independent set of G c ) iscalled consistent . Let Σ be an implicational base over X and a G c = ( X, E c ) consistency-graph. We denote by maxCC ( Σ, G c ) the set of maximal consistentclosed sets of F , that is maxCC ( Σ, G c ) = max ⊆ ( F ∩ IS ( G c )) . An example of im-plicational base along with a consistency-graph is given in Figure 1. Our problemis the following. Maximal Consistent closed-sets Enumeration (MCCEnum)
Input:
An implicational base Σ over X , a non-empty consistency-graph G c = ( X, E c ) . Output:
The set maxCC ( Σ, G c ) of maximal consistent closed sets of F withrespect to G c .Remark that X is part of the input. If G c is empty, MCCEnum is easyto solve as X is the unique element of maxCC ( Σ, G c ) . Hence, we will assumewithout loss of generality that G c is not empty. If Σ is empty, then MCCEnum isequivalent to the enumeration of maximal independent sets of a graph which canbe efficiently solved [23]. If premises of Σ have size , the problem also reducesto maximal independent sets enumeration [22, 25]. In [21] the authors identify,for a fixed Σ , the consistency-graphs G c such that MIS ( G c ) = maxCC ( Σ, G c ) .We conclude with a recall on enumeration algorithms [23]. Let A be analgorithm with input x and output a set of solutions R ( x ) . We denote by | R ( x ) | the number of solutions in R ( x ) . We assume that each solution in R ( x ) has size numerating maximal consistent closed sets 5 poly ( | x | ) . The algorithm A is running in output-polynomial time if its executiontime is bounded by poly ( | x | + | R ( x ) | ) . It is incremental-polynomial if for any ≤ i ≤ | R ( x ) | , the time spent between the i -th and i + 1 -th output is boundedby poly ( | x | + i ) , and the algorithm stops in time poly ( | x | ) after the last output. Ifthe delay between two solutions output and after the last one is poly ( | x | ) , A has polynomial-delay . Note that if A is running in incremental-polynomial time, it isalso output-polynomial. Finally, we say that A runs in output-quasipolynomial time if is execution time is bounded by N polylog ( N ) where N = | x | + | R ( x ) | . We show that
MCCEnum cannot be solved in output-polynomial time unless P = NP . To do so, we use a reduction from the problem of enumerating co-atomsof a closure system. Co-atoms Enumeration (CE)
Input:
An implicational base Σ Y over Y . Output:
The co-atoms C ( F Y ) of the closure system F Y associated to Σ Y .It is proved by Kavvadias et al. in [25] that CE admits no output-polynomialtime algorithm unless P = NP . Our first step is to prove the following lemma. Lemma 1.
Let Σ Y be an implicational base over Y . Let X = Y ∪ { u, v } , Σ = Σ Y ∪ { Y ! uv } and let G c = ( X, E c = { uv } ) be a consistency-graph.The following equality holds: maxCC ( Σ, G c ) = (cid:91) C ∈C ( F Y ) { C ∪ { u } , C ∪ { v }} (1) Proof.
