Let G = (V , E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of th... more Let G = (V , E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the characteristic vectors of all spanning trees of G. In this paper, we describe all facets of P(G) as a consequence of the facets of the bases polytope P(M) of a matroid M, i.e., the convex hull of the characteristic vectors of all bases of M.
Let G=(V, E) be an undirected graph. The subtours elimination polytope P(G) is the set of x∈R^E s... more Let G=(V, E) be an undirected graph. The subtours elimination polytope P(G) is the set of x∈R^E such that: 0≤ x(e)≤ 1 for any edge e∈ E, x(δ (v))=2 for any vertex v∈ V, and x(δ (U))≥ 2 for any nonempty and proper subset U of V. P(G) is a relaxation of the Traveling Salesman Polytope, i.e., the convex hull of the Hamiton circuits of G. Maurras Maurras 1975 and Grötschel and Padberg Grotschel and Padberg 1979b characterize the facets of P(G) when G is a complete graph. In this paper we generalize their result by giving a minimal description of P(G) in the general case and by presenting a short proof of it.
Let G=(V, E) be an undirected graph. The subtours elimination polytope P(G) is the set of x∈R^E s... more Let G=(V, E) be an undirected graph. The subtours elimination polytope P(G) is the set of x∈R^E such that: 0≤ x(e)≤ 1 for any edge e∈ E, x(δ (v))=2 for any vertex v∈ V, and x(δ (U))≥ 2 for any nonempty and proper subset U of V. P(G) is a relaxation of the Traveling Salesman Polytope, i.e., the convex hull of the Hamiton circuits of G. Maurras Maurras 1975 and Grötschel and Padberg Grotschel and Padberg 1979b characterize the facets of P(G) when G is a complete graph. In this paper we generalize their result by giving a minimal description of P(G) in the general case and by presenting a short proof of it.
Let E be a finite set and P, S, L three classes of subsets of E, and r a function defined on 2^E.... more Let E be a finite set and P, S, L three classes of subsets of E, and r a function defined on 2^E. In this paper, we give an algorithm for testing if the quadruple ( P, S, L, r) is the locked structure of a given matroid, i.e., recognizing if ( P, S, L, r) defines a matroid. This problem is intractable. Our algorithm improves the running time complexity of a previous algorithm due to Spinrad. We deduce a polynomial time algorithm for recognizing large classes of matroids called polynomially locked matroids and uniform matroids.
Let E be a finite set and P, S, L three classes of subsets of E, and r a function defined on 2^E.... more Let E be a finite set and P, S, L three classes of subsets of E, and r a function defined on 2^E. In this paper, we give an algorithm for testing if the quadruple ( P, S, L, r) is the locked structure of a given matroid, i.e., recognizing if ( P, S, L, r) defines a matroid. This problem is intractable. Our algorithm improves the running time complexity of a previous algorithm due to Spinrad. We deduce a polynomial time algorithm for recognizing large classes of matroids called polynomially locked matroids and uniform matroids.
Given a graph G=(V, E), a connected sides cut (U, V U) or δ (U) is the set of edges of E linking ... more Given a graph G=(V, E), a connected sides cut (U, V U) or δ (U) is the set of edges of E linking all vertices of U to all vertices of V U such that the induced subgraphs G[U] and G[V U] are connected. Given a positive weight function w defined on E, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut Ω such that w(Ω) is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.
Given a graph G=(V, E), a connected sides cut (U, V U) or δ (U) is the set of edges of E linking ... more Given a graph G=(V, E), a connected sides cut (U, V U) or δ (U) is the set of edges of E linking all vertices of U to all vertices of V U such that the induced subgraphs G[U] and G[V U] are connected. Given a positive weight function w defined on E, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut Ω such that w(Ω) is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.
Let M be a matroid defined on a finite set E and L⊂ E. L is locked in M if M|L and M^*|(E\\ L) ar... more Let M be a matroid defined on a finite set E and L⊂ E. L is locked in M if M|L and M^*|(E\\ L) are 2-connected, and min{r(L), r^*(E\\ L)}≥ 2. Locked subsets characterize nontrivial facets of the bases polytope. In this paper, we give a new axiom system for matroids based on locked subsets. We deduce an algorithm for recognizing matroids improving the running time complexity of the best known till today. This algorithm induces a polynomial time algorithm for recognizing uniform matroids. This latter problem is intractable if we use an independence oracle.
