# Ticket #12716: trac_12716_MILP.patch

File trac_12716_MILP.patch, 21.9 KB (added by dcoudert, 9 years ago)
• ## sage/graphs/graph_decompositions/vertex_separation.pyx

# HG changeset patch
# User dcoudert <david.coudert@inria.fr>
# Date 1333889402 -7200
# Node ID 70b3aa8dd7faf486e7370c2e517e74a9928c8b54
diff --git a/sage/graphs/graph_decompositions/vertex_separation.pyx b/sage/graphs/graph_decompositions/vertex_separation.pyx
 a r""" Vertex separation This module implements several algorithms to compute the vertex separation of a digraph and the corresponding ordering of the vertices. Given an ordering v_1, ..., v_n of the vertices of V(G), its *cost* is defined as: This module implements several algorithms to compute the vertex separation of a digraph and the corresponding ordering of the vertices. It also implements tests functions for evaluation the width of a linear orderings. Given an ordering v_1,\cdots, v_n of the vertices of V(G), its *cost* is defined as: .. MATH:: c'(S) = |N^+_G(S)\backslash S| The *vertex separation* of a digraph G is equal to the minimum cost of an ordering of its vertices. The *vertex separation* of a digraph G is equal to the minimum cost of an ordering of its vertices. **Vertex separation and pathwidth** G a digraph D with the same vertex set, and in which each edge uv of G is replaced by two edges uv and vu in D. The vertex separation of D is equal to the pathwidth of G, and the corresponding ordering of the vertices of D encodes an optimal path-decomposition of G. D, also called a *layout*, encodes an optimal path-decomposition of G. This is a result of Kinnersley [Kin92]_ and Bodlaender [Bod98]_. **References** .. [Kin92] *The vertex separation number of a graph equals its path-width*, Nancy G. Kinnersley, Information Processing Letters, Volume 42, Issue 6, Pages 345-350, 24 July 1992 **This module contains the following methods** .. [Bod98] *A partial k-arboretum of graphs with bounded treewidth*, Hans L. Bodlaender, Theoretical Computer Science, Volume 209, Issues 1-2, Pages 1-45, 6 December 1998 ===================================  =========================================== **Authors** :meth:path_decomposition           Returns the pathwidth of the given graph and the ordering of the vertices resulting in a corresponding path decomposition - Nathann Cohen :meth:vertex_separation            Returns an optimal ordering of the vertices and its cost for vertex-separation :meth:vertex_separation_MILP       Computes the vertex separation of G and the optimal ordering of its vertices using an MILP formulation :meth:lower_bound                  Returns a lower bound on the vertex separation of G :meth:is_valid_ordering            Test if the linear vertex ordering L is valid for (di)graph G :meth:width_of_path_decomposition  Returns the width of the path decomposition induced by the linear ordering L of the vertices of G ===================================  =========================================== Exponential algorithm for vertex separation ------------------------------------------- \max c'(\{v_1\}),c'(\{v_1,v_2\}),...,c'(\{v_1,...,v_n\})\geq\max c'_1,...,c'_n where c_i is the minimum cost of a set S on i vertices. Evaluating the c_i can take time (and in particular more than the previous exact algorithm), but it does not need much memory to run. where c_i is the minimum cost of a set S on i vertices. Evaluating the c_i can take time (and in particular more than the previous exact algorithm), but it does not need much memory to run. Methods MILP formulation for the vertex separation ------------------------------------------ We describe bellow a mixed integer linear program (MILP) for determining an optimal layout for the vertex separation of G. This MILP is an improved version of the formulation proposed in [SP10]_. VARIABLES: - x_v^t -- Variable set to 1 if either v has an in-neighbor u such that u\in S_{t'}=\{v_1, v_2,\cdots, v_{t'}\} and v\in N_G^+(S_{t'})\backslash S_{t'} with t'\leq t, or v\in S_t. It is set to 0 otherwise. - y_v^t -- Variable set to 1 if v\in S_t, and 0 otherwise. The order of v in the layout is the smallest t such that y_v^t==1. - u_v^t -- Variable set to 1 if v\in N_G^+(S_t)\backslash S_t has an in-neighbor in S_t, and set to 0 otherwise. - z -- Objective value to minimize. It is equal to the maximum over all step t of the number of vertices such that u_v^t==1. MILP formulation: .. MATH:: :nowrap: \begin{alignat}{2} \intertext{Minimize:} &z&\\ \intertext{Such that:} x_v^t &\leq x_v^{t+1}& \forall