5.1 Monotone Integer Programming [12–14,36]
Consider the following system of linear inequalities
(*) Ax ≥ b, ¯0 ≤ x ≤ c, where A ∈ Rm×n, b ∈ Rm+, c ∈ Rn+ (that is, coordinates of ccan take value +∞),x∈Zn+, and ¯0∈Rn+ is the origin.
System (*) is calledmonotoneifAx≥b⇒Ax ≥b ∀x, x|¯0≤x≤x≤c.
Complexity of Generation 9 Non-negativity of the matrix,A∈Rm×n+ , is sufficient for monotonicity, but not necessary. Note that without monotonicity, already verification of solvability of (*) is NP-complete, while for monotone systems it is trivial. In this case we will consider monotone generation.
Problem (K): generate all minimal integer solutionsZ(A, b, c) to (*).
If A is a (0,1) matrix, xis a (0,1) n-vector, and b =e is the vector of m ones, then (K) is the dualization of the hypergraphH defined by the rows ofA;
then, a (minimal) solutionxto (*) is a (minimal) transversal toH.
We generalize, replacing a (0,1) vectorxby a non-negative integer one,b=e by an arbitrary b ∈ Rm+, and finally, H(A, b, c) by Z(A, b, c), thus, replacing the Boolean cube by the direct product of discrete intervals (a box). These replacements keeps all basic properties [9].
Now, let us Consider the dual problem.
Problem (L): generate M IS(Z(A, b, c)) = Zdc(A, b, c), in other words, all maximalx∈Zn+not feasible to the monotone system (*), that is,x∈Z(A, b, c).
Problem (K) is uniformly DB, satisfying the inequality inequality
|Zd(A, b, c)| ≤mn|Z(A, b, c)|,
which is “almost” precise. Namely, there exists an example with
|Zd(A, b, c)| ≥ mn
2log2m|Z(A, b, c)|.
An incremental QP generation algorithm for Z(A, b, c) is provided by the method of joint generation, while generatingZd(A, b, c) is NP-complete [12].
5.2 Maximal Frequent and Minimal Infrequent Sets in Data Mining [1,17,24]
Given a binary m×n matrix A and positive integer t, a set C of columns is called t-frequent if there exist at leasttrows such that in the obtained t× |C|
submatrix all entries are 1s; otherwise C is called t-infrequent. Let α and β denote the numbers of the maximal t-frequent and minimal t-infrequent sets, respectively. The DB inequality α≤(m−t+ 1)β, implies that generating all minimalt-infrequent sets is incremental QP. In contrast, generating all maximal t-frequent sets is NP-complete. See [17] for more details.
5.3 Minimal Strongly Connected Subgraphs and Dicuts [35]
Given a (strongly connected) digraphG= (V, E), generate all
– (K) minimal edge-setsE⊆Esuch that the digraphG= (V, E) is strongly connected;
– (L) minimal dicuts, that is, minimal edge-setsE⊆E such that the digraph G= (V, E\E) is not strongly connected.
These dual problems are not DB; the size of each set may be exponential in the size of the other. Problem (K) is solved in incremental P time by the supergraph method, while (L) is NP-complete [35]. The proof of the latter claim is based on the “sausage lemma” (Sect.2.4and [25]) but requires extra tricks [35].
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5.4 Generating Generalized Paths, Cuts, and Spanning Sets in Graphs [32,33,36,45]
Given a (non-directed connected) graphG= (V, E) andk edge-subsetsE1, . . . , Ek ⊆E, for every I ⊆[k] define the graph GI = (V,∪i∈IEi) and consider the following two pairs of problems. Generate all
– (K1) generalized spanning sets, that is, all minimalI⊆[k] ={1, . . . , k}such that graphGI is connected;
– (L1) generalized complementary cuts, that is, all maximalI⊆[k] such that graphGI is not connected.
Given also two poless, t∈V,generate all
– (K2) generalized s-t-paths, that is, all minimal I ⊆ [k] such that s and t belong to one connected component ofGI.