Let M be a matroid defined on a finite set E and L⊂ E. L is locked in M if M|L and M^*|(E\ L) are... more Let M be a matroid defined on a finite set E and L⊂ E. L is locked in M if M|L and M^*|(E\ L) are 2-connected, and min{r(L), r^*(E\ L)}≥ 2. Locked subsets characterize nontrivial facets of the bases polytope. In this paper, we give a new axiom system for matroids based on locked subsets. We deduce an algorithm for recognizing matroids improving the running time complexity of the best known till today. This algorithm induces a polynomial time algorithm for recognizing uniform matroids. This latter problem is intractable if we use an independence oracle.
Let G=(V, E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the a... more Let G=(V, E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the all spanning trees of G. In this paper, we describe all facets of P(G) as a consequence of the facets of the bases polytope of a matroid.
Let G=(V, E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the a... more Let G=(V, E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the all spanning trees of G. In this paper, we describe all facets of P(G) as a consequence of the facets of the bases polytope of a matroid.
networks, Quality-of-Service routing, virtual sub-networks, series parallel graphs, packing of sp... more networks, Quality-of-Service routing, virtual sub-networks, series parallel graphs, packing of spanning
networks, Quality-of-Service routing, virtual sub-networks, series parallel graphs, packing of sp... more networks, Quality-of-Service routing, virtual sub-networks, series parallel graphs, packing of spanning
Given a graph G=V,E, a connected sides cut U,V\U or δU is the set of edges of E linking all verti... more Given a graph G=V,E, a connected sides cut U,V\U or δU is the set of edges of E linking all vertices of U to all vertices of V\U such that the induced subgraphs GU and GV\U are connected. Given a positive weight function w defined on E, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut Ω such that wΩ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.
This paper gives a characterization of uniform matroids by means of locked subsets. Locked subset... more This paper gives a characterization of uniform matroids by means of locked subsets. Locked subsets are 2-connected subsets, their complements are 2-connected in the dual, and the minimum rank of both is 2. Locked subsets give the nontrivial facets of the bases polytope.
Given an undirected graph $G = (V, E)$, and a vertex $r\in V$, an $r$-acyclic orientation of $G$ ... more Given an undirected graph $G = (V, E)$, and a vertex $r\in V$, an $r$-acyclic orientation of $G$ is an orientation $OE$ of the edges of $G$ such that the digraph $OG = (V, OE)$ is acyclic and $r$ is the unique vertex with indegree equal to 0. For $w\in \mathbb{R}^E_+$, $k(G, w)$ is the value of the $w$-maximum packing of $r$-arborescences for all $r\in V$ and all $r$-acyclic orientations $OE$ of $G$. In this case, the Broadcast Routing (in Computers Networks) Problem (BRP) is to compute $k(G, w)$, by finding an optimal $r$ and an optimal $r$-acyclic orientation. BRP is a mathematical formulation of multipath broadcast routing in computer networks. In this paper, we provide a polynomial time algorithm to solve BRP in outerplanar graphs. Outerplanar graphs are encountered in many applications such as computational geometry, robotics, etc.
In this paper we present a max flow multipath routing algorithm (MFMP) that is designed to reduce... more In this paper we present a max flow multipath routing algorithm (MFMP) that is designed to reduce latency, provide high throughput and balance traffic load. The max flow multipath algorithm is based on a Ford-Fulkerson algorithm .It consists of determining a set of disjoints path that are loop free with maximum flow, then splitting network traffic among those paths on a round robin fashion. Through simulation we show that our algorithm performs well than a multi shortest path (MSP) and a single shortest path (SSP).
Soient n un entier non negatif et H un ensemble fini de vecteurs a n composantes entieres. Le con... more Soient n un entier non negatif et H un ensemble fini de vecteurs a n composantes entieres. Le cone engendre par H, note cone(H), est l'ensemble des vecteurs qui sont combinaisons lineaires non negatives des vecteurs de H. Le treillis engendre par H, note lat(H), est l'ensemble des combinaisons lineaires entieres des vecteurs de H. On dit que H est un systeme generateur de Hilbert si tout vecteur qui appartient a cone(H) et a lat(H) peut s'exprimer comme combinaison lineaire de vecteurs de H avec des coefficients entiers et non negatifs. Si H est minimal pour la propriete precedente, le meme cone et le meme treillis, alors on dit que H est une base de Hilbert. Un exemple important de base de Hilbert, etudie dans cette these, est l'ensemble B(M) des (vecteurs caracteristiques des) bases d'un matroide M. Cook, Fonlupt et Schrijver ont montre que, si H est une base de Hilbert, tout vecteur entier qui est dans cone(H) et dans lat(H) est combinaison lineaire de 2n-1 ve...