v\in V,\ 0\leq t\leq n-2\\ y_v^t &\leq y_v^{t+1}& \forall v\in V,\ 0\leq t\leq n-2\\ y_v^t &\leq x_w^t& \forall v\in V,\ \forall w\in N^+(v),\ 0\leq t\leq n-1\\ \sum_{v \in V} y_v^0 &\leq 1&\\ \sum_{v \in V} y_v^{t+1} &\leq 1+\sum_{v \in V} y_v^t& 0\leq t\leq n-2\\ \sum_{v \in V} y_v^{N-1} &= n&\\ x_v^t-y_v^t&\leq u_v^t & \forall v \in V,\ 0\leq t\leq n-1\\ \sum_{v \in V} u_v^t &\leq z& 0\leq t\leq n-1\\ 0 \leq x_v^t &\leq 1& \forall v\in V,\ 0\leq t\leq n-1\\ 0 \leq u_v^t &\leq 1& \forall v\in V,\ 0\leq t\leq n-1\\ y_v^t &\in \{0,1\}& \forall v\in V,\ 0\leq t\leq n-1\\ 0 \leq z &\leq n& \end{alignat} The vertex separation of G is given by the value of z, and the order of vertex v in the optimal layout is given by the smallest t for which y_v^t==1. REFERENCES ---------- .. [Bod98] *A partial k-arboretum of graphs with bounded treewidth*, Hans L. Bodlaender, Theoretical Computer Science 209(1-2):1-45, 1998. .. [Kin92] *The vertex separation number of a graph equals its path-width*, Nancy G. Kinnersley, Information Processing Letters 42(6):345-350, 1992. .. [SP10] *Lightpath Reconfiguration in WDM networks*, Fernando Solano and Michal Pioro, IEEE/OSA Journal of Optical Communication and Networking 2(12):1010-1021, 2010. AUTHORS ------- - Nathann Cohen (2011-10): Initial version and exact exponential algorithm - David Coudert (2012-04): MILP formulation and tests functions METHODS ------- """ documentation). .. NOTE:: This method runs in exponential time but has no memory constraint. return min #################### # Exact algorithms # #################### ################################ # Exact exponential algorithms # ################################ def path_decomposition(G, verbose = False): def path_decomposition(G, algorithm = "exponential", verbose = False): r""" Returns the pathwidth of the given graph and the ordering of the vertices resulting in a corresponding path decomposition. - G -- a digraph - algorithm -- (default: "exponential") Specify the algorithm to use among - exponential -- Use an exponential time and space algorithm. This algorithm only works of graphs on less than 32 vertices. - MILP -- Use a mixed integer linear programming formulation. This algorithm has no size restriction but could take a very long time. - verbose (boolean) -- whether to display information on the computations. .. NOTE:: Because of its current implementation, this algorithm only works on graphs on less than 32 vertices. This can be changed to 54 if necessary, but 32 vertices already require 4GB of memory. Because of its current implementation, this exponential algorithm only works on graphs on less than 32 vertices. This can be changed to 54 if necessary, but 32 vertices already require 4GB of memory. EXAMPLE: The vertex separation of a circuit is equal to 2:: The vertex separation of a cycle is equal to 2:: sage: from sage.graphs.graph_decompositions.vertex_separation import path_decomposition sage: g = graphs.CycleGraph(6) sage: path_decomposition(g) (2, [0, 1, 2, 3, 4, 5]) sage: pw, L = path_decomposition(g); pw 2 sage: pwm, Lm = path_decomposition(g, algorithm = "MILP"); pwm 2 TEST: from sage.graphs.graph import Graph if not isinstance(G, Graph): raise ValueError("The parameter must be a Graph.") from sage.graphs.digraph import DiGraph return vertex_separation(DiGraph(G), verbose = verbose) if algorithm == "exponential": return vertex_separation(DiGraph(G), verbose = verbose) else: return vertex_separation_MILP(DiGraph(G), verbosity = (1 if verbose else 0)) def vertex_separation(G, verbose = False): r""" EXAMPLE: The vertex separation of a circuit is equal to 2:: The vertex separation of a circuit is equal to 1:: sage: from sage.graphs.graph_decompositions.vertex_separation import vertex_separation sage: g = digraphs.Circuit(6) sage: vertex_separation(g) (1, [0, 1, 2, 3, 4, 5]) (1, [0, 1, 2, 3, 4, 5]) TEST: Graphs with non-integer vertices:: sage: from sage.graphs.graph_decompositions.vertex_separation import vertex_separation sage: D=digraphs.DeBruijn(2,3) sage: vertex_separation(D) (2, ['000', '001', '100', '010', '101', '011', '110', '111']) return a else: return b ################################################################# # Function for testing the validity of a linear vertex ordering # ################################################################# def is_valid_ordering(G, L): r""" Test if the linear vertex ordering L is valid for (di)graph G. A linear ordering L of the vertices of a (di)graph G is valid if all vertices of G are in