int igraph_i_maximum_bipartite_matching_unweighted(const igraph_t* graph,
const igraph_vector_bool_t* types, igraph_integer_t* matching_size,
igraph_vector_long_t* matching) {
long int i, j, k, n, no_of_nodes = igraph_vcount(graph);
long int num_matched; /* number of matched vertex pairs */
igraph_vector_long_t match; /* will store the matching */
igraph_vector_t labels; /* will store the labels */
igraph_vector_t neis; /* used to retrieve the neighbors of a node */
igraph_dqueue_long_t q; /* a FIFO for push ordering */
igraph_bool_t smaller_set; /* denotes which part of the bipartite graph is smaller */
long int label_changed = 0; /* Counter to decide when to run a global relabeling */
long int relabeling_freq = no_of_nodes / 2;
/* We will use:
* - FIFO push ordering
* - global relabeling frequency: n/2 steps where n is the number of nodes
* - simple greedy matching for initialization
*/
/* (1) Initialize data structures */
IGRAPH_CHECK(igraph_vector_long_init(&match, no_of_nodes));
IGRAPH_FINALLY(igraph_vector_long_destroy, &match);
IGRAPH_VECTOR_INIT_FINALLY(&labels, no_of_nodes);
IGRAPH_VECTOR_INIT_FINALLY(&neis, 0);
IGRAPH_CHECK(igraph_dqueue_long_init(&q, 0));
IGRAPH_FINALLY(igraph_dqueue_long_destroy, &q);
/* (2) Initially, every node is unmatched */
igraph_vector_long_fill(&match, -1);
/* (3) Find an initial matching in a greedy manner.
* At the same time, find which side of the graph is smaller. */
num_matched = 0; j = 0;
for (i = 0; i < no_of_nodes; i++) {
if (VECTOR(*types)[i])
j++;
if (MATCHED(i))
continue;
IGRAPH_CHECK(igraph_neighbors(graph, &neis, (igraph_integer_t) i,
IGRAPH_ALL));
n = igraph_vector_size(&neis);
for (j = 0; j < n; j++) {
k = (long int) VECTOR(neis)[j];
if (UNMATCHED(k)) {
/* We match vertex i to vertex VECTOR(neis)[j] */
VECTOR(match)[k] = i;
VECTOR(match)[i] = k;
num_matched++;
break;
}
}
}
smaller_set = (j <= no_of_nodes/2);
/* (4) Set the initial labeling -- lines 1 and 2 in the tech report */
IGRAPH_CHECK(igraph_i_maximum_bipartite_matching_unweighted_relabel(
graph, types, &labels, &match, smaller_set));
/* (5) Fill the push queue with the unmatched nodes from the smaller set. */
for (i = 0; i < no_of_nodes; i++) {
if (UNMATCHED(i) && VECTOR(*types)[i] == smaller_set)
IGRAPH_CHECK(igraph_dqueue_long_push(&q, i));
}
/* (6) Main loop from the referenced tech report -- lines 4--13 */
label_changed = 0;
while (!igraph_dqueue_long_empty(&q)) {
long int v = igraph_dqueue_long_pop(&q); /* Line 13 */
long int u = -1, label_u = 2 * no_of_nodes;
long int w;
if (label_changed >= relabeling_freq) {
/* Run global relabeling */
IGRAPH_CHECK(igraph_i_maximum_bipartite_matching_unweighted_relabel(
graph, types, &labels, &match, smaller_set));
label_changed = 0;
}
debug("Considering vertex %ld\n", v);
/* Line 5: find row u among the neighbors of v s.t. label(u) is minimal */
IGRAPH_CHECK(igraph_neighbors(graph, &neis, (igraph_integer_t) v,
IGRAPH_ALL));
n = igraph_vector_size(&neis);
for (i = 0; i < n; i++) {
if (VECTOR(labels)[(long int)VECTOR(neis)[i]] < label_u) {
