Compression with wildcards: All exact or all minimal hitting sets

1 Introduction

Let W be a finite set (such as all sets in this article) and P ( W ) its powerset. Given a hypergraph (=  set-system) H ⊆ P ( W ) , a ( H -)hitting set is a set X ⊆ W such that X ∩ H ≠ ∅ for all hyperedges H ∈ H . Let HS ( H ) be the set of all hitting sets, and MHS ( H ) the subset of all (inclusion-)minimal hitting sets, henceforth called MHSes. The famous minimal hitting set problem is as follows: given H ⊆ P ( W ) , is it possible to enumerate MHS ( H ) in polynomial total time,^[1] i.e., polynomial in w ≔ ∣ W ∣ , h ≔ ∣ H ∣ , and mhs ≔ ∣ MHS ( H ) ∣ ? We refer to [1,2] for the history and the state of the art concerning this problem. Suffice it to say that polynomial total time is possible for various fixed parameters, e.g., if the size of hyperedges is bounded from above. And that Hagen in his thesis [2] managed to show that three popular algorithms do provably not run in total polynomial time. There is no point to further review “old-school” (i.e., one-by-one) enumeration because for this article, it being dedicated to the compression of MHS ( H ) , there is little to compare it with. As to “little,” the only work the author is aware of is due to Toda [3]. In Section 11, it will be compared with our approach.

The latter can be described in picturesque ways as follows. For fixed H , identify the MHSes with diamonds and the ordinary hitting sets (i.e., the members of HS ( H ) \ MHS ( H ) ) with worthless pebbles, which, however, may be hard to distinguish from diamonds. Some friendly sponsor provides R many nonempty boxes that are filled with both kinds of stones. All diamonds are distributed among the boxes but usually not all pebbles (which is just as well). Our main quest is to retrieve all diamonds (and only them) as efficiently as possible. A box is good if it contains at least one diamond, and bad otherwise. A box 100% filled with diamonds is very-good. As will be seen, depending on the structure of H , very-good boxes can be both numerous and heavy! Furthermore, the number of diamonds in a very-good box is found at once, and the diamonds themselves are arranged in a pleasant, compressed manner.

To obtain a first impression of the quality of boxes, the Monte-Carlo method picks (say) 20 stones at random from each box ρ and determines the number α ( ρ ) of diamonds among them. If 0 < α ( ρ ) < 20 , then ρ is merely-good, i.e., good but not very-good. However, if α ( ρ ) = 0 , then ρ is only likely-bad in the sense that picking 20 random pebbles does not imply the absence of diamonds in that box, but it makes it rather likely.^[2] Similarly, if α ( ρ ) = 20 , then ρ is likely-very-good. If a likely-very-good box contains thousands of stones, then classifying the stones one-by-one is time-consuming. We will provide three criteria for very-goodness that settle the issue faster. Efficient criteria for badness are harder to come by but an elegant sufficient condition exists. As to merely-good boxes ρ , there are two approaches, each with benefits and drawbacks. The first is to classify the stones one-by-one. The second uses subtle machinery but has the benefit that the diamonds in ρ get repackaged into brandnew very-good boxes.

Here comes the section break-up, phrased in more mathematical terms. The preliminaries in Section 2 concern Boolean functions and three kinds of wildcards: the e -, the n -, and the g -wildcard. All of them generalize the don’t-care symbol ∗ familiar from describing partial models of Boolean functions. Strings of mentioned symbols yield various kinds of “rows,” e.g., 012e-rows or 01g-rows. If the type is clear, we simply speak of rows. Furthermore, we adopt the vertical layout (VL) technique used in data mining. It substitutes set operations that involve many small sets by set operations with few large sets. Section 3 discloses the sponsor of boxes as the transversal e -algorithm of [4]. And Section 4 discloses the mathematical nature of the R boxes as 01g-rows. We then establish (Theorem 1) the pleasant fact that all minimum-cardinality MHSes occur in very-good rows, which, moreover, can be pinpointed at once among the R rows in total.