Let C ∈ C ( F Y ) . We show that C ∪ { u } and C ∪ { v } are in maxCC ( Σ, G c ) .As no implication of Σ has u or v in its premise, we have that C ∪{ u } and C ∪{ v } are consistent and closed with respect to Σ . Let y ∈ Y \ C . As C is a co-atomof F Y , it must be that φ Y ( C ∪ { y } ) = Y . As Y ! uv is an implication of Σ , itfollows that uv ⊆ φ ( C ∪ { u, y } ) . Thus, for any x ∈ X \ ( C ∪ { u } ) , φ ( C ∪ { u, x } ) is inconsistent. We conclude that C ∪ { u } ∈ maxCC ( Σ, G c ) . Similarly we obtain C ∪ { v } ∈ maxCC ( Σ, G c ) .Let S ∈ maxCC ( Σ, G c ) . We show that S can be written as C ∪ { u } or C ∪ { v } for some co-atom C of F Y . First, let F be a consistent closed set in F such that u / ∈ F and v / ∈ F . As Σ has no implication with u or v in its premise, it followsthat both F ∪ { u } and F ∪ { v } are closed and consistent. Hence, either u ∈ S or v ∈ S . Without loss of generality, let us assume u ∈ S . Let C = S \ { u } .As S ∈ maxCC ( Σ, G c ) , it is closed with respect to Σ Y and does not contain Y .Thus, C ∈ F Y and C ⊂ Y . Let y ∈ Y \ C . As S ∈ maxCC ( Σ, G c ) , it must bethat φ ( S ∪ { y } ) contains the inconsistent pair uv of G c . Hence, Y ⊆ φ ( S ∪ { y } ) by construction of Σ . Consequently, we have that Y = φ Y ( C ∪ { y } ) for any y ∈ Y \ C . Hence, we conclude that C ∈ C ( F Y ) as expected. (cid:117)(cid:116) L. Nourine, S. Vilmin
Therefore, if there is an algorithm solving
MCCEnum in output-polynomialtime, it can be used to solve CE within the same running time using the reductionof Lemma 1. Consequently, we obtain the following theorem. Theorem 1.
The problem
MCCEnum cannot be solved in output-polynomialtime unless P = NP . In fact, we can strengthen the preceding theorem by a careful analysis of theclosure system used in the reduction in [25]. More precisely, we show that theproblem remains untractable for lower bounded closure systems. These have beenintroduced with the doubling construction in [9] and then studied in [1, 6, 16].A characterization of lower bounded lattices is given in [16] in terms of the D -relation. This relations relies on J ( F ) and we say that x depends on y , denotedby xDy (recall that we identified X with J ( F ) ) if there exists a meet-irreducibleelement M ∈ M ( F ) such that x " M y . A D -cycle is a sequence x , . . . , x k ⊆ X such that x Dx D . . . Dx k Dx . Theorem 2. (Reformulated from Corollary 2.39, [16]) A closure system F islower bounded if and only if it contains no D -cycle. Corollary 1.
The problem
MCCEnum cannot be solved in output-polynomialtime unless P = NP , even in lower bounded closure systems.Proof. Consider a positive 3-CNF over n variables and m clauses ψ ( x , . . . , x n ) = m (cid:94) i =1 C i = m (cid:94) i =1 ( x i, ∨ x i, ∨ x i, ) Let Y = { x , . . . x n , y , . . . , y m , z } and consider the following sets of implications: – Σ = { x i,k x j,k ! z | i ∈ [ m ] , k ∈ [3] } , – Σ = { y i ! z | i ∈ [ m ] } , – Σ = { x i,k z ! y i | i ∈ [ m ] , k ∈ [3] } .And let Σ Y = Σ ∪ Σ ∪ Σ . In [25] the authors show that CE is alreadyintractable for these instances.Therefore, applying the reduction from Lemma 1, we obtain that MCCEnum cannot be solved in output-polynomial time in the following case: X = Y ∪{ u, v } , Σ = Σ Y ∪ { Y ! uv } , G c = ( X, E c = { uv } ) .Let us show that F , the closure system associated to Σ , is indeed lowerbounded. We proceed by analysing the D -relation. Observe first that F is stan-dard. We begin with u, v . Let t ∈ X \{ u } and M ∈ M ( F ) such that t " M . As nopremise of Σ contains u , it must be that u ∈ M . Hence for any t ∈ X \{ u } , t doesnot depend on u . Applying the same reasoning on v , we obtain that no D -cyclecan contain u or v . Let x i ∈ X , i ∈ [ n ] . As x i is the conclusion of no implicationin Σ , we have that the unique meet-irreducible element M i satisfying x i " M i is X \ x i . Therefore, there is no element in X \ { x i } on which x i depends, so thatno D -cycle can contain x i , for any i ∈ [ n ] . Let us move to z . As y j ! z ∈ Σ for numerating maximal consistent closed sets 7 any j ∈ [ m ] , we have y j ∗ = φ ( y j ) ∗ = { z } . Hence, zDy j cannot hold since M y j implies z ∈ M , for any M ∈ M ( F ) . Thus, z only depends on some of the x i ’s, i ∈ [ n ] , and no D -cycle can contain z either.Henceforth, the only possible D -cycles must be contained in { y , . . . , y m } .We show that for any i, k ∈ [ m ] , y i Dy k does not hold. For any y i , i ∈ [ m ] , wehave y i ∗ = { z } as y i ! z ∈ Σ . Hence, a meet-irreducible element M i satisfying y i " M i y k must contain z . Let F ∈ F be any closed set satisfying y i / ∈ F but z ∈ F . Assume there exists some y k such that y k / ∈ F . Then F ∪ { y k } ∈ F , as y k ! z is the only implication having y k in its premise, and z ∈ F . Therefore, it must bethat for any M i ∈ M ( F ) such that z ∈ M i and y i / ∈ M i , { y , . . . , y m }\{ y i } ⊆ M i is verified, so that y i " M i y k is not possible. As a consequence y i Dy k cannothold, for any i, k ∈ [ m ] . We conclude that F has no D -cycles and that it is lowerbounded by Theorem 2. (cid:117)(cid:116) Therefore, there is no algorithm solving
MCCEnum in output-polynomialtime unless P = NP even when restricted to lower bounded closure systems. Inthe next section, we consider classes of closure systems where MCCEnum canbe solved in incremental-polynomial time.
Let Σ be an implicational base over X and G c a non-empty consistency-graph.Observe that IS ( G c ) ∪ { X } is a closure system where a set F ⊆ X is closed ifand only if F = X or it is an independent set of G c . From this point of view,elements of maxCC ( Σ, G c ) , are those maximal proper subsets of X that are bothclosed in F and IS ( G c ) ∪ { X } . Consequently, the maximal consistent closed setsof F with respect to G c are exactly the co-atoms of F ∩ ( IS ( G c ) ∪ { X } ) . Now,if we can guarantee that K , the keys of F ∩ ( IS ( G c ) ∪ { X } ) , has polynomial sizewith respect to Σ , X and G c , we can derive an incremental-polynomial timealgorithm computing maxCC ( Σ, G c ) in two steps:1. Compute the set of keys K which has polynomial size with respect to X ,2. Compute MIS ( K ) = maxCC ( Σ, G c ) .To identify cases where K has polynomial size with respect to Σ, X and G c ,the first step is to characterize its elements. To do so, we have to guaranteethat a set Y ⊂ X contains a key of K whenever Y or φ ( Y ) is inconsistent withrespect to G c . Looking at G c is sufficient to distinguish between consistent andinconsistent closed sets of F . However, there may be consistent (non-closed)sets Y such that φ ( Y ) contains an edge of G c . These will not be seen by justconsidering G c . Thus, if uv is the edge of G c contained in φ ( Y ) , we deduce thatthere must be a minimal generator A u of u contained in Y , possibly A u = { u } .Similarly, Y contains a minimal generator A v of v . In particular, keys in K willshare the following property. Proposition 1.
Let K ∈ K . Then there exists uv ∈ E c , a minimal generator A u of u , and a minimal generator A v of v such that K = A u ∪ A v . L. Nourine, S. Vilmin
Proof.
Let K ∈ K . By assumption, φ ( K ) contains an edge uv of G c . Thus, thereexists minimal generators A u of u and A v of v such that A u ∪ A v ⊆ K . Assumethat A u ∪ A v ⊂ K and let x ∈ K \ ( A u ∪ A v ) . As u ∈ φ ( A u ) and v ∈ φ ( A v ) , weget uv ∈ φ ( K \ { x } ) , a contradiction with the minimality of K . (cid:117)(cid:116) Example 1.