L, and if L contains no other vertex and no duplicated vertices. INPUT: - G -- a Graph or a DiGraph. - L -- an ordered list of the vertices of G. OUTPUT: Returns True if L is a valid vertex ordering for G, and False oterwise. EXAMPLE: Path decomposition of a cycle:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: G = graphs.CycleGraph(6) sage: L = [u for u in G.vertices()] sage: vertex_separation.is_valid_ordering(G, L) True sage: vertex_separation.is_valid_ordering(G, [1,2]) False TEST: Giving anything else than a Graph or a DiGraph:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: vertex_separation.is_valid_ordering(2, []) Traceback (most recent call last): ... ValueError: The input parameter must be a Graph or a DiGraph. Giving anything else than a list:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: G = graphs.CycleGraph(6) sage: vertex_separation.is_valid_ordering(G, {}) Traceback (most recent call last): ... ValueError: The second parameter must be of type 'list'. """ from sage.graphs.graph import Graph from sage.graphs.digraph import DiGraph if not isinstance(G, Graph) and not isinstance(G, DiGraph): raise ValueError("The input parameter must be a Graph or a DiGraph.") if not isinstance(L, list): raise ValueError("The second parameter must be of type 'list'.") return len( set( G.vertices() ).symmetric_difference( set(L) ) ) == 0 #################################################################### # Measurement functions of the widths of some graph decompositions # #################################################################### def width_of_path_decomposition(G, L): r""" Returns the width of the path decomposition induced by the linear ordering L of the vertices of G. If G is an instance of Graph, this function returns the width pw_L(G) of the path decomposition induced by the linear ordering L of the vertices of G. If G is a DiGraph, it returns instead the width vs_L(G) of the directed path decomposition induced by the linear ordering L of the vertices of G, where .. MATH:: vs_L(G) & =  \max_{0\leq i< |V|-1} | N^+(L[:i])\setminus L[:i] |\\ pw_L(G) & =  \max_{0\leq i< |V|-1} | N(L[:i])\setminus L[:i] |\\ INPUT: - G -- a Graph or a DiGraph - L -- a linear ordering of the vertices of G EXAMPLES: Path decomposition of a cycle:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: G = graphs.CycleGraph(6) sage: L = [u for u in G.vertices()] sage: vertex_separation.width_of_path_decomposition(G, L) 2 Directed path decomposition of a circuit:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: G = digraphs.Circuit(6) sage: L = [u for u in G.vertices()] sage: vertex_separation.width_of_path_decomposition(G, L) 1 TESTS: Path decomposition of a BalancedTree:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: G = graphs.BalancedTree(3,2) sage: pw, L = vertex_separation.path_decomposition(G) sage: pw == vertex_separation.width_of_path_decomposition(G, L) True sage: L.reverse() sage: pw == vertex_separation.width_of_path_decomposition(G, L) False Directed path decomposition of a circuit:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: G = digraphs.Circuit(8) sage: vs, L = vertex_separation.vertex_separation(G) sage: vs == vertex_separation.width_of_path_decomposition(G, L) True sage: L = [0,4,6,3,1,5,2,7] sage: vs == vertex_separation.width_of_path_decomposition(G, L) False Giving a wrong linear ordering:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: G = Graph() sage: vertex_separation.width_of_path_decomposition(G, ['a','b']) Traceback (most recent call last): ... ValueError: The input linear vertex ordering L is not valid for G. """ if not is_valid_ordering(G, L): raise ValueError("The input linear vertex ordering L is not valid for G.") vsL = 0 S = [] neighbors_of_S_in_V_minus_S = [] for u in L: # We remove u from the list of neighbors of S if u in neighbors_of_S_in_V_minus_S: neighbors_of_S_in_V_minus_S.remove(u) # We add vertex u to the set S S.append(u) if G._directed: Nu = G.neighbors_out(u) else: Nu = G.neighbors(u) # We add the (out-)neighbors of u to the list of neighbors of S for v in Nu: if (not v in S) and (not v in neighbors_of_S_in_V_minus_S): neighbors_of_S_in_V_minus_S.append(v) # We update the cost of the vertex separation vsL = max( vsL, len(neighbors_of_S_in_V_minus_S) ) return vsL ########################################## # MILP formulation for vertex separation # ########################################## def vertex_separation_MILP(G, integrality = False, solver = None, verbosity = 0): r""" Computes