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 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;
}
void ParBMAlgorithm::find_new_hses_fork (const Hypergraph& H,
const Hypergraph& G,
const Hypergraph& C,
Hypergraph::EdgeQueue& results) const {
/**
Find any new hitting sets of H with respect to G,
splitting the work over the full cover C of Tr(H) and queueing
the results.
**/
Hypergraph::Edge V = C.verts_covered();
for (auto c: C) {
#pragma omp task shared(H, G, results)
{
assert(c.is_subset_of(V));
find_new_hses(H, G, c, results);
}
}
#pragma omp taskwait
}
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;
}
void ParBMAlgorithm::find_new_hses (const Hypergraph& H,
const Hypergraph& G,
const Hypergraph::Edge& c,
Hypergraph::EdgeQueue& results) const {
/**
Find any new hitting sets of H^c with respect to G_c, queueing the
results.
**/
// Construct the reduced hypergraphs
Hypergraph Hc = H.contraction(c, false);
Hypergraph Gc = G.restriction(c);
BOOST_LOG_TRIVIAL(trace) << "Starting transversal build with |Hc| = " << Hc.num_edges()
<< " and |Gc| = " << Gc.num_edges();
// Initialize a candidate hitting set to store intermediate results
Hypergraph::Edge new_hs = Hc.verts_covered();
// Step 1: Initialize Gc if it is empty by returning the set of all vertices
if (Gc.num_edges() == 0) {
// If any edge of Hc is empty, this is a dead end
for (const auto& e: Hc) {
if (e.none()) {
return;
}
}
// Otherwise, the support of Hc is a new hitting set
results.enqueue(new_hs);
return;
}
// Step 2: Handle |Gc| = 1
if (Gc.num_edges() == 1) {
assert(Hc.num_edges() != 0);
// Check whether H has a singleton for every vertex it covers
Hypergraph::Edge Hc_verts_to_cover = Gc[0];
for (const auto& e: Hc) {
if (e.count() == 1) {
Hc_verts_to_cover -= e;
}
}
if (Hc_verts_to_cover.none()) {
// If so, this is a dead end
return;
} else {
// If not, we choose some vertex that was hit by Gc but was
// not a singleton in Hc
Hypergraph::EdgeIndex uncovered_vertex = Hc_verts_to_cover.find_first();
new_hs.reset(uncovered_vertex);
results.enqueue(new_hs);
return;
}
// We definitely shouldn't ever be here!
throw std::runtime_error("invalid state in |Gc|=1 case.");
}
// Step 3: Split up subcases using a full cover
// Construct I according to lemmas 7 and 8
Hypergraph::Edge I = Gc.vertices_with_degree_above_threshold(0.5);
// Per lemma 7, look for an edge that does not intersect I
Hypergraph::Edge missed_edge = find_missed_edge(Hc, I);
// If found, form a full cover
if (missed_edge.any()) {
Hypergraph C = l4_full_cover(Hc, missed_edge);
find_new_hses_fork(H, G, C, results);
return;
}
// Per lemma 8, look for a transversal that covers I
Hypergraph::Edge subtrans_edge = find_subset_edge(Gc, I);
// If found, form a full cover
if (subtrans_edge.any()) {
Hypergraph C = l5_full_cover(Hc, subtrans_edge);
find_new_hses_fork(H, G, C, results);
return;
}
// If we reach this point, I itself is a new HS
results.enqueue(I);
return;
}