Unfortunately, in the remainder, we face deeper waters in our quest to find all MHSes. For starters, Section 5 elaborates the first aforementioned approach towards merely-good 01g-rows ρ by offering four algorithms to classify one-by-one the bitstrings (=  stones) in ρ . Algorithm 1 relies on the diamonds (= MHSes) retrieved so far, whereas Algorithm 2 only relies on the knowledge of H . By definition, X ⊆ W is an MC-set if each x ∈ X has a private hyperedge H ∈ H , i.e., one that cuts x sharply in the sense that H ∩ X = { x } . The set-system MC ( H ) of all MC-sets is dual to HS ( H ) in that the former is a set-ideal and the latter a set-filter. It holds (proven in [5], recast in Theorem 2) that MC ( H ) ∩ HS ( H ) = MHS ( H ) . Algorithm 3 exploits this, and Algorithm 4 additionally uses facts that are fully justified only in Section 9.

Section 6 elaborates the second aforementioned approach towards merely-good rows. It relies on Sections 7 and 8 that deliver two criteria for very-goodness, each of which is sufficient and necessary. The first is based on inclusion–exclusion, and the second on matroid theory (Rado’s theorem).

As to Section 9, the subsets of W that are not MC, yet all their proper subsets are MC, are nice to know. They are collected in the set-system MinNotMC ( H ) . For instance, it allows us to calculate the cardinality ∣ MHS ( H ) ∣ without knowing MHS ( H ) . Section 10 calculates MinNotMC ( H ) . It exploits the fact that minimal set-coverings are cryptomorphic to minimal hitting sets and can hence be handled with the transversal e -algorithm.

Section 11 features numerical experiments with Mathematica. In a nutshell, our compression with wildcards (specifically the number and thickness of very-good rows) is the more impressive the fewer and the larger the hyperedges are. Although some ideas of previous sections have not yet been implemented in Mathematica, in 11.6, we attempt a preliminary comparison of our methods with the two winners [3] and [5] of a competition carried out in [1].

Section 12 at first seems to abandon minimal hitting sets and turn to exact hitting sets (EHSes). Are they that different? By definition, Y ⊆ W is an EHS for H if ∣ Y ∩ H ∣ = 1 for all H ∈ H . Under the mild assumption that ⋃ H = W each EHS must be an MHS, yet the converse fails severely in that some hypergraphs have plenty MHSes but no EHSes. Nevertheless, our previously used g -wildcards can sometimes compress the set-system EHS ( H ) of all EHSes. As to “sometimes,” any fixed hypergraph H ⊆ P ( W ) induces a natural, apparently novel equivalence relation ∼ on W . It turns out that compressing EHS ( H ) is possible iff ∼ is nontrivial. Furthermore, Knuth’s popular Dancing-Link algorithm appears in Section 12, and in Theorem 4, we enumerate the perfect matchings of any graph without K 3 , 3 -minor in polynomial total time.

2 Preliminaries on Boolean functions, partial models, wildcards, and VL

After Boolean functions (Section 2.1), we turn to e -wildcards (2.2–2.3), followed by n -wildcards and g -wildcards (2.4). In 2.5, we sieve the minimal members of any set-system S ⊆ P ( W ) and 2.6 introduces VL.

Throughout the article for any integer w ≥ 1 , we put [ w ] ≔ { 1 , 2 , … , w } . For convenience, usually W ≔ [ w ] . If the powerset is concerned, we write P [ w ] instead of P ( [ w ] ) . Furthermore, we use the shorthand “iff” for “if and only if,” and write ⊂ (as opposed to ⊆ ) for proper inclusion.

2.1 We freely identify bitstrings of length w (also called 01-rows) with subsets of [ w ] in the usual way; thus, X = { 2 , 4 , 5 } (viewed, say, as subset of [6]) matches x = ( 0 , 1 , 0 , 1 , 1 , 0 , 0 ) . Depending on circumstances, one or the other view is preferable. We now extend 01-rows to 012-rows such as

The following type of notation that refers to the positions of the various symbols will be used throughout:

While 01-rows encode sets, 012-rows encode set-systems because “2” is viewed^[3] as don’t-care symbol that can be freely replaced by 0 or 1. Thus, r = ( 0 , 2 , 2 , 1 , 0 , 2 ) encodes, and in fact will be identified ^[4] with, the set-system:

which, with obvious shorthand notation (that will only be applied to sets of 1-digit numbers), can also be rendered (since elements of sets can be listed in arbitrary order) as:

As to a general 012-row r , if it is viewed as a set-system, this set-system is { ones ( r ) ∪ S : S ⊆ twos ( r ) } . While zeros ( r ) does not come up here, the 0’s are as important as the 1’s in the sequel (ponder what would become of r = ( 0 , 2 , 2 , 1 , 0 , 2 ) without the 0’s).