the vertex separation of G and the optimal ordering of its vertices using an MILP formulation. This function uses a mixed integer linear program (MILP) for determining an optimal layout for the vertex separation of G. This MILP is an improved version of the formulation proposed in [SP10]_. See the module's documentation for more details on this MILP formulation. INPUTS: - G -- a DiGraph - integrality -- (default: False) Specify if all variables must be integral of if some variables can be relaxed. - solver -- (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:solve of the class :class:MixedIntegerLinearProgram. - verbose -- integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet. OUTPUT: A pair (cost, ordering) representing the optimal ordering of the vertices and its cost. EXAMPLE: Vertex separation of a De Bruijn digraph:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: G = digraphs.DeBruijn(2,3) sage: vs, L = vertex_separation.vertex_separation_MILP(G); vs 2 sage: vs == vertex_separation.width_of_path_decomposition(G, L) True sage: vse, Le = vertex_separation.vertex_separation(G); vse 2 The vertex separation of a circuit is 1:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: G = digraphs.Circuit(6) sage: vs, L = vertex_separation.vertex_separation_MILP(G); vs 1 TESTS: Comparison with exponential algorithm:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: for i in range(20): ...       G = digraphs.RandomDirectedGNP(10,0.2) ...       ve,le = vertex_separation.vertex_separation(G) ...       vm,lm = vertex_separation.vertex_separation_MILP(G) ...       if ve != vm: ...          print "The solution is not optimal!" Giving anything else than a DiGraph:: sage: from sage.graphs.graph_decompositions import vertex_separation sage: vertex_separation.vertex_separation_MILP([]) Traceback (most recent call last): ... ValueError: The first input parameter must be a DiGraph. """ from sage.graphs.digraph import DiGraph if not isinstance(G, DiGraph): raise ValueError("The first input parameter must be a DiGraph.") from sage.numerical.mip import MixedIntegerLinearProgram, Sum, MIPSolverException p = MixedIntegerLinearProgram( maximization = False, solver = solver ) # Declaration of variables. x = p.new_variable( integer = integrality, dim = 2 ) u = p.new_variable( integer = integrality, dim = 2 ) y = p.new_variable( integer = True, dim = 2 ) z = p.new_variable( integer = True, dim = 1 ) N = G.num_verts() V = G.vertices() #  (2) x[v][t] <= x[v][t+1]   for all v in V, and for t:=0..N-2 #  (3) y[v][t] <= y[v][t+1]   for all v in V, and for t:=0..N-2 for v in V: for t in xrange(N-1): p.add_constraint( x[v][t] - x[v][t+1], max = 0 ) p.add_constraint( y[v][t] - y[v][t+1], max = 0 ) #  (4) y[v][t] <= x[w][t]  for all v in V, for all w in N^+(v), and for all t:=0..N-1 for v in V: for w in G.neighbors_out(v): for t in xrange(N): p.add_constraint( y[v][t] - x[w][t], max = 0 ) #  (5) sum_{v in V} y[v][0] <= 1 p.add_constraint( Sum([ y[v][0] for v in V ]), max = 1 ) #  (6) sum_{v in V} y[v][t+1] <= sum_{v in V} y[v][t] + 1  for t:=0..N-2 for t in xrange(N-1): p.add_constraint( Sum([ y[v][t+1] - y[v][t] for v in V ]), max = 1 ) #  (7) sum_{v in V} y[v][N-1] = N p.add_constraint( Sum([ y[v][N-1] for v in V ]), min = N ) p.add_constraint( Sum([ y[v][N-1] for v in V ]), max = N ) #  (8) u[v][t] >= x[v][t]-y[v][t]    for all v in V, and for all t:=0..N-1 for v in V: for t in xrange(N): p.add_constraint( x[v][t] - y[v][t] - u[v][t], max = 0 ) #  (9) z >= sum_{v in V} u[v][t]   for all t:=0..N-1 for t in xrange(N): p.add_constraint( Sum([ u[v][t] for v in V ]) - z['z'], max = 0 ) # (10) 0 <= x[v][t] and u[v][t] <= 1 # (11) y[v][t] in {0,1} for v in V: for t in xrange(N): p.add_constraint( x[v][t], min = 0 ) p.add_constraint( x[v][t], max = 1 ) p.add_constraint( u[v][t], min = 0 ) p.add_constraint( u[v][t], max = 1 ) p.set_binary( y[v][t] ) # (12) 0 <= z <= |V| p.add_constraint( z['z'], min = 0 ) p.add_constraint( z['z'], max = N ) #  (1) Minimize z p.set_objective( z['z'] ) try: obj = p.solve( log=verbosity ) taby = p.get_values( y ) tabz = p.get_values( z ) # since exactly one vertex is processed per step, we can reconstruct the sequence seq = [] for t in xrange(N): for v in V: if (taby[v][t] > 0) and (not v in seq): seq.append(v) break vs = int(round( tabz['z'] )) except MIPSolverException: if integrality: raise ValueError("Unbounded or unexpected error") else: raise ValueError("Unbounded or unexpected error. Try with 'integrality = True'.") del p return vs, seq