u = (long int) VECTOR(neis)[i];
label_u = (long int) VECTOR(labels)[u];
label_changed++;
}
}
debug(" Neighbor with smallest label: %ld (label=%ld)\n", u, label_u);
if (label_u < no_of_nodes) { /* Line 6 */
VECTOR(labels)[v] = VECTOR(labels)[u] + 1; /* Line 7 */
if (MATCHED(u)) { /* Line 8 */
w = VECTOR(match)[u];
debug(" Vertex %ld is matched to %ld, performing a double push\n", u, w);
if (w != v) {
VECTOR(match)[u] = -1; VECTOR(match)[w] = -1; /* Line 9 */
IGRAPH_CHECK(igraph_dqueue_long_push(&q, w)); /* Line 10 */
debug(" Unmatching & activating vertex %ld\n", w);
num_matched--;
}
}
VECTOR(match)[u] = v; VECTOR(match)[v] = u; /* Line 11 */
num_matched++;
VECTOR(labels)[u] += 2; /* Line 12 */
label_changed++;
}
printf("MATCH: ");
igraph_vector_long_print(&match);
printf("LABELS ");
igraph_vector_print(&labels);
}
/* Fill the output parameters */
if (matching != 0) {
IGRAPH_CHECK(igraph_vector_long_update(matching, &match));
}
if (matching_size != 0) {
*matching_size = (igraph_integer_t) num_matched;
}
/* Release everything */
igraph_dqueue_long_destroy(&q);
igraph_vector_destroy(&neis);
igraph_vector_destroy(&labels);
igraph_vector_long_destroy(&match);
IGRAPH_FINALLY_CLEAN(4);
return IGRAPH_SUCCESS;
}
/**
* Finding maximum bipartite matchings on bipartite graphs using the
* Hungarian algorithm (a.k.a. Kuhn-Munkres algorithm).
*
* The algorithm uses a maximum cardinality matching on a subset of
* tight edges as a starting point. This is achieved by
* \c igraph_i_maximum_bipartite_matching_unweighted on the restricted
* graph.
*
* The algorithm works reliably only if the weights are integers. The
* \c eps parameter should specity a very small number; if the slack on
* an edge falls below \c eps, it will be considered tight. If all your
* weights are integers, you can safely set \c eps to zero.
*/
int igraph_i_maximum_bipartite_matching_weighted(const igraph_t* graph,
const igraph_vector_bool_t* types, igraph_integer_t* matching_size,
igraph_real_t* matching_weight, igraph_vector_long_t* matching,
const igraph_vector_t* weights, igraph_real_t eps) {
long int i, j, k, n, no_of_nodes, no_of_edges;
igraph_integer_t u, v, w, msize;
igraph_t newgraph;
igraph_vector_long_t match; /* will store the matching */
igraph_vector_t slack; /* will store the slack on each edge */
igraph_vector_t parent; /* parent vertices during a BFS */
igraph_vector_t vec1, vec2; /* general temporary vectors */
igraph_vector_t labels; /* will store the labels */
igraph_dqueue_long_t q; /* a FIFO for BST */
igraph_bool_t smaller_set; /* denotes which part of the bipartite graph is smaller */
long int smaller_set_size; /* size of the smaller set */
igraph_real_t dual; /* solution of the dual problem */
igraph_adjlist_t tight_phantom_edges; /* adjacency list to manage tight phantom edges */
igraph_integer_t alternating_path_endpoint;
igraph_vector_t* neis;
igraph_vector_int_t *neis2;
igraph_inclist_t inclist; /* incidence list of the original graph */
/* The Hungarian algorithm is originally for complete bipartite graphs.