2.1.1 That leads us to { 0 , 1 } viewed as Boolean algebra^[5] and to Boolean functions f : { 0 , 1 } n → { 0 , 1 } whose basic features are assumed to be familiar to the reader, so that we only need to fix notation here. Any bitstring x ∈ { 0 , 1 } n with f ( x ) = 1 is a model of f . Apart from other means, Boolean functions can be defined by Boolean formulas. Thus, by writing f ( x ) ≔ x 1 ∨ x 2 ∨ x 3 , we define^[6] a Boolean function f : { 0 , 1 } 3 → { 0 , 1 } that, e.g., satisfies f ( ( 0 , 1 , 1 ) ) = 0 ∨ 1 ∨ 1 = 1 . It is clear that only ( 0 , 0 , 0 ) fails to be a model, and so the modelset is

The union on the righthand side is not disjoint since, e.g., ( 1 , 0 , 1 ) ∈ ( 1 , 2 , 2 ) ∩ ( 2 , 2 , 1 ) . Fortunately, this can be cured as follows (here and henceforth ⊎ signifies disjoint union):

This idea is long known, and its visualization has been coined Abraham-flag in [7]. Thus, a general n × n Abraham-flag has 1’s in the main diagonal, 0’s below it, and 2’s above it. The row-cardinalities sum up to 2 n − 1 + 2 n − 2 + ⋯ + 1 , which equals 2 n − 1 , as is to be expected. In connection with Boolean functions, 012-rows usually describe partial models. For instance, ( 1 , 2 , 2 ) is a partial model of x 1 ∨ x 2 ∨ x 3 in the sense that replacing the 2’s by 0 or 1 in any way results in a model of x 1 ∨ x 2 ∨ x 3 .

2.2 In addition to the don’t-care symbol “2,” we will use three further wildcards. For starters, instead^[7] of using an s × s Abraham-flag to spell out Mod ( x 1 ∨ ⋯ ∨ x s ) , we can, better still, simply define

Roughly speaking, s symbols e (not necessarily adjacent) demand bitstrings to have “at least one 1 in that area.” Combining such e -wildcards (distinguished by subscripts) gives rise to 012e-rows like

which, by definition, consists of those subsets S ⊆ [ 8 ] that satisfy

• 2 , 7 ∉ S (because zeros ( r ′ ) = { 2 , 7 } ),

• 6 ∈ S (because ones ( r ′ ) = { 6 } ),

• { 1 , 4 } ∩ S ≠ ∅ (because of e 1 , e 1 ),

• { 5 , 8 } ∩ S ≠ ∅ (because of e 2 , e 2 ).

The fact that twos ( r ′ ) = { 3 , 9 , 10 } reflects the fact that 3, 9, and 10 do not occur in any of the conditions. By e-bubble, we mean the position-set of any given e -wildcard. Thus, the e 2 -bubble of the e 2 -wildcard in (2) is { 5 , 8 } . It is easy to see that

and that 2 2 − 1 generalizes to 2 s − 1 for e -bubbles of size s .

Alternatively (but clumsier), r ′ in (2) could be defined^[8] as follows:

2.2.1 Observe that the intersection ρ ∩ ρ ′ of an 012e-row ρ with an 012-row ρ ′ is either empty (when 0’s and 1’s clash) or is again a 012e-row, which arises in obvious ways:

2.2.2 The set of all minimal ^[9] members contained in a 012e-row will play a crucial role. One checks that the set-system Min ( r ′ ) of all minimal members of the set-system r ′ in (2) equals

Generally, if the 012e-row r has t ≥ 1 many e -wildcards of cardinalities ε 1 , … , ε t , then^[10] each X ∈ Min ( r ) is of type X = ones ( r ) ∪ T , where T cuts each e -bubble in exactly one element. Thus, ∣ T ∣ = t . If we define the degree of r as:

then

For general set-systems S , it will be more demanding (2.5) to sieve Min ( S ) from S . Nevertheless, (5) will keep coming back even in that context.

