Michele Conforti - Academia.edu (original) (raw)
Papers by Michele Conforti
Mathematical Programming, 1990
In this paper we show that, if G is a Berge graph such that neither G nor its complement G contai... more In this paper we show that, if G is a Berge graph such that neither G nor its complement G contains certain induced subgraphs, named proper wheels and long prisms, then either G is a basic perfect graph (a bipartite graph, a line graph of a bipartite graph or the complement of such graphs) or it has a skew partition that cannot occur in a minimally imperfect graph. This structural result implies that G is perfect.
Journal of Combinatorial Theory, 2003
We prove that the strong perfect graph conjecture holds for graphs that do not contain parachutes... more We prove that the strong perfect graph conjecture holds for graphs that do not contain parachutes or proper wheels. This is done by showing the following theorem:
Discrete Mathematics, 2006
Journal of Graph Theory, 1999
It is an old problem in graph theory to test whether a graph contains a chordless cycle of length... more It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β-perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even-signable. Graphs that can be signed so that every triangle is odd and every hole is odd are called odd-signable. We derive from a theorem due to Truemper co-NP characterizations of even-signable and odd-290 JOURNAL OF GRAPH THEORY signable graphs. A graph is strongly even-signable if it can be signed so that every cycle of length ≥ 4 with at most one chord is even and every triangle is odd. Clearly a strongly even-signable graph is even-signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd-signable. Every strongly oddsignable graph is odd-signable. We give co-NP characterizations for both strongly even-signable and strongly odd-signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (capfree graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well-studied subclass of cap-free graphs. If a graph is strongly even-signable or strongly odd-signable, then it is cap-free. In fact, strongly even-signable graphs are those cap-free graphs that are even-signable. From our decomposition theorem, we derive decomposition results for strongly odd-signable and strongly even-signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes.
Mathematics of Operations Research, 1987
Mathematical Programming, 1990
Theequipartition problem is defined as follows: given a graphG = (V, E) and edge weightsc e , par... more Theequipartition problem is defined as follows: given a graphG = (V, E) and edge weightsc e , partition the setV into two sets of ⌈1/2|V|⌉ and ⌊1/2|V|⌋ nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized. Anequicut is a feasible solution of the above problem and theequicut polytope Q(G) is the convex hull of the incidence vectors of equicuts inG. In this paper we describe some facet inducing inequalities of this polytope.
Mathematical Programming, 1995
A 0, ±1-matrixA is balanced if, in every submatrix with two nonzero entries per row and column, t... more A 0, ±1-matrixA is balanced if, in every submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of four. This definition was introduced by Truemper (1978) and generalizes the notion of a balanced 0, 1-matrix introduced by Berge (1970). In this paper, we extend a bicoloring theorem of Berge (1970) and total dual integrality results of Fulkerson, Hoffman and Oppenheim (1974) to balanced 0, ±1-matrices.
Mathematical Programming, 1990
The following basic clustering problem arises in different domains, ranging from physics, statist... more The following basic clustering problem arises in different domains, ranging from physics, statistics and Boolean function minimization. Given a graphG = (V, E) and edge weightsc e , partition the setV into two sets of ⌈1/2|V|⌉ and ⌊1/2|V|⌋ nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized. Anequicut is a feasible solution of the above problem and theequicut polytope Q(G) is the convex hull of the incidence vectors of equicuts inG. In this paper we give some integer programming formulations of the equicut problem, study the dimension of the equicut polytope and describe some basic classes of facet-inducing inequalities forQ(G).
Journal of Graph Theory, 1999
It is an old problem in graph theory to test whether a graph contains a chordless cycle of length... more It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β-perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even-signable. Graphs that can be signed so that every triangle is odd and every hole is odd are called odd-signable. We derive from a theorem due to Truemper co-NP characterizations of even-signable and odd-290 JOURNAL OF GRAPH THEORY signable graphs. A graph is strongly even-signable if it can be signed so that every cycle of length ≥ 4 with at most one chord is even and every triangle is odd. Clearly a strongly even-signable graph is even-signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd-signable. Every strongly oddsignable graph is odd-signable. We give co-NP characterizations for both strongly even-signable and strongly odd-signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (capfree graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well-studied subclass of cap-free graphs. If a graph is strongly even-signable or strongly odd-signable, then it is cap-free. In fact, strongly even-signable graphs are those cap-free graphs that are even-signable. From our decomposition theorem, we derive decomposition results for strongly odd-signable and strongly even-signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes.
