Skip to content

mis.pipeline.kernelization

[docs] module mis.pipeline.kernelization

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
import abc

import networkx as nx
from networkx.classes.reportviews import DegreeView
from mis.pipeline.preprocessor import BasePreprocessor
from mis.shared.graphs import is_independent


class BaseKernelization(BasePreprocessor, abc.ABC):
    """
    Shared base class for kernelization.
    """

    def __init__(self, graph: nx.Graph) -> None:
        # The latest version of the graph.
        # We rewrite it progressively to decrease the number of
        # nodes and edges.
        self.kernel: nx.Graph = graph.copy()
        self.initial_number_of_nodes = self.kernel.number_of_nodes()
        self.rule_application_sequence: list[BaseRebuilder] = []

        # An index used to generate new node numbers.
        self._new_node_gen_counter: int = 1
        if self.initial_number_of_nodes > 0:
            self._new_node_gen_counter = max(self.kernel.nodes()) + 1

        # Get rid of any node with a self-loop (a node that is its own
        # neighbour), as it cannot be part of a solution and we rely upon
        # their absence in rule applications.
        for node in list(self.kernel.nodes()):
            if self.kernel.has_edge(node, node):
                self.kernel.remove_node(node)

    @abc.abstractmethod
    def preprocess(self) -> nx.Graph:
        # Invariant: from this point, `self.kernel` does not contain any
        # self-loop.
        ...

    """
    Apply all the rules, in every possible order, until the graph cannot
    be reduced further.

    This method is left abstract as the list of rules may differ for
    various kinds of graphs (e.g. unweighted vs. weighted).
    """

    def rebuild(self, partial_solution: set[int]) -> set[int]:
        """
        Rebuild a MIS solution to the original graph from
        a partial MIS solution on the reduced graph obtained
        by kernelization.
        """
        partial_solution = set(partial_solution)
        for rule_app in reversed(self.rule_application_sequence):
            rule_app.rebuild(partial_solution)
        return partial_solution

    def is_independent(self, nodes: list[int]) -> bool:
        """
        Determine if a set of nodes represents an independent set
        within a given graph.

        Returns:
            True if the nodes in `nodes` represent an independent
                set within `graph`.
            False otherwise, i.e. if there's at least one connection
                between two nodes of `nodes`
        """
        return is_independent(self.kernel, nodes)

    def is_subclique(self, nodes: list[int]) -> bool:
        """
        Determine whether a list of nodes represents a clique
        within the graph, i.e. whether every pair of nodes is connected.
        """
        for i, u in enumerate(nodes):
            for v in nodes[i + 1 :]:
                if not self.kernel.has_edge(u, v):
                    return False
        return True

    def is_isolated(self, node: int) -> bool:
        """
        Determine whether a node is isolated, i.e. this node + its neighbours
        represent a clique.
        """
        closed_neighborhood: list[int] = list(self.kernel.neighbors(node))
        closed_neighborhood.append(node)
        if self.is_subclique(nodes=closed_neighborhood):
            return True
        return False

    def _add_node(self) -> int:
        """
        Add a new node with a unique index.
        """
        node = self._new_node_gen_counter
        self._new_node_gen_counter += 1
        self.kernel.add_node(node)
        return node


class Kernelization(BaseKernelization):
    """
    Apply well-known transformations to the graph to reduce its size without
    compromising the result.

    This algorithm is adapted from e.g.:
    https://schulzchristian.github.io/thesis/masterarbeit_demian_hespe.pdf

    Unless you are experimenting with your own preprocessors, you should
    probably use Kernelization in your pipeline.
    """

    def preprocess(self) -> nx.Graph:
        """
        Apply all rules, exhaustively, until the graph cannot be reduced
        further, storing the rules for rebuilding after the fact.
        """
        while (kernel_size_start := self.kernel.number_of_nodes()) > 0:
            self.search_rule_isolated_node_removal()
            self.search_rule_twin_reduction()
            self.search_rule_node_fold()
            self.search_rule_unconfined_and_diamond()
            kernel_size_end: int = self.kernel.number_of_nodes()
            if kernel_size_start - kernel_size_end == 0:
                # We didn't find any rule to apply, time to stop.
                break
        return self.kernel

