Hypergraph FKAlgorithmA::transversal (const Hypergraph& H) const {
BOOST_LOG_TRIVIAL(debug) << "Starting FKA. Hypergraph has "
<< H.num_verts() << " vertices and "
<< H.num_edges() << " edges.";
Hypergraph G (H.num_verts());
Hypergraph Hmin = H.minimization();
Hypergraph::Edge V = Hmin.verts_covered();
bool still_searching_for_transversals = true;
while (still_searching_for_transversals) {
Hypergraph::Edge omit_set = find_omit_set(Hmin, G);
if (omit_set.none() and G.num_edges() > 0) {
BOOST_LOG_TRIVIAL(debug) << "Received empty omit_set, so we're done.";
still_searching_for_transversals = false;
} else {
Hypergraph::Edge new_hs = V - omit_set;
Hypergraph::Edge new_mhs = minimize_new_hs(H, G, new_hs);
BOOST_LOG_TRIVIAL(trace) << "Received witness."
<< "\nomit_set:\t" << omit_set
<< "\nMHS:\t\t" << new_mhs;
G.add_edge(new_mhs, true);
BOOST_LOG_TRIVIAL(debug) << "New G size: " << G.num_edges();
}
}
return G;
}
Hypergraph ParBMAlgorithm::transversal (const Hypergraph& H) const {
BOOST_LOG_TRIVIAL(debug) << "Starting BM with " << num_threads
<< " threads. Hypergraph has "
<< H.num_verts() << " vertices and "
<< H.num_edges() << " edges.";
// Set up inputs
Hypergraph Hmin = H.minimization();
Hypergraph G (H.num_verts());
Hypergraph::Edge V = Hmin.verts_covered();
// Initialize using any HS we can find
Hypergraph::Edge first_hs = FKAlgorithm::minimize_new_hs(Hmin, G, V);
G.add_edge(first_hs);
// Grow G until it covers all vertices
bool G_has_full_coverage = false;
while (not G_has_full_coverage) {
Hypergraph::Edge new_hs = V - coverage_condition_check(H, G);
if (new_hs.is_proper_subset_of(V)) {
Hypergraph::Edge new_mhs = FKAlgorithm::minimize_new_hs(Hmin, G, new_hs);
G.add_edge(new_mhs);
} else {
G_has_full_coverage = true;
}
}
// Apply the BM algorithm repeatedly, generating new transversals
// until duality is confirmed
bool still_searching_for_transversals = true;
Hypergraph::EdgeQueue new_hses, new_mhses;
#pragma omp parallel shared(Hmin, G, new_hses, new_mhses) num_threads(num_threads)
#pragma omp master
while (still_searching_for_transversals) {
find_new_hses(Hmin, G, Hmin.verts_covered(), new_hses);
if (new_hses.size_approx() == 0) {
still_searching_for_transversals = false;
} else {
minimize_new_hses(Hmin, G, new_hses, new_mhses);
Hypergraph::Edge new_mhs;
while (new_mhses.try_dequeue(new_mhs)) {
// The results will all be inclusion-minimal, but
// there may be some overlap. Thus, we try to add
// them...
try {
G.add_edge(new_mhs, true);
}
// But ignore any minimality_violated_exception
// that is thrown.
catch (minimality_violated_exception& e) {}
}
BOOST_LOG_TRIVIAL(debug) << "New |G|: " << G.num_edges();
}
}
return G;
}
Hypergraph::Edge ParBMAlgorithm::find_subset_edge (const Hypergraph& H,
const Hypergraph::Edge& I) {
/**
Search for an edge of H which is a subset of I.
If found, return it; if not, return an empty edge as a signal.
**/
for (auto& edge: H) {
if (edge.is_subset_of(I)) {
return edge;
}
}
// If we make it this far, no edge was found
Hypergraph::Edge empty_edge (H.num_verts());
return empty_edge;
}
Hypergraph::Edge ParBMAlgorithm::find_missed_edge (const Hypergraph& H,
const Hypergraph::Edge& I) {
/**
Search for an edge of H which does not intersect I.
If found, return it; if not, return an empty edge as a signal.
**/
for (auto& edge: H) {
if (not edge.intersects(I)) {
return edge;
}
}
// If we make it this far, no edge was found
Hypergraph::Edge empty_edge (H.num_verts());
return empty_edge;
}
Hypergraph ParBMAlgorithm::l5_full_cover (const Hypergraph& H,
const Hypergraph::Edge& base_transversal) {
/**
Find a full cover of the dual of H from the given base_transversal
in accordance with lemma 5 of BM.