Journal of The ACM, 1995
... inference can be solved efficiently for Horn clauses, clauses with at most two literals and s... more ... inference can be solved efficiently for Horn clauses, clauses with at most two literals and several related classes [Chandru and Hooker 1991; Truemper 1990]. ... Q'(A) = {x~@'*:Ax> 1 n(A), A'x= 1 n(A'), O<x S 1) R(A) = {( X,,S) GLZ+m : Ax+s21-n(A), osx,.ss l}. ...
Journal of Combinatorial Theory, 1999
A0# 1 matrix is balanced if it does not contain a square submatrix of odd order withtwo ones per ... more A0# 1 matrix is balanced if it does not contain a square submatrix of odd order withtwo ones per row and per column. Weshow that a balanced 0,1 matrix is either totallyunimodular or its bipartite representation has a cutset consisting of two adjacentnodesand some of their neighbors. This result yields a polytime recognition algorithm forbalancedness. To prove the result, we
Mathematical Programming, 1991
Theequipartition problem is defined as follows: given a graphG = (V, E) and edge weightsce, parti... more Theequipartition problem is defined as follows: given a graphG = (V, E) and edge weightsce, partition the setV into two sets of ?1/2|V|? and ?1/2|V|? nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized.
Mathematical Programming, 1993
A (0, 1) matrix is linear if it does not contain a 2×2 submatrix of all ones. In this paper we gi... more A (0, 1) matrix is linear if it does not contain a 2×2 submatrix of all ones. In this paper we give polynomial algorithms to test whether a linear matrix is balanced or perfect. The algorithms are based on decomposition results previously obtained by the authors.
Mathematical Programming, 1987
A (0, ±1) matrix A is restricted unimodular if every matrix obtained from A by setting to zero an... more A (0, ±1) matrix A is restricted unimodular if every matrix obtained from A by setting to zero any subset of its entries is totally unimodular. Restricted unimodular matrices are also known as matrices without odd cycles. They have been studied by Commoner and recently Yannakakis has given a polynomial algorithm to recognize when a matrix belongs to this class. A matrix A is strongly unimodular if any matrix obtained from A by setting at most one of its entries to zero is totally unimodular. Crama et al. have shown that (0,1) matrix A is strongly unimodular if and only if any basis of (A, 1) is triangular, whereI is an identity matrix of suitable dimensions. In this paper we give a very simple algorithm to test whether a matrix is restricted unimodular and we show that all strongly unimodular matrices can be obtained by composing restricted unimodular matrices with a simple operation.
Mathematical Programming, 1991
Polyak's subgradient algorithm for nondifferentiable optimization problem... more Polyak's subgradient algorithm for nondifferentiable optimization problems requires prior knowledge of the optimal value of the objective function to find an optimal solution. In this paper we extend the convergence properties of the Polyak's subgradient algorithm with a fixed target value to a more general case with variable target values. Then a target value updating scheme is provided which finds
Mathematical Programming, 1992
In this paper we define wheel matrices and characterize some properties of matrices that are perf... more In this paper we define wheel matrices and characterize some properties of matrices that are perfect but not balanced. A consequence of our results is that if a matrixA is perfect then we can construct a polynomial number of submatricesA I,⋯,A n ofA such thatA is balanced if and only if all the submatricesA 1,⋯,A n ofA are perfect. This implies that if the problem of testing perfection is polynomially solvable, then so is the problem of testing balancedness.
Mathematical Programming, 1992
Claude Berge defines a (0, 1) matrix A to be linear ifA does not contain a 2 × 2 submatrix of all... more Claude Berge defines a (0, 1) matrix A to be linear ifA does not contain a 2 × 2 submatrix of all ones.A(0, 1) matrixA is balanced ifA does not contain a square submatrix of odd order with two ones per row and column. The contraction of a rowi of a matrix consists of the removal of rowi and all the columns that have a 1 in the entry corresponding to rowi. In this paper we show that if a linear balanced matrixA does not belong to a subclass of totally unimodular matrices, thenA orA T contains a rowi such that the submatrix obtained by contracting rowi has a block-diagonal structure.