We consider Σ , X and G c of Figure 1. We have that φ (135) = 1235 is inconsistent as it contains . However is consistent with respect to G c .For this example, we will have K = { , , , } . Note that can bedecomposed following Proposition 1 as the minimal generator of , and asa trivial minimal generator for itself.Remark that E c (cid:42) K in the general case, as there may be an implication u ! v in Σ for some inconsistent pair uv ∈ E c . Thus u is a key which satisfiesProposition 1 with A u = A v = { u } . It also follows from Proposition 1 that c( F ) plays an important role for MCCEnum . When no restriction on c( F ) holds, K can have exponential size with respect to Σ and G c . The next example drawnfrom [25] illustrates this exponential growth. Example 2.
Let X = { x , . . . , x n , y , . . . , y n , u, v } and Σ = { x i ! y i | i ∈ [ n ] } ∪{ y . . . y n ! uv } . The consistency-graph is G c = ( X, { uv } ) . The set of non-trivialminimal generators of u and v is { z . . . z n | z i ∈ { x i , y i } , i ∈ [ n ] } . Moreover,minimal generators of u and v are also the keys of F ∩ ( IS ( G c ) ∪ { X } ) . Thus, | K | = 2 n , which is exponential with respect to Σ and G . Observe that Σ isacyclic [19, 32]: for any x, y ∈ X if y belongs to some minimal generator of x ,then x is never contained in a minimal generator of y .Hence, computing maxCC ( Σ, G c ) through the intermediary of K is in generalimpossible in output-polynomial time. In fact, this exponential blow up occurseven for small classes of closure systems where the Carathéodory number c( F ) is unbounded. In Example 2 for instance, the closure system induced by Σ isacyclic [19, 32], a particular case of lower boundedness [1].On the other hand, let us assume now that c( F ) is bounded by some con-stant k ∈ (cid:78) . By Proposition 1, every key in K has at most × k elements. Asa consequence we show in the next theorem that the two-steps algorithm wedescribed can be conducted in incremental-polynomial time. Theorem 3.
Let Σ be an implicational base over X with induced F , and G c =( X, E c ) a consistency-graph. If c( F ) ≤ k for some constant k ∈ (cid:78) , the problem MCCEnum can be solved in incremental-polynomial time.Proof.
The set of keys K can be computed in incremental-polynomial time withrespect to K , Σ , X and G c using the algorithm of Lucchesi and Osborn [27]with input Σ (cid:48) = Σ ∪ { uv ! X | uv ∈ E c } . Observe that the closure systemassociated to Σ (cid:48) is exactly F ∩ { IS ( G c ) ∪ { X }} . Indeed, a consistent closed setof F models Σ (cid:48) and a subset F ⊆ X which satisfies Σ (cid:48) must also satisfy Σ andbeing an independent set of G c if F ⊂ X . Note that K is then computed intime poly ( | Σ | + | X | + | G c | + | K | ) . As the total size of K is bounded by | X | k byProposition 1, we get that K is computed in time poly ( | Σ | + | X | + | G c | ) . numerating maximal consistent closed sets 9 Then, we apply the algorithm of Eiter and Gottlob [13] to compute
MIS ( K ) = maxCC ( Σ, G c ) which runs in incremental polynomial time. Since K has polyno-mial size with respect to | X | , the delay between the i -th and ( i +1) -th solution of maxCC ( Σ, G c ) output is bounded by poly ( | X | k + i ) , that is poly ( | X | + i ) . Further-more, the delay after the last output is also bounded by poly ( | X | k ) = poly ( | X | ) .As the time spent before the first solution output is poly ( | Σ | + | X | + | G c | ) , thewhole algorithm has incremental delay as expected. (cid:117)(cid:116) To conclude this section, we show that Theorem 3 applies to various classesof closure systems present in the theory of convex geometries [26].A closure system F is distributive if for any F , F ∈ F , F ∪ F ∈ F .Implicational bases of distributive closure systems have premises of size one [18].Let P = ( X, ≤ ) be a partially ordered set, or poset. A subset Y ⊆ X is convex in P if for any triple x ≤ y ≤ z , x, z ∈ Y implies y ∈ Y . The family { Y ⊆ X | Y is convex in P } is known to be closure system over X [8, 24].Let G = ( X, E ) be a graph. We say that G is chordal if every it has noinduced cycle of size ≥ . A chord in a path from x to y is an edge connecting tonon-adjacent vertices of the path. A subset Y of X is monophonically convex in G if for every pair x, y of elements in Y , every z ∈ X which lies on a chordless pathfrom x to y is in Y . The family { Y ⊆ X | Y is monophonically convex in G } isa closure system [14, 26].Finally, let X ⊆ (cid:82) n , n ∈ (cid:78) , be a finite set of points, and denote by ch ( Y ) the convex hull of Y . The set system { ch ( Y ) | Y ⊆ X } forms a closure system [26]usually known as an affine convex geometry . Corollary 2.