    # -----------------isolated_node_removal---------------------------
    def apply_rule_isolated_node_removal(self, isolated: int) -> None:
        rule_app = RebuilderIsolatedNodeRemoval(isolated)
        self.rule_application_sequence.append(rule_app)
        neighborhood = list(self.kernel.neighbors(isolated))
        self.kernel.remove_nodes_from(neighborhood)
        self.kernel.remove_node(isolated)

    def search_rule_isolated_node_removal(self) -> None:
        """
        Remove any isolated node (see `is_isolated` for a definition).
        """
        for node in list(self.kernel.nodes()):
            # Since we're modifying `self.kernel` while iterating, we're
            # calling `list()` to make sure that we still have some kind
            # of valid iterator.
            if not self.kernel.has_node(node):
                # This might be possible if our iterator has not
                # been invalidated but our operation caused the node to
                # disappear from `self.kernel`.
                continue
            if self.is_isolated(node):
                self.apply_rule_isolated_node_removal(node)

    # -----------------unweighted_node_folding---------------------------

    def _fold_three(self, v: int, u: int, x: int, v_prime: int) -> None:
        """
        Fold three nodes V, U and X into a new single node V'.
        """
        neighbors_v_prime = set(self.kernel.neighbors(u)) | set(self.kernel.neighbors(x))
        for node in neighbors_v_prime:
            self.kernel.add_edge(v_prime, node)
        self.kernel.remove_nodes_from([v, u, x])

    def apply_rule_node_fold(self, v: int, u: int, x: int) -> None:
        v_prime = self._add_node()
        rule_app = RebuilderNodeFolding(v, u, x, v_prime)
        self.rule_application_sequence.append(rule_app)
        self._fold_three(v, u, x, v_prime)

    def search_rule_node_fold(self) -> None:
        """
        If a node V has exactly two neighbours U and X and there is no edge
        between U and X, fold U, V and X and into a single node.
        """
        if self.kernel.number_of_nodes() == 0:
            return
        assert isinstance(self.kernel.degree, DegreeView)
        for v in list(self.kernel.nodes()):
            # Since we're modifying `self.kernel` while iterating, we're
            # calling `list()` to make sure that we still have some kind
            # of valid iterator.
            if not self.kernel.has_node(v):
                # This might be possible if our iterator has not
                # been invalidated but our operation caused `v` to
                # disappear from `self.kernel`.
                continue
            if self.kernel.has_node(v):
                if self.kernel.degree(v) == 2:
                    [u, x] = self.kernel.neighbors(v)
                    if not self.kernel.has_edge(u, x):
                        self.apply_rule_node_fold(v, u, x)

    # -----------------unconfined reduction---------------------------
    def aux_search_confinement(
        self, neighbors_S: set[int], S: set[int]
    ) -> tuple[int, int, set[int]]:
        min: int = -1
        min_value: int = self.initial_number_of_nodes + 2
        min_set_diff: set[int] = set()
        for u in neighbors_S:
            neighbors_u: set[int] = set(self.kernel.neighbors(u))
            inter: set[int] = neighbors_u & S
            if len(inter) == 1:
                if len(neighbors_u - neighbors_S - S) < min_value:
                    min = u
                    min_set_diff = neighbors_u - neighbors_S - S
                    min_value = len(min_set_diff)
        return min, min_value, min_set_diff

    def apply_rule_unconfined(self, v: int) -> None:
        rule_app = RebuilderUnconfined()
        self.rule_application_sequence.append(rule_app)
        self.kernel.remove_node(v)