* For non-complete bipartite graphs, a phantom edge of weight zero must be
* added between every pair of non-connected vertices. We don't do this
* explicitly of course. See the comments below about how phantom edges
* are taken into account. */
no_of_nodes = igraph_vcount(graph);
no_of_edges = igraph_ecount(graph);
if (eps < 0) {
IGRAPH_WARNING("negative epsilon given, clamping to zero");
eps = 0;
}
/* (1) Initialize data structures */
IGRAPH_CHECK(igraph_vector_long_init(&match, no_of_nodes));
IGRAPH_FINALLY(igraph_vector_long_destroy, &match);
IGRAPH_CHECK(igraph_vector_init(&slack, no_of_edges));
IGRAPH_FINALLY(igraph_vector_destroy, &slack);
IGRAPH_VECTOR_INIT_FINALLY(&vec1, 0);
IGRAPH_VECTOR_INIT_FINALLY(&vec2, 0);
IGRAPH_VECTOR_INIT_FINALLY(&labels, no_of_nodes);
IGRAPH_CHECK(igraph_dqueue_long_init(&q, 0));
IGRAPH_FINALLY(igraph_dqueue_long_destroy, &q);
IGRAPH_VECTOR_INIT_FINALLY(&parent, no_of_nodes);
IGRAPH_CHECK(igraph_adjlist_init_empty(&tight_phantom_edges,
(igraph_integer_t) no_of_nodes));
IGRAPH_FINALLY(igraph_adjlist_destroy, &tight_phantom_edges);
IGRAPH_CHECK(igraph_inclist_init(graph, &inclist, IGRAPH_ALL));
IGRAPH_FINALLY(igraph_inclist_destroy, &inclist);
/* (2) Find which set is the smaller one */
j = 0;
for (i = 0; i < no_of_nodes; i++) {
if (VECTOR(*types)[i] == 0)
j++;
}
smaller_set = (j > no_of_nodes / 2);
smaller_set_size = smaller_set ? (no_of_nodes - j) : j;
/* (3) Calculate the initial labeling and the set of tight edges. Use the
* smaller set only. Here we can assume that there are no phantom edges
* among the tight ones. */
dual = 0;
for (i = 0; i < no_of_nodes; i++) {
igraph_real_t max_weight = 0;
if (VECTOR(*types)[i] != smaller_set) {
VECTOR(labels)[i] = 0;
continue;
}
neis = igraph_inclist_get(&inclist, i);
n = igraph_vector_size(neis);
for (j = 0, k = 0; j < n; j++) {
if (VECTOR(*weights)[(long int)VECTOR(*neis)[j]] > max_weight) {
k = (long int) VECTOR(*neis)[j];
max_weight = VECTOR(*weights)[k];
}
}
VECTOR(labels)[i] = max_weight;
dual += max_weight;
}
igraph_vector_clear(&vec1);
IGRAPH_CHECK(igraph_get_edgelist(graph, &vec2, 0));
#define IS_TIGHT(i) (VECTOR(slack)[i] <= eps)
for (i = 0, j = 0; i < no_of_edges; i++, j+=2) {
u = (igraph_integer_t) VECTOR(vec2)[j];
v = (igraph_integer_t) VECTOR(vec2)[j+1];
VECTOR(slack)[i] = VECTOR(labels)[u] + VECTOR(labels)[v] - VECTOR(*weights)[i];
if (IS_TIGHT(i)) {
IGRAPH_CHECK(igraph_vector_push_back(&vec1, u));
IGRAPH_CHECK(igraph_vector_push_back(&vec1, v));
}
}
igraph_vector_clear(&vec2);
/* (4) Construct a temporary graph on which the initial maximum matching
* will be calculated (only on the subset of tight edges) */
IGRAPH_CHECK(igraph_create(&newgraph, &vec1,
(igraph_integer_t) no_of_nodes, 0));
IGRAPH_FINALLY(igraph_destroy, &newgraph);
IGRAPH_CHECK(igraph_maximum_bipartite_matching(&newgraph, types, &msize, 0, &match, 0, 0));
igraph_destroy(&newgraph);
IGRAPH_FINALLY_CLEAN(1);
/* (5) Main loop until the matching becomes maximal */
while (msize < smaller_set_size) {
igraph_real_t min_slack, min_slack_2;
igraph_integer_t min_slack_u, min_slack_v;
/* (7) Fill the push queue with the unmatched nodes from the smaller set. */
igraph_vector_clear(&vec1);
igraph_vector_clear(&vec2);
igraph_vector_fill(&parent, -1);
for (i = 0; i < no_of_nodes; i++) {
if (UNMATCHED(i) && VECTOR(*types)[i] == smaller_set) {
IGRAPH_CHECK(igraph_dqueue_long_push(&q, i));
VECTOR(parent)[i] = i;
IGRAPH_CHECK(igraph_vector_push_back(&vec1, i));
}
}
#ifdef MATCHING_DEBUG
debug("Matching:");
igraph_vector_long_print(&match);
debug("Unmatched vertices are marked by non-negative numbers:\n");
igraph_vector_print(&parent);
debug("Labeling:");
igraph_vector_print(&labels);
debug("Slacks:");
igraph_vector_print(&slack);
#endif
/* (8) Run the BFS */
alternating_path_endpoint = -1;
while (!igraph_dqueue_long_empty(&q)) {
v = (int) igraph_dqueue_long_pop(&q);
debug("Considering vertex %ld\n", (long int)v);
/* v is always in the smaller set. Find the neighbors of v, which
* are all in the larger set. Find the pairs of these nodes in
* the smaller set and push them to the queue. Mark the traversed
* nodes as seen.