– (L2) generalized complementarys-t-cuts, that is, all maximal I ⊆[k] such thatsandtare in distinct connected component ofGI.
Problem (K1) is QP uniformly DB and, hence, can be solved in incremental QP time. Three other problems are NP-complete; all three proofs are based on the sausage lemma.
5.5 Spanning Linear Spaces by Linear Subspaces [10,34]
LetL={L1, . . . , Lk}be a set of linear subspaces in a spaceL=Fdof dimension dover a fieldF and t≤dbe a positive integer threshold, for every I⊆[k] set LI =∪i∈ILi, or in other words,LI is spanned byLi, i∈I;
Consider the following pair of dual problems. Generate all – (K) minimal subsetsI⊆[k] such thatdim(LI)≥t;
– (L) maximal subsetsI⊆[k] such that dim(LI)< t.
Ift=dthenLI =L, that is,Li, i∈I,span the whole space. Problem (K) is QP uniformly DB and, hence, is incremental QP, while (L) is NP-complete.
Let us fix a basisB in L and assume that subspaces Li, i∈ I, are defined by subsets of B. In this case problem (K) is reduced to generating all minimal covers ofB by these subsets. This problem is equivalent to dualization.
5.6 Polymatroid Functions and Systems of Polymatroid Inequalities [10,34,38,41]
An integer non-negative set-functionf : 2V →Z+ is calledpolymatroid if it is submodular,f(I∪I) +f(I∩I)≤f(I) +f(I) for anyI, I∈V, monotone non-decreasing, andf(∅) = 0.
For example, f(I) = dim(LI) from the previous section is a polymatroid function. A special case of this example is the number of vertices minus the
Complexity of Generation 11 number of connected components of a graph. In other words, given a graph G = (V, E) and a family I of its edge-sets {Ei ⊆ E | i ∈ I}, we set f(I) =
|V| −C(G(V, EI)), where EI =∪i∈IEi and C(G) is the number of connected components of G. Another example is the number of degrees of freedom of a mechanical system. See [38] for more examples and details.
Given a system of polymatroid inequalities
(*) fj(I) ≥tj, j ∈ [r] = {1, . . . , r}, where functions fj are polymatroid and thresholdstj are QP inn=|V|, generate all
(K) minimalI⊆V satisfying (*); (L) maximalI⊆V not satisfying (*).
Problem (K) is QP uniformly DB, and, hence, it is incremental QP, while (L) is NP-complete.
5.7 Uniformly DB Inequalities for Polymatroid Systems [10,34]
LetH(f, t) be the family of all minimal subsets ofV satisfying (*).
Standardly,M IS(H(f, t)) denote the family of all maximal subsets ofV not satisfying (*). Let β =|H(f, t)| andα=|M IS(H(f, t)| denote the numbers of sets in these families. Then,
α≤βlogt/c(n,β)forβ≥2 andα≤nwhenβ = 1, where c=c(n, β) is the (unique) root of the equation
2c(nc/logβ−1) = 1.
This is a DB inequality for (K). It is QP whenevertis QP inn=|V|. This DB inequality can be replaced by a much simpler, although slightly weaker, one:
α≤(nβ)logt.
This bound can be viewed as a generalization, from graphs to hypergraphs, of an upper bound for the number of MIS. For graphs we have
2p≤ |M IS(G)| ≤δp+ 1,
where p=p(G) is the maximum size of the induced matchings in Gand δ = δ(G) is the maximum number of pairs of vertices at distance 2; in particular, δ≤n−1
2
, where the equality is achieved only on stars. This bound was proven in [4]; slightly weaker results were obtained in [2,43].
For the systems ofrpolymatroid inequalities, we generalize as follows:
α≤rmax(n, βlogt/c(n,β)), where t= max(t1, . . . , tr).
Interestingly, the coefficient 1/c= 1/c(n, β) in the exponent is accurate and cannot be reduced.
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