    def unconfined_loop(self, v: int, S: set[int], neighbors_S: set[int]) -> bool:
        min: int = 0
        min_value: int = 0
        min_set_diff: set[int] = set()
        min, min_value, min_set_diff = self.aux_search_confinement(neighbors_S, S)
        next_loop: bool = False
        # If there is no such node, then v is confined.
        if min == -1:
            pass
            """
            if self.find_diamond_reduction(neighbors_S, S):
                self.apply_rule_diamond(v)
            """
        # If N(u)\N[S] = ∅, then v is unconfined.
        if min_value == 0:
            self.apply_rule_unconfined(v)
        # If N (u)\ N [S] is a single node w,
        # then add w to S and repeat the algorithm.
        elif min_value == 1:
            w = list(min_set_diff)[0]
            S.add(w)
            neighbors_S |= set(self.kernel.neighbors(w))
            neighbors_S -= {w}
            next_loop = True
        # Otherwise, v is confined.
        else:
            pass
        return next_loop

    def search_rule_unconfined_and_diamond(self) -> None:
        if self.kernel.number_of_nodes() == 0:
            return
        for v in list(self.kernel.nodes()):
            # Since we're modifying `self.kernel` while iterating, we're
            # calling `list()` to make sure that we still have some kind
            # of valid iterator.
            if not self.kernel.has_node(v):
                continue
            # First, initialize S = {v}.
            S: set[int] = {v}
            neighbors_S: set[int] = set(self.kernel.neighbors(v))
            go_to_next_loop: bool = True
            while go_to_next_loop:
                # Then find u∈N(S) such that |N(u) ∩ S| = 1
                # and |N(u)\N[S]| is minimized
                go_to_next_loop = self.unconfined_loop(v, S, neighbors_S)

    # -----------------twin reduction---------------------------
    def fold_twin(self, u: int, v: int, v_prime: int, neighbors_u: list[int]) -> None:
        w_0: int = neighbors_u[0]
        w_1: int = neighbors_u[1]
        w_2: int = neighbors_u[2]
        neighbors_w_0 = set(self.kernel.neighbors(w_0))
        neighbors_w_1 = set(self.kernel.neighbors(w_1))
        neighbors_w_2 = set(self.kernel.neighbors(w_2))
        neighbors_v_prime = neighbors_w_0 | neighbors_w_1 | neighbors_w_2
        for node in neighbors_v_prime:
            self.kernel.add_edge(node, v_prime)
        self.kernel.remove_nodes_from([u, v, w_0, w_1, w_2])

    def find_twin(self, v: int) -> int | None:
        """
        Find a twin of a node, i.e. another node with the same
        neighbours.
        """
        neighbors_v: set[int] = set(self.kernel.neighbors(v))
        for u in list(self.kernel.nodes()):
            # Note: It might be sufficient to walk through
            # the neighbours of neighbours of neighbours of v.
            # Unclear whether this would be faster.
            if u == v:
                continue
            if not self.kernel.has_node(u):
                # FIXME: Can this happen?
                continue
            if not self.kernel.has_node(v):
                # FIXME: Can this happen?
                continue
            neighbors_u: set[int] = set(self.kernel.neighbors(u))
            if neighbors_u == neighbors_v:
                # Note: Since there are no self-loops, we can deduce
                # that U and V are also not neighbours.
                return int(u)
        return None

    def apply_rule_twin_independent(self, v: int, u: int, neighbors_u: list[int]) -> None:
        v_prime = self._add_node()
        rule_app = RebuilderTwinIndependent(
            v, u, neighbors_u[0], neighbors_u[1], neighbors_u[2], v_prime
        )
        self.rule_application_sequence.append(rule_app)
        self.fold_twin(u, v, v_prime, neighbors_u)

    def apply_rule_twin_has_dependency(self, v: int, u: int, neighbors_u: list[int]) -> None:
        rule_app = RebuilderTwinHasDependency(v, u)
        self.rule_application_sequence.append(rule_app)
        self.kernel.remove_nodes_from(neighbors_u)
        self.kernel.remove_nodes_from([u, v])