2.3 Let us introduce higher-level Abraham-flags, i.e., constituted by certain 012e-rows as opposed to the 012-rows in 2.1. Consider

Soon, we need to be able to, e.g., sieve those bitstrings ( x 1 , … , x 9 ) from r that have at least one 1 among { x 1 , … , x 5 } . In other words, we need to “impose” ( e , e , e , e , e ) upon r , i.e., putting r ′ ≔ ( e , e , e , e , e , 2 , 2 , 2 , 2 ) the intersection r ∩ r ′ of two 012e-rows must be rewritten in a handy format. The answer is ( e 1 , e 1 , e 2 , e 2 , e 3 , e 1 , e 2 , e 2 , e 3 ) ∩ ( e , e , e , e , e , 2 , 2 , 2 , 2 ) = r 1 ⊎ r 2 ⊎ r 3 , where

The first part of the righthand side is a novel 3 × 3 Abraham-flag in the sense that the boldface main diagonal entries are either 1 (as in 2.1) or full e -wildcards. Likewise, the entries below the main diagonal are again 0’s. What happens above the main diagonal, and how all of this affects the last four columns in (7), has been discussed^[11] in [4] (see also Section 3.1).

2.4 Dually to e -wildcards, we will encounter n -wildcards that demand “at least one 0 here.” Thus, for instance,

Mutatis mutandis as in 2.2, we define n -bubbles and 012 n -rows.

2.4.1 Apart from e -wildcards and n -wildcards,^[12] a third type of wildcard takes care of the requirement “exactly one 1 here.” Namely, by definition:

One trivial application of these g -wildcards (and coupled g-bubbles) is the compression of MHS ( H ) for hypergraphs with disjoint hyperedges. Thus, if H 1 = { 123 , 45 , 6789 } , then MHS ( H 1 ) = ( g 1 , g 1 , g 1 , g 2 , g 2 , g 3 , g 3 , g 3 , g 3 ) . Slightly more subtle and important later, one checks that r ′ = ( e 1 , 0 , 2 , e 1 , e 2 , 1 , 0 , e 2 , 2 , 2 ) from (2) has Min ( r ′ ) = ( g 1 , 0 , 0 , g 1 , g 2 , 1 , 0 , g 2 , 0 , 0 ) . Expressions like this are called 01g-rows.

2.5 Let S ⊆ P [ w ] be any set-system. The problem to obtain^[13] the set-system Min ( S ) of all minimal members of S occurs frequently in discrete mathematics. The naive way to proceed is to decide for each X ∈ S whether there is another Y ∈ S with Y ⊂ X . Clearly, X belongs to Min ( S ) iff no such Y exists. Since deciding whether or not Y ⊂ X costs O ( w ) , the overall cost is O ( ∣ S ∣ 2 w ) .

To the author’s best knowledge (and surprise), the following refinement has not appeared in the literature before. Start by grouping the members of S according to their cardinalities m 1 < m 2 < ⋯ < m s (often m i + 1 = m i + 1 ). This induces the decomposition S = S [ 1 ] ⊎ S [ 2 ] ⊎ ⋯ ⊎ S [ s ] and costs O ( ∣ S ∣ w ) . It suffices to show how to calculate Min [ i ] ≔ S [ i ] ∩ Min ( S ) for all 1 ≤ i ≤ s .

Clearly, Min [ 1 ] = S [ 1 ] since minimum-cardinality implies minimal. Set S ′ [ i ] ≔ S [ i ] for 2 ≤ i ≤ s . Throughout the remainder, we will have Min [ i ] ⊆ S ′ [ i ] ⊆ S [ i ] and the set-systems S ′ [ i ] keep shrinking until they reach S ′ [ i ] = Min [ i ] . To begin with, pick any X ∈ Min [ 1 ] and remove all^[14]  Y ∈ S ′ [ i ] ( i ≥ 2 ) from S ′ [ i ] whenever X ⊂ Y . This costs O ( ∣ S ∣ w ) . Doing the same for all members X ′ ∈ Min [ 1 ] costs O ( ∣ S ∣ w ⋅ ∣ Min [ 1 ] ∣ ) = O ( ∣ S ∣ w ⋅ min ) , where min ≔ ∣ Min ( S ) ∣ . It is clear that afterward, S ′ [ 2 ] = Min [ 2 ] . Next for each X ∈ Min [ 2 ] and all Y ∈ S ′ [ i ] ( i ≥ 3 ) remove Y from S ′ [ i ] whenever X ⊂ Y . Clearly, afterward, S ′ [ 3 ] = Min [ 3 ] . And so it goes on until we obtain S ′ [ s ] = Min [ s ] . The overall cost is O ( ∣ S ∣ w ⋅ min ⋅ s ) = O ( ∣ S ∣ w 2 ⋅ min ) .