*
* Here we have to be careful as there are two types of incident
* edges on v: real edges and phantom ones. Real edges are
* given by igraph_inclist_get. Phantom edges are not given so we
* (ab)use an adjacency list data structure that lists the
* vertices connected to v by phantom edges only. */
neis = igraph_inclist_get(&inclist, v);
n = igraph_vector_size(neis);
for (i = 0; i < n; i++) {
j = (long int) VECTOR(*neis)[i];
/* We only care about tight edges */
if (!IS_TIGHT(j))
continue;
/* Have we seen the other endpoint already? */
u = IGRAPH_OTHER(graph, j, v);
if (VECTOR(parent)[u] >= 0)
continue;
debug(" Reached vertex %ld via edge %ld\n", (long)u, (long)j);
VECTOR(parent)[u] = v;
IGRAPH_CHECK(igraph_vector_push_back(&vec2, u));
w = (int) VECTOR(match)[u];
if (w == -1) {
/* u is unmatched and it is in the larger set. Therefore, we
* could improve the matching by following the parents back
* from u to the root.
*/
alternating_path_endpoint = u;
break; /* since we don't need any more endpoints that come from v */
} else {
IGRAPH_CHECK(igraph_dqueue_long_push(&q, w));
VECTOR(parent)[w] = u;
}
IGRAPH_CHECK(igraph_vector_push_back(&vec1, w));
}
/* Now do the same with the phantom edges */
neis2 = igraph_adjlist_get(&tight_phantom_edges, v);
n = igraph_vector_int_size(neis2);
for (i = 0; i < n; i++) {
u = (igraph_integer_t) VECTOR(*neis2)[i];
/* Have we seen u already? */
if (VECTOR(parent)[u] >= 0)
continue;
/* Check if the edge is really tight; it might have happened that the
* edge became non-tight in the meanwhile. We do not remove these from
* tight_phantom_edges at the moment, so we check them once again here.
*/
if (fabs(VECTOR(labels)[(long int)v] + VECTOR(labels)[(long int)u]) > eps)
continue;
debug(" Reached vertex %ld via tight phantom edge\n", (long)u);
VECTOR(parent)[u] = v;
IGRAPH_CHECK(igraph_vector_push_back(&vec2, u));
w = (int) VECTOR(match)[u];
if (w == -1) {
/* u is unmatched and it is in the larger set. Therefore, we
* could improve the matching by following the parents back
* from u to the root.
*/
alternating_path_endpoint = u;
break; /* since we don't need any more endpoints that come from v */
} else {
IGRAPH_CHECK(igraph_dqueue_long_push(&q, w));
VECTOR(parent)[w] = u;
}
IGRAPH_CHECK(igraph_vector_push_back(&vec1, w));
}
}
/* Okay; did we have an alternating path? */
if (alternating_path_endpoint != -1) {
#ifdef MATCHING_DEBUG
debug("BFS parent tree:");
igraph_vector_print(&parent);
#endif
/* Increase the size of the matching with the alternating path. */
v = alternating_path_endpoint;
u = (igraph_integer_t) VECTOR(parent)[v];
debug("Extending matching with alternating path ending in %ld.\n", (long int)v);
while (u != v) {
w = (int) VECTOR(match)[v];
if (w != -1)
VECTOR(match)[w] = -1;
VECTOR(match)[v] = u;
VECTOR(match)[v] = u;
w = (int) VECTOR(match)[u];
if (w != -1)
VECTOR(match)[w] = -1;
VECTOR(match)[u] = v;
v = (igraph_integer_t) VECTOR(parent)[u];
u = (igraph_integer_t) VECTOR(parent)[v];
}
msize++;
#ifdef MATCHING_DEBUG
debug("New matching after update:");
igraph_vector_long_print(&match);
debug("Matching size is now: %ld\n", (long)msize);
#endif
continue;
}
#ifdef MATCHING_DEBUG
debug("Vertices reachable from unmatched ones via tight edges:\n");
igraph_vector_print(&vec1);
igraph_vector_print(&vec2);
#endif
/* At this point, vec1 contains the nodes in the smaller set (A)
* reachable from unmatched nodes in A via tight edges only, while vec2
* contains the nodes in the larger set (B) reachable from unmatched
* nodes in A via tight edges only. Also, parent[i] >= 0 if node i
* is reachable */
/* Check the edges between reachable nodes in A and unreachable
* nodes in B, and find the minimum slack on them.