    def search_rule_twin_reduction(self) -> None:
        """
        If a node has exactly 3 neighbours and a twin (another
        node with the exact same neighbours), we can merge the
        5 nodes.
        """
        if self.kernel.number_of_nodes() == 0:
            return
        assert isinstance(self.kernel.degree, DegreeView)
        for v in list(self.kernel.nodes()):
            # Since we're modifying `self.kernel` while iterating, we're
            # calling `list()` to make sure that we still have some kind
            # of valid iterator.
            if not self.kernel.has_node(v):
                continue
            if self.kernel.degree(v) != 3:
                continue
            u: int | None = self.find_twin(v)
            if u is None:
                continue
            neighbors_u: list[int] = list(self.kernel.neighbors(u))
            if self.is_independent(neighbors_u):
                self.apply_rule_twin_independent(v, u, neighbors_u)
            else:
                self.apply_rule_twin_has_dependency(v, u, neighbors_u)


class BaseRebuilder(abc.ABC):
    """
    The pre-processing operations attempt to remove edges
    and/or vertices from the original graph. Therefore,
    when we build a MIS for these reduced graphs (the
    "partial solution"), we may end up with a solution
    that does not work for the original graph.

    Each rebuilder corresponds to one of the operations
    that previously reduced the size of the graph, and is
    charged with adapting the MIS solution to the greater graph.
    """

    @abc.abstractmethod
    def rebuild(self, partial_solution: set[int]) -> None: ...

    """
    Convert a solution `partial_solution` that is valid on a reduced
    graph to a solution that is valid on the graph prior to this
    reduction step.
    """


class RebuilderIsolatedNodeRemoval(BaseRebuilder):
    def __init__(self, isolated: int):
        self.isolated = isolated

    def rebuild(self, partial_solution: set[int]) -> None:
        partial_solution.add(self.isolated)


class RebuilderNodeFolding(BaseRebuilder):
    def __init__(self, v: int, u: int, x: int, v_prime: int):
        self.v = v
        self.u = u
        self.x = x
        self.v_prime = v_prime

    def rebuild(self, partial_solution: set[int]) -> None:
        if self.v_prime in partial_solution:
            partial_solution.add(self.u)
            partial_solution.add(self.x)
            partial_solution.remove(self.v_prime)
        else:
            partial_solution.add(self.v)


class RebuilderUnconfined(BaseRebuilder):
    def rebuild(self, partial_solution: set[int]) -> None:
        pass


class RebuilderTwinIndependent(BaseRebuilder):
    def __init__(self, v: int, u: int, w_0: int, w_1: int, w_2: int, v_prime: int):
        """
        Invariants:
         - U has exactly 3 neighbours W0, W1, W2;
         - V has exactly the same neighbours as U;
         - there is no self-loop around U or V (hence U and V are not
            neighbours);
         - there is no edge between W1, W2, W3;
         - V' is the node obtained by merging U, V, W1, W2, W3.
        """
        self.v: int = v
        self.u: int = u
        self.w_0: int = w_0
        self.w_1: int = w_1
        self.w_2: int = w_2
        self.v_prime: int = v_prime

    def rebuild(self, partial_solution: set[int]) -> None:
        if self.v_prime in partial_solution:
            # Since V' is part of the solution, none of its
            # neighbours is part of the solution. Consequently,
            # either U and V can be added to grow the solution
            # or W0, W1, W2 can be added to grow the solution,
            # without affecting the rest of the system.
            partial_solution.add(self.w_0)
            partial_solution.add(self.w_1)
            partial_solution.add(self.w_2)
            partial_solution.remove(self.v_prime)
        else:
            # The only neighbours of U and V are represented
            # by V'. Since V' is not part of the solution,
            # and since U and V are not neighbours, we can
            # always add U and V.
            partial_solution.add(self.u)
            partial_solution.add(self.v)


class RebuilderTwinHasDependency(BaseRebuilder):
    def __init__(self, v: int, u: int):
        """
        Invariants:
         - U has exactly 3 neighbours;
         - V has exactly the same neighbours as U;
         - there is no self-loop around U;
         - there is at least one connection between two neighbours of U.
        """
        self.v: int = v
        self.u: int = u

    def rebuild(self, partial_solution: set[int]) -> None:
        # Because of the invariants, U and V are always part of the solution.
        partial_solution.add(self.u)
        partial_solution.add(self.v)