2.6 The operations ∨ and ∧ on { 0 , 1 } extend to operations on { 0 , 1 } m (and they match union/intersection of sets in P [ m ] ). Adopting Mathematica terminology, we call the extended operations BitOr and BitAnd. For instance, referring to the columns of the 8 × 6 matrix A with rows Z 1 to Z 8 (Table 1), it holds that BitAnd ( col 2 , col 6 ) = ( 0 , 0 , 1 , 1 , 0 , 0 , 0 , 1 ) T (where the T means “transposed”).

2.6.1 What is this good for? The fact that BitAnd ( col 2 , col 6 ) had a component 1 exactly on the 3th, 4th, and 8th position tells us that among the sets Z 1 , … , Z 8 , the ones that contain the set { 2 , 6 } are exactly Z 3 , Z 4 , and Z 8 . This is, e.g., relevant for speeding up the method of 2.5.

2.6.2 Here comes another application. Consider the set-system

The straightforward (=  “horizontal”) way to see whether X = { 1 , 2 , 5 } is a G -transversal checks whether any intersection X ∩ Y ( Y ∈ G ) is empty. In contrast, VL demands^[15] to build the 8 × 6 matrix A ( G ) whose i th row Y i ′ is the characteristic bitstring of the i th set Y i listed in (8). It happens that A ( G ) is rendered in Table 1. A moment’s reflection confirms the following. The fact that BitOr ( col 1 , col 2 , col 5 ) = ( 1 , 1 , 1 , 1 , 1 , 1 , 0 , 1 ) T does not equal ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) T is tantamount to X not being a G -hitting set ( X ∩ Y 7 = ∅ ). Although the formal complexities of the horizontal and vertical way coincide, in practice, VL is the faster the more (small) sets G contain. Simply put, computer hardware prefers doing few operations with long bitstrings to doing many operations with short bitstrings.

3 Review of the transversal e -algorithm

We survey the transversal e -algorithm (3.1) and adapt it to count or generate hitting sets of fixed cardinality (3.2). In 3.3, we indicate how the transversal e -algorithm dualizes to the noncover n -algorithm.

3.1 Consider the task to enumerate the set HS ( H 2 ) of all hitting sets of the hypergraph H 2 whose five hyperedges X ⊆ [ 6 ] are

One idea is to first compute the hitting sets of the hypergraph { H 1 } , then the ones of { H 1 , H 2 } , and so forth until we obtain the hitting sets of { H 1 , … , H 5 } = H 2 . Calculating HS ( { H 1 } ) is easy in view of 2.2. It consists of all bitstrings (= subsets of [6]) that have at least 1 on the positions 1, 2, 5, and so HS ( { H 1 } ) = ( e , e , 2 , 2 , e , 2 ) . Likewise, HS ( { H 1 , H 2 } ) = ( e 1 , e 1 , e 2 , e 2 , e 1 , 2 ) ≕ r ′ .

Now, it gets trickier because H 3 intersects H 1 and H 2 , i.e., the e 3 -wildcard supposed to be modeling H 3 interferes with existing e -wildcards. In 2.3, we indicated how this is to be handled. Recall that the row in (6), which suffered the same predicament as r ′ mentioned earlier, had to be replaced by three candidate sons in (7). The essence of the transversal e -algorithm is to keep on picking the topmost row r ′ of a “to do” stack of 012e-rows and to impose some e -wildcard upon r ′ , which in turn can trigger up^[16] to t candidate sons. Each candidate son r i must be feasible in the sense that r i ∩ HS ( H ) ≠ ∅ , for otherwise further processing of r i cannot possibly yield any hitting sets. The feasible candidate sons are put on top of the last-in-first-out (LIFO)-stack, and the others are discarded. Fortunately, deciding feasibility is easy:

The effect of discarding infeasible candidate sons is that in each set of candidate sons, at least one will be feasible. This in turn is the reason that the e -algorithm runs in total polynomial time, in fact in O ( R h 2 w 2 ) time. For the fine details of this transversal e -algorithm,^[17] the reader is referred to [4]. To summarize, for any given hypergraph H ⊆ P [ w ] , the transversal e -algorithm renders HS ( H ) as a disjoint union of R many 012e-rows; thus,

3.1.1 Applied to H 2 , the transversal e -algorithm yields HS ( H 2 ) = ρ 1 ¯ ⊎ ⋯ ⊎ ρ 4 ¯ , where the ρ i ¯ ’s are defined in Table 2.

In view of 2.2, we conclude that

3.2 Let μ ≔ μ ( H ) be the minimum cardinality achieved by any hitting set of the hypergraph H . In our scenario, μ gets usually known^[18] only after (11) has been obtained. For all c ∈ { μ , μ + 1 , … , w } , we put

Of particular interest is the set-system MCHS ( H ) of all minimum-cardinality hitting set, i.e.

3.2.1 In some circumstances (e.g., in [10]), it is irrelevant whether the H -hitting sets are minimal; just their cardinality matters. Let us hence calculate ∣ HS ( H , c ) ∣ for any fixed c ≥ μ . Viewing (11) for any such c , let I ¯ be the set of indices i ≤ R such that the 012e-row ρ i ¯ has degree ≤ c (that is because ρ i ¯ ∩ HS ( H , c ) = ∅ if i ∉ I ¯ ). Putting S ( ρ i ¯ ) ≔ { X ∈ ρ i ¯ : ∣ X ∣ = c } , we obtain ∣ HS ( H , c ) ∣ by summing up the numbers ∣ S ( ρ i ¯ ) ∣ ( i ∈ I ¯ ) . It is easy to calculate the numbers ∣ S ( ρ i ¯ ) ∣ with inclusion–exclusion; for a faster way, see [4, Thm.1].

3.2.2 Suppose the set HS ( H , c ) itself needs to be calculated. By the aforementioned each fixed set-family HS ( H , c ) is the disjoint union of all sets S ( ρ i ¯ ) ( i ∈ I ¯ ) . But sieving S ( ρ i ¯ ) from ρ i ¯ is more cumbersome than calculating ∣ S ( ρ i ¯ ) ∣ . Leaving ways of compression to the future, we only note that if S ( ρ i ¯ ) has α elements, then by [4, Thm.2], it can be enumerated one-by-one in total polynomial time O ( α w 2 ) .

3.2.3 If HS ( H , c ) is of interest cardinality-wise (or the members themselves) for all values μ ≤ c ≤ w , then upon running the transversal e -algorithm, each c gets processed as discussed in 3.2.1 (or 3.2.2). However, if only values c ≤ d for some bound d are relevant, then it pays to adjust the transversal e -algorithm as follows. In addition to (10), the arising candidate sons should also satisfy deg ( ρ ¯ ) ≤ d . That is because deg ( ρ ¯ ) > d implies that all successor rows ρ i ¯ of ρ ¯ will have deg ( ρ i ¯ ) ≥ deg ( ρ ¯ ) > d and so cannot contain any members of HS ( H , c ) . Problem is that, in contrast to the remarks after (10), it can now happen that some rows loose all their candidate sons. Nevertheless, performance in practice may be good.

3.3 The family HS ( H ) of all H -hitting sets is a set-filter ℱ in the sense that ( X ∈ ℱ and X ⊆ Y ) ⇒ Y ∈ ℱ . Now, let S ⊆ P [ w ] be a set-system. Call Z ∈ P [ w ] a S -noncover if Z ⊉ Y for all Y ∈ S . Then the family N C ( S ) of all S -noncovers is a set-ideal J in the sense that ( X ∈ J and Y ⊆ X ) ⇒ Y ∈ J . Consider any set-filter ℱ ⊆ P [ w ] . The minimal members of ℱ are called its generators, and they determine ℱ uniquely. Likewise, for any set-ideal^[19] J ⊆ P [ w ] , the maximal members of J are called its facets, and they determine J uniquely. Furthermore, let ℱ and J be complementary set-systems in the sense that ℱ ⊎ J = P [ w ] . It then holds that ℱ is a set-filter iff J is a set-ideal.