Let Σ be an implicational base over X and G c = ( X, E c ) . MC-CEnum can be solved in incremental-polynomial time in the following cases: – F is distributive, – F is the family of convex subsets of a poset, – F is the family of monophonically convex subsets of a chordal graph, – F is an affine convex geometry in (cid:82) k for a fixed constant k .Proof. Distributive lattices have Carathéodory number as they can be rep-resented by implicational bases with singleton premises. The family of convexsubsets of a poset has Carathéodory number 2 [24] (Corollary 13). The family ofmonophonically convex subsets of a chordal graphs have Carathéodory numberat most 2 [14] (Corollary 3.4). The Carathéodory number of an affine convexgeometry in (cid:82) k is k − (see for instance [26], p. 32). (cid:117)(cid:116) In the distributive case, the algorithm can perform in polynomial delay usingthe algorithm of [23] since K will be a graph by Proposition 1. This connectswith previous results on distributive closure systems by Kavvadias et al. [25]. In this section, we are interested in biatomic atomistic closure systems. Namely,we show that when minimal generators obey an independence condition, the size of X is exponential with respect to c( F ) . To do so, we show that in biatomicatomistic closure systems, each subset of a minimal generator is itself a minimalgenerator. This result applies to atomistic modular closure systems, which canbe represented by implications with premises of size at most [31]. This suggeststhat MCCEnum becomes more difficult when implications have binary premises.First, we need to define atomistic biatomic closure systems. let F be a closuresystem over X with associated closure operator φ . We say that F is atomistic if for any x ∈ X , φ ( x ) = { x } . Equivalently, F is atomistic if its join-irreducibleelements equal its atoms. Note that in a standard closure system, an atom is asingleton element. Biatomic closure systems have been studied by Birkhoff andBennett in [5, 8]. We reformulate their definition in terms of closure systems. Aclosure system F is biatomic if for every closed sets F , F ∈ F and any atom { x } ∈ F , x ∈ φ ( F ∪ F ) implies the existence of atoms { x } ⊆ F , { x } ⊆ F such that x ∈ φ ( x x ) . In atomistic closure systems in particular, the biatomiccondition applies to every element of X . Hence the next property of biatomicatomistic closure systems. Proposition 2.
Let F be a biatomic atomistic closure system. Let F ∈ F and x, y ∈ X with x, y / ∈ F . If y ∈ φ ( F ∪ { x } ) , then there exists an element z ∈ F such that y ∈ φ ( xz ) .Proof. In atomistic closure systems, every element of X is closed, therefore weapply the definition to the closed sets F and { x } . (cid:117)(cid:116) We will also make use of the following folklore result about minimal genera-tors. We give a proof for self-containment.