Mathematical Programming, 1990
In this paper we show that, if G is a Berge graph such that neither G nor its complement G contai... more In this paper we show that, if G is a Berge graph such that neither G nor its complement G contains certain induced subgraphs, named proper wheels and long prisms, then either G is a basic perfect graph (a bipartite graph, a line graph of a bipartite graph or the complement of such graphs) or it has a skew partition that cannot occur in a minimally imperfect graph. This structural result implies that G is perfect.
Journal of Combinatorial Theory, 2003
We prove that the strong perfect graph conjecture holds for graphs that do not contain parachutes... more We prove that the strong perfect graph conjecture holds for graphs that do not contain parachutes or proper wheels. This is done by showing the following theorem:
Discrete Mathematics, 2006
Journal of Graph Theory, 1999
It is an old problem in graph theory to test whether a graph contains a chordless cycle of length... more It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β-perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even-signable. Graphs that can be signed so that every triangle is odd and every hole is odd are called odd-signable. We derive from a theorem due to Truemper co-NP characterizations of even-signable and odd-290 JOURNAL OF GRAPH THEORY signable graphs. A graph is strongly even-signable if it can be signed so that every cycle of length ≥ 4 with at most one chord is even and every triangle is odd. Clearly a strongly even-signable graph is even-signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd-signable. Every strongly oddsignable graph is odd-signable. We give co-NP characterizations for both strongly even-signable and strongly odd-signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (capfree graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well-studied subclass of cap-free graphs. If a graph is strongly even-signable or strongly odd-signable, then it is cap-free. In fact, strongly even-signable graphs are those cap-free graphs that are even-signable. From our decomposition theorem, we derive decomposition results for strongly odd-signable and strongly even-signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes.
Mathematics of Operations Research, 1987
Mathematical Programming, 1990
Theequipartition problem is defined as follows: given a graphG = (V, E) and edge weightsc e , par... more Theequipartition problem is defined as follows: given a graphG = (V, E) and edge weightsc e , partition the setV into two sets of ⌈1/2|V|⌉ and ⌊1/2|V|⌋ nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized. Anequicut is a feasible solution of the above problem and theequicut polytope Q(G) is the convex hull of the incidence vectors of equicuts inG. In this paper we describe some facet inducing inequalities of this polytope.
Mathematical Programming, 1995
A 0, ±1-matrixA is balanced if, in every submatrix with two nonzero entries per row and column, t... more A 0, ±1-matrixA is balanced if, in every submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of four. This definition was introduced by Truemper (1978) and generalizes the notion of a balanced 0, 1-matrix introduced by Berge (1970). In this paper, we extend a bicoloring theorem of Berge (1970) and total dual integrality results of Fulkerson, Hoffman and Oppenheim (1974) to balanced 0, ±1-matrices.
Mathematical Programming, 1990
The following basic clustering problem arises in different domains, ranging from physics, statist... more The following basic clustering problem arises in different domains, ranging from physics, statistics and Boolean function minimization. Given a graphG = (V, E) and edge weightsc e , partition the setV into two sets of ⌈1/2|V|⌉ and ⌊1/2|V|⌋ nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized. Anequicut is a feasible solution of the above problem and theequicut polytope Q(G) is the convex hull of the incidence vectors of equicuts inG. In this paper we give some integer programming formulations of the equicut problem, study the dimension of the equicut polytope and describe some basic classes of facet-inducing inequalities forQ(G).
Journal of Graph Theory, 1999
It is an old problem in graph theory to test whether a graph contains a chordless cycle of length... more It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β-perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even-signable. Graphs that can be signed so that every triangle is odd and every hole is odd are called odd-signable. We derive from a theorem due to Truemper co-NP characterizations of even-signable and odd-290 JOURNAL OF GRAPH THEORY signable graphs. A graph is strongly even-signable if it can be signed so that every cycle of length ≥ 4 with at most one chord is even and every triangle is odd. Clearly a strongly even-signable graph is even-signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd-signable. Every strongly oddsignable graph is odd-signable. We give co-NP characterizations for both strongly even-signable and strongly odd-signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (capfree graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well-studied subclass of cap-free graphs. If a graph is strongly even-signable or strongly odd-signable, then it is cap-free. In fact, strongly even-signable graphs are those cap-free graphs that are even-signable. From our decomposition theorem, we derive decomposition results for strongly odd-signable and strongly even-signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes.
