arXiv subjects
Publications and source records attributed to Brahim Chaourar.
Let $n$ be a nonnegative integer, and $f(n)$ the number of unlabeled finite topologies on $n$ points. We prove that $f(n+m) \geq f(n) f(m)$ both for the labeled and unlabeled cases. Moreover, we prove a similar inequality for labeled and unlabeled $T_0$ topologies.
In this paper, we give a new axioms system based on nonseparable flats with their ranks to define a matroid. We deduce a polynomial time algorithm for deciding if a given matroid (respectively, arbitrary structure) is an uniform matroid. This problem is intractable if we use an independence or an equivalent oracle.
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 $M$.
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.
Let $M$ be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. 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.
Given a graph $G=(V, E)$, a connected cut $δ(U)$ is the set of edges of E linking all vertices of U to all vertices of $V\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\backslash U]$ are connected. Given a positive weight function $w$ defined on $E$, the connected maximum cut problem (CMAX CUT) is to find a connected cut $Ω$ such that $w(Ω)$ is maximum among all connected cuts. CMAX CUT is NP-hard even for planar graphs. In this paper, we prove that CMAX CUT is polynomial for graphs without $K_5\backslash e$ as a minor. We deduce a quadratic time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.
Let $G=(V, E)$ be an undirected graph. The subtours elimination polytope $P(G)$ is the set of $x\in \mathbb{R}^E$ such that: $0\leq x(e)\leq 1$ for any edge $e\in E$, $x(δ(v))=2$ for any vertex $v\in V$, and $x(δ(U))\geq 2$ 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 \cite{Maurras 1975} and Grötschel and Padberg \cite{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.
Given an undirected graph $G=(V, E)$ with a weight function $c\in R^E$, and a positive integer $K$, the Kth Traveling Salesman Problem (KthTSP) is to find $K$ Hamilton cycles $H_1, H_2, , ..., H_K$ such that, for any Hamilton cycle $H\not \in \{H_1, H_2, , ..., H_K \}$, we have $c(H)\geq c(H_i), i=1, 2, ..., K$. This problem is NP-hard even for $K$ fixed. We prove that KthTSP is pseudopolynomial when TSP is polynomial.
Given a graph $G=(V, E)$, a connected sides cut $(U, V\backslash U)$ or $δ(U)$ is the set of edges of E linking all vertices of U to all vertices of $V\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\backslash U]$ 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 $M$ to be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. In this paper, we prove that the nontrivial facets of the bases polytope of $M$ are described by the locked subsets. We deduce that finding the maximum--weight basis of $M$ is a polynomial time problem for matroids with a polynomial number of locked subsets. This class of matroids is closed under 2-sums and contains the class of uniform matroids, the Vámos matroid and all the excluded minors of 2-sums of uniform matroids. We deduce also a matroid oracle for testing uniformity of matroids after one call of this oracle.
Let M to be a matroid defined on a finite set E. A subset L of E is locked in M if L is 2-connected in M, the complement of L is 2-connected in the dual M*, and min{r(L), r*(complement of L)} is greater than 1. In this paper, we prove that the nontrivial facets of the bases polytope of M are described by the locked subsets. We deduce that finding the maximum-weight basis of M is a polynomial problem for matroids with a polynomial number of locked subsets. This class of matroids is closed under 2-sums and contains uniform matroids.