Proposition 3. If A x is a minimal generator of x ∈ X , then φ ( A ) ∩ A x = A for any A ⊆ A x .Proof. First, we have that A ⊆ φ ( A ) ∩ A x as A ⊆ φ ( A ) and A ⊆ A x . Nowsuppose that there exists a ∈ φ ( A ) ∩ A x such that a / ∈ A . Then, a ∈ φ ( A x \ { a } ) as A ⊆ A x \{ a } . Hence, φ ( A x ) = φ ( A x \{ a } ) and x ∈ φ ( A x \{ a } ) , a contradictionwith A x being a minimal generator of x . (cid:117)(cid:116) Our first step is to show that in a biatomic atomistic closure system, if A x is a minimal generator for some x ∈ X , then every non-empty subset A of A x isitself a minimal generator for some y ∈ X . We prove this statement in Lemmas2 and 3. Recall that an element x ∈ X is a (trivial) minimal generator of itself. Lemma 2.
Let x ∈ X and let A x be a minimal generator of x with size k ≥ .Then for any a i ∈ A x , i ∈ [ k ] , there exists y i ∈ X such that A x \ { a i } is aminimal generator of y i .Proof. Let A x = { a , . . . , a k } be a minimal generator of x such that k ≥ . Then,for any a i ∈ A x , i ∈ [ k ] , we have a i / ∈ φ ( A x \ { a i } ) by Proposition 3. However,we have x ∈ φ ( { a i } ∪ φ ( A x \ { a i } )) = φ ( A x ) . Thus, by Proposition 2, there mustexists y i ∈ φ ( A x \ { a i } ) such that x ∈ φ ( a i y i ) . numerating maximal consistent closed sets 11 Let us show that A x \ { a i } is a minimal generator of y i . Assume for contra-diction this is not the case. As y i ∈ φ ( A x \ { a i } ) , there must be a proper subset A of A x \ { a i } which is a minimal generator for y i . Note that since A x has atleast elements, at least one proper subset of A x \ { a i } exists. As A ⊂ A x \ { a i } ,there exists a j ∈ A x , a j (cid:54) = a i , such that a j / ∈ A . Therefore, A ⊆ A x \ { a j } and φ ( A ) ⊆ φ ( A x \ { a j } ) . More precisely, y i ∈ φ ( A ) and hence y i ∈ φ ( A x \ { a j } ) .However, we also have that a i ∈ φ ( A x \ { a j } ) , and since x ∈ φ ( a i y i ) , we musthave x ∈ φ ( A x \ { a j } ) , a contradiction with A x being a minimal generator of x . Thus, we deduce that A x \ { a i } is a minimal generator for y i , concluding theproof. (cid:117)(cid:116) In the particular case where A x has only two elements, say a and a , then A x \ { a } = { a } and the element a is a trivial minimal generator of itself. Byusing inductively Lemma 2 on the size of A x , one can derive the next straight-forward lemma. Lemma 3.
Let F be a biatomic atomistic closure system. Let A x be a minimalgenerator of some x ∈ X . Then, for any A ⊆ A x with A (cid:54) = ∅ , there exists y ∈ X such that A is a minimal generator of y . Thus, for a given minimal generator A x of x , any non-empty subset A of A x is associated to some y ∈ X . We show next than when A x also satisfiesan independence condition, A will be the unique subset of A x associated to y .Following [18], we reformulate the definition of independence in an atomisticclosure system F , but restricted to its atoms. A subset Y of X is independent in F if for any Y , Y ⊆ Y , φ ( Y ∩ Y ) = φ ( Y ) ∩ φ ( Y ) . Lemma 4.