Given H ⊆ P [ w ] , the transversal e -algorithm renders the set-filter HS ( H ) in the convenient format (11). Since set-filter and set-ideal are dual concepts, and so are e -wildcards and n -wildcards, it comes as no surprise that some noncover n-algorithm (see, e.g., [6]), fed with S renders the set-ideal N C ( S ) as a disjoint union of R ′ many 012n-rows:

4 From minimum-cardinality toward inclusion-minimal

We argue why all 012e-rows ρ i ¯ in (11) should get “shaved” and become certain 01g-rows ρ i ⊆ ρ i ¯ . Thus, (11) will improve to (17). It pleasantly turns out that MCHS ( H ) in (13) is the union of some such rows ρ i . In 4.2, we comment on situations where moreover MCHS ( H ) = MHS ( H ) , and in 4.4, resume the Monte-Carlo method of Section 1 in order to obtain an estimate for ∣ MHS ( H ) ∣ .

4.1 Let H ⊆ P [ w ] be a hypergraph. In the remainder of this article, we assume that the transversal e -algorithm has rendered HS ( H ) as a disjoint union of R many 012e-rows as in (11). Different from [4] where these rows were coined “final,” here the availability of them is not the end but only the beginning. That is why, we henceforth call them

Suppose X ⊆ [ w ] is any minimal H -hitting set. Then, X is contained in some semifinal 012e-row ρ i ¯ because of (11). Being minimal within HS ( H ) , a fortiori X is minimal within the smaller set-system ρ i ¯ ⊆ HS ( H ) , i.e., X ∈ Min ( ρ i ¯ ) . In view of (5), it follows that for all 1 ≤ i ≤ R :

In particular, consider Y ∈ MCHS ( H ) ⊆ MHS ( H ) (see (13)). As before, Y ∈ Min ( ρ j ¯ ) for some j ≤ R . But all sets in Min ( ρ j ¯ ) have the same cardinality as Y , and so are themselves in MCHS ( H ) . Hence, ⊆ in (14) becomes =. To summarize:

To illustrate, consider HS ( H 2 ) = ρ 1 ¯ ⊎ ⋯ ⊎ ρ 4 ¯ in Table 2. One checks that all these rows happen to have degree 3, and so μ = 3 . It follows from Theorem 1 and the fact (see 2.4.1) that sets of type Min ( ρ i ¯ ) can conveniently be rendered by single 01 g -rows ρ i that

where the ρ i ’s are defined as follows:

4.2 As opposed to (15) where incidentally MCHS ( H 2 ) = MHS ( H 2 ) , for general hypergraphs H , only a few semifinal 012e-rows ρ i ¯ will have degree μ ! If only MCHS ( H ) is sought, then all rows ρ i ¯ with deg ( ρ i ¯ ) > μ are superfluous. Yet to avoid them, one cannot proceed as in 3.2.3 because usually, d ≔ μ is not known in advance. However, guessing and working with some slightly larger d > μ will still beat computing all R rows. (If it happens that one guesses a d < μ , then the proposed method will not deliver any semifinal 012e-rows. But it will improve the next guess, and with binary search, one can even pin down μ .)

4.2.1 In the following set-up μ , is known^[20] in advance; it even happens that MHS ( H ) = MCHS ( H ) . Namely, if H is the family of all cocircuits [8, p. 653] of a matroid, then MHS ( H ) is the set of all matroid bases and μ is easy to come by. In arXiv:2002.09707 (submitted), this has been implemented for the scenario where the cocircuits are the minimal cutsets of a graph G , in which case MHS ( H ) = MCHS ( H ) is the set of all spanning trees of G .

4.2.2 Suppose that μ is known, be it by binary search or by theoretical reasoning as in 4.2.1. Then, one still sits with the problem (mentioned in 3.2.3) that some top-rows of the LIFO-stack may loose all their candidate sons. That this cannot happen in 4.2.1 is one of the (numerically well-supported) conjectures raised in arXiv:2002.09707. In another vein, if all H ∈ H have ∣ H ∣ = 2 , so H is the edge-set of a graph, then instead of MHSes, one rather speaks of minimal vertex-covers. In this scenario, μ remains hard to compute, but at least “loosing all candidate sons” can be avoided (work in progress).