*
* Since the weights are positive, we do no harm if we first
* assume that there are no "real" edges between the two sets
* mentioned above and determine an upper bound for min_slack
* based on this. */
min_slack = IGRAPH_INFINITY;
min_slack_u = min_slack_v = 0;
n = igraph_vector_size(&vec1);
for (i = 0; i < no_of_nodes; i++) {
if (VECTOR(*types)[i] == smaller_set)
continue;
if (VECTOR(labels)[i] < min_slack) {
min_slack = VECTOR(labels)[i];
min_slack_v = (igraph_integer_t) i;
}
}
min_slack_2 = IGRAPH_INFINITY;
for (i = 0; i < n; i++) {
u = (igraph_integer_t) VECTOR(vec1)[i];
/* u is surely from the smaller set, but we are interested in it
* only if it is reachable from an unmatched vertex */
if (VECTOR(parent)[u] < 0)
continue;
if (VECTOR(labels)[u] < min_slack_2) {
min_slack_2 = VECTOR(labels)[u];
min_slack_u = u;
}
}
min_slack += min_slack_2;
debug("Starting approximation for min_slack = %.4f (based on vertex pair %ld--%ld)\n",
min_slack, (long int)min_slack_u, (long int)min_slack_v);
n = igraph_vector_size(&vec1);
for (i = 0; i < n; i++) {
u = (igraph_integer_t) VECTOR(vec1)[i];
/* u is a reachable node in A; get its incident edges.
*
* There are two types of incident edges: 1) real edges,
* 2) phantom edges. Phantom edges were treated earlier
* when we determined the initial value for min_slack. */
debug("Trying to expand along vertex %ld\n", (long int)u);
neis = igraph_inclist_get(&inclist, u);
k = igraph_vector_size(neis);
for (j = 0; j < k; j++) {
/* v is the vertex sitting at the other end of an edge incident
* on u; check whether it was reached */
v = IGRAPH_OTHER(graph, VECTOR(*neis)[j], u);
debug(" Edge %ld -- %ld (ID=%ld)\n", (long int)u, (long int)v, (long int)VECTOR(*neis)[j]);
if (VECTOR(parent)[v] >= 0) {
/* v was reached, so we are not interested in it */
debug(" %ld was reached, so we are not interested in it\n", (long int)v);
continue;
}
/* v is the ID of the edge from now on */
v = (igraph_integer_t) VECTOR(*neis)[j];
if (VECTOR(slack)[v] < min_slack) {
min_slack = VECTOR(slack)[v];
min_slack_u = u;
min_slack_v = IGRAPH_OTHER(graph, v, u);
}
debug(" Slack of this edge: %.4f, min slack is now: %.4f\n",
VECTOR(slack)[v], min_slack);
}
}
debug("Minimum slack: %.4f on edge %d--%d\n", min_slack, (int)min_slack_u, (int)min_slack_v);
if (min_slack > 0) {
/* Decrease the label of reachable nodes in A by min_slack.