Journal of The ACM, 1995
... inference can be solved efficiently for Horn clauses, clauses with at most two literals and s... more ... inference can be solved efficiently for Horn clauses, clauses with at most two literals and several related classes [Chandru and Hooker 1991; Truemper 1990]. ... Q'(A) = {x~@'*:Ax> 1 n(A), A'x= 1 n(A'), O<x S 1) R(A) = {( X,,S) GLZ+m : Ax+s21-n(A), osx,.ss l}. ...
Journal of Combinatorial Theory, 1999
A0# 1 matrix is balanced if it does not contain a square submatrix of odd order withtwo ones per ... more A0# 1 matrix is balanced if it does not contain a square submatrix of odd order withtwo ones per row and per column. Weshow that a balanced 0,1 matrix is either totallyunimodular or its bipartite representation has a cutset consisting of two adjacentnodesand some of their neighbors. This result yields a polytime recognition algorithm forbalancedness. To prove the result, we
Mathematical Programming, 1991
Theequipartition problem is defined as follows: given a graphG = (V, E) and edge weightsce, parti... more Theequipartition problem is defined as follows: given a graphG = (V, E) and edge weightsce, partition the setV into two sets of ?1/2|V|? and ?1/2|V|? nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized.
Mathematical Programming, 1993
A (0, 1) matrix is linear if it does not contain a 2×2 submatrix of all ones. In this paper we gi... more A (0, 1) matrix is linear if it does not contain a 2×2 submatrix of all ones. In this paper we give polynomial algorithms to test whether a linear matrix is balanced or perfect. The algorithms are based on decomposition results previously obtained by the authors.
Mathematical Programming, 1987
A (0, ±1) matrix A is restricted unimodular if every matrix obtained from A by setting to zero an... more A (0, ±1) matrix A is restricted unimodular if every matrix obtained from A by setting to zero any subset of its entries is totally unimodular. Restricted unimodular matrices are also known as matrices without odd cycles. They have been studied by Commoner and recently Yannakakis has given a polynomial algorithm to recognize when a matrix belongs to this class. A matrix A is strongly unimodular if any matrix obtained from A by setting at most one of its entries to zero is totally unimodular. Crama et al. have shown that (0,1) matrix A is strongly unimodular if and only if any basis of (A, 1) is triangular, whereI is an identity matrix of suitable dimensions. In this paper we give a very simple algorithm to test whether a matrix is restricted unimodular and we show that all strongly unimodular matrices can be obtained by composing restricted unimodular matrices with a simple operation.
Mathematical Programming, 1991
Polyak's subgradient algorithm for nondifferentiable optimization problem... more Polyak's subgradient algorithm for nondifferentiable optimization problems requires prior knowledge of the optimal value of the objective function to find an optimal solution. In this paper we extend the convergence properties of the Polyak's subgradient algorithm with a fixed target value to a more general case with variable target values. Then a target value updating scheme is provided which finds
Mathematical Programming, 1992
In this paper we define wheel matrices and characterize some properties of matrices that are perf... more In this paper we define wheel matrices and characterize some properties of matrices that are perfect but not balanced. A consequence of our results is that if a matrixA is perfect then we can construct a polynomial number of submatricesA I,⋯,A n ofA such thatA is balanced if and only if all the submatricesA 1,⋯,A n ofA are perfect. This implies that if the problem of testing perfection is polynomially solvable, then so is the problem of testing balancedness.
Mathematical Programming, 1992
Claude Berge defines a (0, 1) matrix A to be linear ifA does not contain a 2 × 2 submatrix of all... more Claude Berge defines a (0, 1) matrix A to be linear ifA does not contain a 2 × 2 submatrix of all ones.A(0, 1) matrixA is balanced ifA does not contain a square submatrix of odd order with two ones per row and column. The contraction of a rowi of a matrix consists of the removal of rowi and all the columns that have a 1 in the entry corresponding to rowi. In this paper we show that if a linear balanced matrixA does not belong to a subclass of totally unimodular matrices, thenA orA T contains a rowi such that the submatrix obtained by contracting rowi has a block-diagonal structure.