Let G = (V , E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of th... more Let G = (V , E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the characteristic vectors of all spanning trees of G. In this paper, we describe all facets of P(G) as a consequence of the facets of the bases polytope P(M) of a matroid M, i.e., the convex hull of the characteristic vectors of all bases of M.
Let G=(V, E) be an undirected graph. The subtours elimination polytope P(G) is the set of x∈R^E s... more Let G=(V, E) be an undirected graph. The subtours elimination polytope P(G) is the set of x∈R^E such that: 0≤ x(e)≤ 1 for any edge e∈ E, x(δ (v))=2 for any vertex v∈ V, and x(δ (U))≥ 2 for any nonempty and proper subset U of V. P(G) is a relaxation of the Traveling Salesman Polytope, i.e., the convex hull of the Hamiton circuits of G. Maurras Maurras 1975 and Grötschel and Padberg Grotschel and Padberg 1979b characterize the facets of P(G) when G is a complete graph. In this paper we generalize their result by giving a minimal description of P(G) in the general case and by presenting a short proof of it.
Let G=(V, E) be an undirected graph. The subtours elimination polytope P(G) is the set of x∈R^E s... more Let G=(V, E) be an undirected graph. The subtours elimination polytope P(G) is the set of x∈R^E such that: 0≤ x(e)≤ 1 for any edge e∈ E, x(δ (v))=2 for any vertex v∈ V, and x(δ (U))≥ 2 for any nonempty and proper subset U of V. P(G) is a relaxation of the Traveling Salesman Polytope, i.e., the convex hull of the Hamiton circuits of G. Maurras Maurras 1975 and Grötschel and Padberg Grotschel and Padberg 1979b characterize the facets of P(G) when G is a complete graph. In this paper we generalize their result by giving a minimal description of P(G) in the general case and by presenting a short proof of it.
Let E be a finite set and P, S, L three classes of subsets of E, and r a function defined on 2^E.... more Let E be a finite set and P, S, L three classes of subsets of E, and r a function defined on 2^E. In this paper, we give an algorithm for testing if the quadruple ( P, S, L, r) is the locked structure of a given matroid, i.e., recognizing if ( P, S, L, r) defines a matroid. This problem is intractable. Our algorithm improves the running time complexity of a previous algorithm due to Spinrad. We deduce a polynomial time algorithm for recognizing large classes of matroids called polynomially locked matroids and uniform matroids.
Let E be a finite set and P, S, L three classes of subsets of E, and r a function defined on 2^E.... more Let E be a finite set and P, S, L three classes of subsets of E, and r a function defined on 2^E. In this paper, we give an algorithm for testing if the quadruple ( P, S, L, r) is the locked structure of a given matroid, i.e., recognizing if ( P, S, L, r) defines a matroid. This problem is intractable. Our algorithm improves the running time complexity of a previous algorithm due to Spinrad. We deduce a polynomial time algorithm for recognizing large classes of matroids called polynomially locked matroids and uniform matroids.
Given a graph G=(V, E), a connected sides cut (U, V U) or δ (U) is the set of edges of E linking ... more Given a graph G=(V, E), a connected sides cut (U, V U) or δ (U) is the set of edges of E linking all vertices of U to all vertices of V U such that the induced subgraphs G[U] and G[V U] are connected. Given a positive weight function w defined on E, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut Ω such that w(Ω) is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.
Given a graph G=(V, E), a connected sides cut (U, V U) or δ (U) is the set of edges of E linking ... more Given a graph G=(V, E), a connected sides cut (U, V U) or δ (U) is the set of edges of E linking all vertices of U to all vertices of V U such that the induced subgraphs G[U] and G[V U] are connected. Given a positive weight function w defined on E, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut Ω such that w(Ω) is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.
Let M be a matroid defined on a finite set E and L⊂ E. L is locked in M if M|L and M^*|(E\\ L) ar... more Let M be a matroid defined on a finite set E and L⊂ E. L is locked in M if M|L and M^*|(E\\ L) are 2-connected, and min{r(L), r^*(E\\ L)}≥ 2. Locked subsets characterize nontrivial facets of the bases polytope. In this paper, we give a new axiom system for matroids based on locked subsets. We deduce an algorithm for recognizing matroids improving the running time complexity of the best known till today. This algorithm induces a polynomial time algorithm for recognizing uniform matroids. This latter problem is intractable if we use an independence oracle.