**/
Hypergraph C (H.num_verts());
Hypergraph::Edge V = H.verts_covered();
C.add_edge(base_transversal, false);
Hypergraph::EdgeIndex i = base_transversal.find_first();
while (i != Hypergraph::Edge::npos) {
V.reset(i);
C.add_edge(V, false);
V.set(i);
i = base_transversal.find_next(i);
}
return C;
}
Hypergraph ParBMAlgorithm::l4_full_cover (const Hypergraph& H,
const Hypergraph::Edge& base_edge) {
/**
Find a full cover of the dual of H from the given base_edge
in accordance with lemma 4 of BM.
**/
Hypergraph C (H.num_verts());
Hypergraph::Edge V = H.verts_covered();
for (auto& edge: H) {
Hypergraph::Edge intersection = edge & base_edge;
Hypergraph::EdgeIndex i = intersection.find_first();
while (i != Hypergraph::Edge::npos) {
Hypergraph::Edge new_edge = V - edge;
new_edge.set(i);
C.add_edge(new_edge, false);
i = intersection.find_next(i);
}
}
return C;
}
Hypergraph::Edge FKAlgorithmA::find_omit_set (const Hypergraph& F,
const Hypergraph& G) {
/**
Test whether F and G are dual.
If so, return an empty edge as a notification.
If not, return a omit_set as per FK.
**/
BOOST_LOG_TRIVIAL(trace) << "Beginning run with "
<< "|F| = " << F.num_edges()
<< " and |G| = " << G.num_edges();
// Input specification
assert(F.num_verts() == G.num_verts());
// Create an empty omit_set to use as temporary storage
Hypergraph::Edge omit_set (F.num_verts());
// FK step 1: initialize if G is empty
if (G.num_edges() == 0) {
BOOST_LOG_TRIVIAL(trace) << "Returning empty omit_set since G is empty.";
return omit_set;
}
// FK step 2: consistency checks
// Check 1.1: hitting condition
omit_set = hitting_condition_check(F, G);
if (omit_set.any()) {
return omit_set;
}
// Check 1.2: same vertices covered
omit_set = coverage_condition_check(F, G);
if (omit_set.any()) {
return omit_set;
}
// Check 1.3: neither F nor G has edges too large
omit_set = edge_size_check(F, G);
if (omit_set.any()) {
return omit_set;
}
// Check 2.1: satisfiability count condition
omit_set = satisfiability_count_check(F, G);
if (omit_set.any()) {
return omit_set;
}
// FK step 3: Check whether F and G are small
// If either hypergraph is empty, they cannot be dual
omit_set = small_hypergraphs_check(F, G);
if (omit_set.any()) {
return omit_set;
}
// FK step 4: Recurse
// Find the most frequently occurring vertex
Hypergraph::EdgeIndex max_freq_vert = most_frequent_vertex(F, G);
// Then we compute the split hypergraphs F0, F1, G0, and G1
std::pair<Hypergraph, Hypergraph> Fsplit, Gsplit;
Hypergraph F0, F1, G0, G1;
Fsplit = split_hypergraph_over_vertex(F, max_freq_vert);
F0 = Fsplit.first;
F1 = Fsplit.second;
Gsplit = split_hypergraph_over_vertex(G, max_freq_vert);
G0 = Gsplit.first;
G1 = Gsplit.second;
// We will also need the unions F0∪F1 and G0∪G1
Hypergraph Fnew = minimized_union(F0, F1);
Hypergraph Gnew = minimized_union(G0, G1);
// And, finally, fire up the two recursions
if (F1.num_edges() > 0 and Gnew.num_edges() > 0) {
BOOST_LOG_TRIVIAL(trace) << "Side 1 recursion.";
Hypergraph::Edge omit_set = find_omit_set(F1, Gnew);
if (omit_set.any()) {
return omit_set;
}
}
if (Fnew.num_edges() > 0 and G1.num_edges() > 0) {
BOOST_LOG_TRIVIAL(trace) << "Side 2 recursion.";
Hypergraph::Edge omit_set = find_omit_set(Fnew, G1);
if (omit_set.any()) {
omit_set.set(max_freq_vert);
return omit_set;
}
}
// If we make it this far, we did not find a nonempty
// omit_set, so the pair is dual
return omit_set;
}