Let F be a biatomic atomistic closure system. Let A x be an indepen-dent minimal generator of x ∈ X , and let A be a non-empty subset of A x . Then,there exists y ∈ X such that A is the unique minimum subset of A x satisfying y ∈ φ ( A ) .Proof. Let A x be an independent minimal generator of x ∈ X , and let A bea non-empty subset of A x . By Lemma 3, there exists y ∈ X such that A is aminimal generator for y , which implies y ∈ φ ( A ) .To prove that A is the unique minimum subset of A x such that y ∈ φ ( A ) , weshow that for any B ⊆ A x such that A (cid:42) B , y ∈ φ ( B ) cannot hold. Consider B ⊆ A x with A (cid:42) B and suppose that y ∈ φ ( B ) . Note that B must exist as theempty set is always a possible choice. Since y ∈ φ ( A ) , we have y ∈ φ ( A ) ∩ φ ( B ) .Furthermore, φ ( A ∩ B ) ⊂ φ ( A ) as A ∩ B ⊂ A and φ ( A ∩ B ) ∩ A x = A ∩ B byProposition 3. Moreover, A x is independent, so that φ ( A ) ∩ φ ( B ) = φ ( A ∩ B ) .Hence, y ∈ φ ( A ∩ B ) ⊂ φ ( A ) , a contradiction with A being a minimal generatorof y . (cid:117)(cid:116) Thus, when A x is independent, each non-empty subset A of A x is the uniqueminimal generator of some y being included in A x . As a consequence, we obtainthe following theorem. Theorem 4.
Let F be a biatomic atomistic closure system. If for any x ∈ X andany minimal generator A x of x , A x is independent, then c( F ) ≤ (cid:100) log ( | X | + 1) (cid:101) .Proof. Let A x be a minimal generator of x , x ∈ X such that c( F ) = | A x | . As A x is a minimal generator, φ ( A ) (cid:54) = φ ( A (cid:48) ) for any distinct A, A (cid:48) ⊆ A x , due toProposition 3. Furthermore A x is independent by assumption. Thus, by Lemma4, for each non-empty subset of A , there exists y ∈ X such that A is the uniqueminimum subset of A x with y ∈ φ ( A ) . Consequently, X must contain at least | A x | − elements in order to cover each non-empty subset of A x , that is | A x | − ≤ | X | , which can be rewritten as | A x | = c( F ) ≤ (cid:100) log ( | X | + 1) (cid:101) as required. (cid:117)(cid:116) Now let F be a biatomic atomistic closure system on X given by some im-plicational base Σ and let G c = ( X, E c ) be a consistency-graph. Assume thatevery minimal generator is independent. By Theorem 4, we have that | X | hasexponential size with respect to c( F ) , and by Proposition 1, it must be that thesize of a key in K cannot exceed × (cid:100) log ( | X | + 1) (cid:101) . Thus, with respect to Σ , G c and X , K will have size quasi-polynomial in the worst case. Using the samealgorithm as in Section 4, we obtain the next theorem. Theorem 5.
Let Σ be an implicational base of a biatomic atomistic closuresystem F over X and G c a consistency-graph. If for any x ∈ X and any minimalgenerator A x of x , A x is independent, then MCCEnum can be solved in output-quasipolynomial time.Proof.
For clarity, we put n = | X | and k as the total size of the output MIS ( K ) . K can be computed in incremental-polynomial time with the algorithm in [27].Furthermore, by Theorem 4, the total size of K is bounded by n log( n ) . Thus,this first step runs in time poly ( | Σ | + | G c | + n + n log( n ) ) , which is bounded by poly ( | Σ | + | G c | + n ) log( n ) being quasipolynomial in the size of Σ , G c , K and X . To compute MIS ( K ) = maxCC ( Σ, G c ) we use the algorithm of Fredman andKhachiyan [15] whose running time is bounded by ( n log( n ) + k ) o (log( n log( n ) + k )) .In our case, we can derive the following upper bounds: ( n log( n ) + k ) o (log( n log( n ) + k )) ≤ ( k + n ) log( n ) × o (log( k + n ) log( n ) ) ≤ ( k + n ) O (log ( k + n )) Thus, the time needed to compute
MIS ( K ) from K is output-quasipolynomial inthe size of X and maxCC ( Σ, G c ) . Consequently, the running time of the wholealgorithm is bounded by poly ( | Σ | + | G c | + n ) log( n ) + ( k + n ) O (log ( k + n )) which is indeed quasipolynomial in the size of the input Σ , X , G c and the output MIS ( K ) = maxCC ( Σ, G c ) . (cid:117)(cid:116) numerating maximal consistent closed sets 13 To conclude this section, we show that atomistic modular closure systems[18,29] satisfy conditions of Theorem 5. Recall that a closure system F is modularif for any F , F , F ∈ F , F ⊆ F implies φ ( F ∪ ( F ∩ F )) = φ ( F ∪ F ) ∩ F .It was proved for instance in [5] (Theorem 7) that atomistic modular closuresystems are biatomic. To show that any minimal generator is independent, wemake use of the following result. Theorem 6. (Reformulated from [18], Theorem 360) Let F be a modular clo-sure system. A subset A = { a , . . . , a k } ⊆ X is independent if and only if φ ( a ) ∩ φ ( a ) = φ ( a a ) ∩ φ ( a ) = · · · = φ ( a . . . a k − ) ∩ φ ( a k ) = ∅ . Proposition 4.