Let M be a matroid defined on a finite set E and L⊂ E. L is locked in M if M|L and M^*|(E\ L) are... more Let M be a matroid defined on a finite set E and L⊂ E. L is locked in M if M|L and M^*|(E\ L) are 2-connected, and min{r(L), r^*(E\ L)}≥ 2. Locked subsets characterize nontrivial facets of the bases polytope. In this paper, we give a new axiom system for matroids based on locked subsets. We deduce an algorithm for recognizing matroids improving the running time complexity of the best known till today. This algorithm induces a polynomial time algorithm for recognizing uniform matroids. This latter problem is intractable if we use an independence oracle.
Let G=(V, E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the a... more Let G=(V, E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the all spanning trees of G. In this paper, we describe all facets of P(G) as a consequence of the facets of the bases polytope of a matroid.
Let G=(V, E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the a... more Let G=(V, E) be an undirected graph. The spanning trees polytope P(G) is the convex hull of the all spanning trees of G. In this paper, we describe all facets of P(G) as a consequence of the facets of the bases polytope of a matroid.
networks, Quality-of-Service routing, virtual sub-networks, series parallel graphs, packing of sp... more networks, Quality-of-Service routing, virtual sub-networks, series parallel graphs, packing of spanning
networks, Quality-of-Service routing, virtual sub-networks, series parallel graphs, packing of sp... more networks, Quality-of-Service routing, virtual sub-networks, series parallel graphs, packing of spanning
Given a graph G=V,E, a connected sides cut U,V\U or δU is the set of edges of E linking all verti... more Given a graph G=V,E, a connected sides cut U,V\U or δU is the set of edges of E linking all vertices of U to all vertices of V\U such that the induced subgraphs GU and GV\U are connected. Given a positive weight function w defined on E, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut Ω such that wΩ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.
This paper gives a characterization of uniform matroids by means of locked subsets. Locked subset... more This paper gives a characterization of uniform matroids by means of locked subsets. Locked subsets are 2-connected subsets, their complements are 2-connected in the dual, and the minimum rank of both is 2. Locked subsets give the nontrivial facets of the bases polytope.
Given an undirected graph $G = (V, E)$, and a vertex $r\in V$, an $r$-acyclic orientation of $G$ ... more Given an undirected graph $G = (V, E)$, and a vertex $r\in V$, an $r$-acyclic orientation of $G$ is an orientation $OE$ of the edges of $G$ such that the digraph $OG = (V, OE)$ is acyclic and $r$ is the unique vertex with indegree equal to 0. For $w\in \mathbb{R}^E_+$, $k(G, w)$ is the value of the $w$-maximum packing of $r$-arborescences for all $r\in V$ and all $r$-acyclic orientations $OE$ of $G$. In this case, the Broadcast Routing (in Computers Networks) Problem (BRP) is to compute $k(G, w)$, by finding an optimal $r$ and an optimal $r$-acyclic orientation. BRP is a mathematical formulation of multipath broadcast routing in computer networks. In this paper, we provide a polynomial time algorithm to solve BRP in outerplanar graphs. Outerplanar graphs are encountered in many applications such as computational geometry, robotics, etc.
In this paper we present a max flow multipath routing algorithm (MFMP) that is designed to reduce... more In this paper we present a max flow multipath routing algorithm (MFMP) that is designed to reduce latency, provide high throughput and balance traffic load. The max flow multipath algorithm is based on a Ford-Fulkerson algorithm .It consists of determining a set of disjoints path that are loop free with maximum flow, then splitting network traffic among those paths on a round robin fashion. Through simulation we show that our algorithm performs well than a multi shortest path (MSP) and a single shortest path (SSP).
Soient n un entier non negatif et H un ensemble fini de vecteurs a n composantes entieres. Le con... more Soient n un entier non negatif et H un ensemble fini de vecteurs a n composantes entieres. Le cone engendre par H, note cone(H), est l'ensemble des vecteurs qui sont combinaisons lineaires non negatives des vecteurs de H. Le treillis engendre par H, note lat(H), est l'ensemble des combinaisons lineaires entieres des vecteurs de H. On dit que H est un systeme generateur de Hilbert si tout vecteur qui appartient a cone(H) et a lat(H) peut s'exprimer comme combinaison lineaire de vecteurs de H avec des coefficients entiers et non negatifs. Si H est minimal pour la propriete precedente, le meme cone et le meme treillis, alors on dit que H est une base de Hilbert. Un exemple important de base de Hilbert, etudie dans cette these, est l'ensemble B(M) des (vecteurs caracteristiques des) bases d'un matroide M. Cook, Fonlupt et Schrijver ont montre que, si H est une base de Hilbert, tout vecteur entier qui est dans cone(H) et dans lat(H) est combinaison lineaire de 2n-1 ve...
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Papers by Brahim Chaourar