* Also update the dual solution */
n = igraph_vector_size(&vec1);
for (i = 0; i < n; i++) {
u = (igraph_integer_t) VECTOR(vec1)[i];
VECTOR(labels)[u] -= min_slack;
neis = igraph_inclist_get(&inclist, u);
k = igraph_vector_size(neis);
for (j = 0; j < k; j++) {
debug(" Decreasing slack of edge %ld (%ld--%ld) by %.4f\n",
(long)VECTOR(*neis)[j], (long)u,
(long)IGRAPH_OTHER(graph, VECTOR(*neis)[j], u), min_slack);
VECTOR(slack)[(long int)VECTOR(*neis)[j]] -= min_slack;
}
dual -= min_slack;
}
/* Increase the label of reachable nodes in B by min_slack.
* Also update the dual solution */
n = igraph_vector_size(&vec2);
for (i = 0; i < n; i++) {
u = (igraph_integer_t) VECTOR(vec2)[i];
VECTOR(labels)[u] += min_slack;
neis = igraph_inclist_get(&inclist, u);
k = igraph_vector_size(neis);
for (j = 0; j < k; j++) {
debug(" Increasing slack of edge %ld (%ld--%ld) by %.4f\n",
(long)VECTOR(*neis)[j], (long)u,
(long)IGRAPH_OTHER(graph, (long)VECTOR(*neis)[j], u), min_slack);
VECTOR(slack)[(long int)VECTOR(*neis)[j]] += min_slack;
}
dual += min_slack;
}
}
/* Update the set of tight phantom edges.
* Note that we must do it even if min_slack is zero; the reason is that
* it can happen that min_slack is zero in the first step if there are
* isolated nodes in the input graph.
*
* TODO: this is O(n^2) here. Can we do it faster? */
for (u = 0; u < no_of_nodes; u++) {
if (VECTOR(*types)[u] != smaller_set)
continue;
for (v = 0; v < no_of_nodes; v++) {
if (VECTOR(*types)[v] == smaller_set)
continue;
if (VECTOR(labels)[(long int)u] + VECTOR(labels)[(long int)v] <= eps) {
/* Tight phantom edge found. Note that we don't have to check whether
* u and v are connected; if they were, then the slack of this edge
* would be negative. */
neis2 = igraph_adjlist_get(&tight_phantom_edges, u);
if (!igraph_vector_int_binsearch(neis2, v, &i)) {
debug("New tight phantom edge: %ld -- %ld\n", (long)u, (long)v);
IGRAPH_CHECK(igraph_vector_int_insert(neis2, i, v));
}
}
}
}
#ifdef MATCHING_DEBUG
debug("New labels:");
igraph_vector_print(&labels);
debug("Slacks after updating with min_slack:");
igraph_vector_print(&slack);
#endif
}
/* Cleanup: remove phantom edges from the matching */
for (i = 0; i < no_of_nodes; i++) {
if (VECTOR(*types)[i] != smaller_set)
continue;
if (VECTOR(match)[i] != -1) {
j = VECTOR(match)[i];
neis2 = igraph_adjlist_get(&tight_phantom_edges, i);
if (igraph_vector_int_binsearch(neis2, j, 0)) {
VECTOR(match)[i] = VECTOR(match)[j] = -1;
msize--;
}
}
}
/* Fill the output parameters */
if (matching != 0) {
IGRAPH_CHECK(igraph_vector_long_update(matching, &match));
}
if (matching_size != 0) {
*matching_size = msize;
}
if (matching_weight != 0) {
*matching_weight = 0;
for (i = 0; i < no_of_edges; i++) {
if (IS_TIGHT(i)) {
IGRAPH_CHECK(igraph_edge(graph, (igraph_integer_t) i, &u, &v));
if (VECTOR(match)[u] == v)
*matching_weight += VECTOR(*weights)[i];
}
}
}
/* Release everything */
#undef IS_TIGHT
igraph_inclist_destroy(&inclist);
igraph_adjlist_destroy(&tight_phantom_edges);
igraph_vector_destroy(&parent);
igraph_dqueue_long_destroy(&q);
igraph_vector_destroy(&labels);
igraph_vector_destroy(&vec1);
igraph_vector_destroy(&vec2);
igraph_vector_destroy(&slack);
igraph_vector_long_destroy(&match);
IGRAPH_FINALLY_CLEAN(9);
return IGRAPH_SUCCESS;
}