Let F be an atomistic modular closure system. Let A x be aminimal generator of some x ∈ X . Then A x is independent.Proof. Let A x = { a , . . . , a k } be a minimal generator for some x ∈ X . Then, byProposition 3, φ ( a . . . a i ) ∩ A x = a . . . a i for any i ∈ [ k ] . Furthermore, φ ( a ) = { a } for any a ∈ X since F is atomistic. Thus we conclude that φ ( a . . . a i ) ∩ φ ( a i +1 ) = ∅ for any i ∈ [ k − as a i +1 / ∈ a . . . a i . It follows by Theorem 6 that A x is indeed independent. (cid:117)(cid:116) Corollary 3.
Let Σ be an implicational base over X and G c = ( X, E c ) . Then MCCEnum can be solved in output-quasipolynomial time if: – F is biatomic atomistic and has Carathéodory number (including convexsubsets of a poset and monophonically convex sets of a chordal graph), – F is atomistic modular.Proof. For the first statement, note that in an atomistic closure system withCarathéodory number , any minimal generator A x contains exactly two ele-ments a , a . Since F is atomistic, a and a are closed and the independence of A x follows.If F is atomistic modular, biatomicity follows from [5] (Theorem 7), andindependence from Proposition 4. (cid:117)(cid:116) Remark 1.
For atomistic modular closure systems, the connection between thesize of X and the Carathéodory number may also be derived from countingarguments on subspaces of vector spaces [30]. In this paper we proved that given a consistency-graph over an implicationalbase, the enumeration of maximal consistent closed sets is impossible in output-polynomial time unless P = NP . Moreover, we showed that this problem, called MCCEnum , is already intractable for the well-known class of lower boundedclosure systems. On the positive side, we proved that when the size of a mini-mal generator is bounded by a constant, the enumeration of maximal consistentclosed sets can be conducted in incremental polynomial time. This result covers various classes of convex geometries. Finally, we proved that in biatomic atom-istic closure systems,
MCCEnum can be solved in output-quasipolynomial timeprovided minimal generators obey an independence condition. This applies inparticular to atomistic modular closure systems. In Figure 2, we summarize ourresults in the hierarchy of closure systems.For future research, we would like to understand which properties or para-maters of closure systems make the problem intractable or solvable in output-polynomial time. For instance, we have seen that a bounded Carathéodory num-ber gives an incremental-polynomial time algorithm, while lower boundednessmakes the problem intractable. Another question is the following: is the prob-lem still hard if the closure system is given by a context (equivalently, its meet-irreducible elements)? The question is particularly interesting for classes such assemidistributive closure systems where we can compute the context in polyno-mial time in the size of an implicational base.
LB Bd. At. Mod.Mod.SDAc.CG
Bool. = BooleanDist. = DistributiveAt. = AtomisticMod. = ModularAc. = AcyclicCG = Convex GeometryBd. = BoundedLB = Lower BoundedSD = Semidistributive
CG, c ( F ) ≤ k . Bool.Dist. incr.-poly K exponential.Intractable. Unknown quasi-poly delay-poly
Fig. 2: The complexity of
MCCEnum in the hierarchy of closure systems
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