4.3 Generalizing Table 2 and (16), each semifinal 012e-row ρ ¯ ≔ ρ i ¯ appearing in (11) yields the

ρ ≔ Min ( ρ ¯ ) , where all 2’s of ρ ¯ have been replaced by 0’s and each e -wildcard of length ε j has been replaced by a g -wildcard of the same length γ j ≔ ε j . Hence, akin to (4) and (5), the semifinal 01g-row ρ has t many g -wildcards, and it holds that the γ 1 γ 2 ⋯ γ t members of ρ all have cardinality ∣ ones ( ρ ) ∣ + t . It follows from (11) that:

Accordingly, we have

The following terminology will be handy as well. A semifinal 01g-row ρ is bad if MHS ( H ) ∩ ρ = ∅ , and good otherwise. Additionally, call ρ very-good if ρ ⊆ MHS ( H ) , and call ρ merely-good if it is good but not very-good. Each X ∈ ρ \ MHS ( H ) a dud. For 012-rows, it holds that good ⇔ very-good .

4.4 Recall from Section 1 that a simple attempt to settle “good or bad?” is the Monte-Carlo way. That is, pick uniformly and at random X ∈ ρ , and check (in whatever way) whether or not X ∈ M H S . If yes, then ρ is good. If no, test some more X . The more often the answer persists to be no, the likelier ρ is bad. Specifically, the density d ≔ ∣ ρ ∩ MHS ( H ) ∣ ∕ ∣ ρ ∣ can be estimated to any desired precision ε as follows. Given ε , δ > 0 , standard statistics yields a value d ′ such that (with error-probability < δ ), it holds that d ∈ [ ( 1 − ε ) d ′ , ( 1 + ε ) d ′ ] . Since ∣ ρ ∣ = γ 1 ⋯ γ t is known, d ′ also yields an estimate for ∣ ρ ∩ MHS ( H ) ∣ , and hence in view of (17) for ∣ MHS ( H ) ∣ .

5 Four algorithms to sieve the MHSes one-by-one from the semifinal 01g-rows

Let ρ be a fixed semifinal 01g-row. In this section, we present four methods (Algorithm 1 to Algorithm 4) to classify all X ∈ ρ one-by-one, i.e., to decide whether X is an MHS or a dud. Algorithm 1 relies on 2.5 and 2.6.1, whereas Algorithm 2 uses the kind of VL in 2.6.2. Algorithms 3 and 4 rely on set-systems PotKi ( ρ ) and MC ( H ) , some of whose features will be postponed to Section 9.

5.1 Referring to 2.5, let m 1 < m 2 < ⋯ < m s be the numbers that occur as cardinalities of H -hitting sets. Then,^[21]  m 1 = μ and m s = w . Putting S ≔ SF ( H ) , we have Min ( S ) = MHS ( H ) , and following 2.5, we obtain MSH ( H ) in time O ( s f ⋅ w ⋅ m h s ⋅ s ) . This method, call it Algorithm 1, compares favorably to future methods if we only care for MHSes of rather small cardinality, say ≤ μ + 2 . Generally, for upper-bounded MHS-size Algorithm 1 costs O ( sf ⋅ w ⋅ m h s ) .

5.2 Let us view the hyperedges of H as bitstrings and take them as the rows of an h × w matrix A . Fix a semifinal 01g-row ρ and put k ≔ deg ( ρ ) . For each fixed Y ∈ ρ (hence a hitting set), it holds that Y ∈ MHS ( H ) iff no set X ≔ Y \ { a } ( a ∈ Y ) is a hitting set. Whether or not VL based on A (see 2.6) is used, the formal cost to classify X is O ( k h ) . Hence, classifying Y costs O ( k 2 w ) . Furthermore, finding ρ ∩ MHS ( H ) costs O ( ∣ ρ ∣ k 2 h ) , and finding MHS ( H ) with this method (call it Algorithm 2) costs O ( s f ⋅ w 2 h ) , since ∣ Y ∣ = k gives way to ∣ Y ∣ ≤ w .