8 research outputs found
Beyond Hypergraph Dualization
International audienceThis problem concerns hypergraph dualization and generalization to poset dualization. A hypergraph H = (V, E) consists of a finite collection E of sets over a finite set V , i.e. E ⊆ P(V) (the powerset of V). The elements of E are called hyperedges, or simply edges. A hypergraph is said simple if none of its edges is contained within another. A transversal (or hitting set) of H is a set T ⊆ V that intersects every edge of E. A transversal is minimal if it does not contain any other transversal as a subset. The set of all minimal transversal of H is denoted by T r(H). The hypergraph (V, T r(H)) is called the transversal hypergraph of H. Given a simple hypergraph H, the hypergraph dualization problem (Trans-Enum for short) concerns the enumeration without repetitions of T r(H). The Trans-Enum problem can also be formulated as a dualization problem in posets. Let (P, ≤) be a poset (i.e. ≤ is a reflexive, antisymmetric, and transitive relation on the set P). For A ⊆ P , ↓ A (resp. ↑ A) is the downward (resp. upward) closure of A under the relation ≤ (i.e. ↓ A is an ideal and ↑ A a filter of (P, ≤)). Two antichains (B + , B −) of P are said to be dual if ↓ B + ∪ ↑ B − = P and ↓ B + ∩ ↑ B − = ∅. Given an implicit description of a poset P and an antichain B + (resp. B −) of P , the poset dualization problem (Dual-Enum for short) enumerates the set B − (resp. B +), denoted by Dual(B +) = B − (resp. Dual(B −) = B +). Notice that the function dual is self-dual or idempotent, i.e. Dual(Dual(B)) = B
Approximation Algorithms for Generalized MST and TSP in Grid Clusters
We consider a special case of the generalized minimum spanning tree problem
(GMST) and the generalized travelling salesman problem (GTSP) where we are
given a set of points inside the integer grid (in Euclidean plane) where each
grid cell is . In the MST version of the problem, the goal is to
find a minimum tree that contains exactly one point from each non-empty grid
cell (cluster). Similarly, in the TSP version of the problem, the goal is to
find a minimum weight cycle containing one point from each non-empty grid cell.
We give a and -approximation
algorithm for these two problems in the described setting, respectively.
Our motivation is based on the problem posed in [7] for a constant
approximation algorithm. The authors designed a PTAS for the more special case
of the GMST where non-empty cells are connected end dense enough. However,
their algorithm heavily relies on this connectivity restriction and is
unpractical. Our results develop the topic further
Guest editors’ foreword
The 26th International Symposium on Algorithms and Computation (ISAAC 2015)\u3cbr/\u3ewas held in Nagoya, Japan, December 9–11, 2015. The program committee received\u3cbr/\u3e180 high-quality submissions, and 65 were accepted for presentation. This special\u3cbr/\u3eissue gathers a selection of six of these accepted papers, which went through the\u3cbr/\u3estandard refereeing process of the International Journal on Computational Geometry\u3cbr/\u3eand Applications
Approximation algorithms for the Euclidean traveling salesman problem with discrete and continuous neighborhoods
In the Euclidean traveling salesman problem with discrete neighborhoods, we are given a set of points P in the plane and a set of n connected regions (neighborhoods), each containing at least one point of P. We seek to find a tour of minimum length which visits at least one point in each region. We give (i) an O(a)-approximation algorithm for the case when the regions are disjoint and a-fat, with possibly varying size; (ii) an O(a3)-approximation algorithm for intersecting a-fat regions with comparable diameters. These results also apply to the case with continuous neighborhoods, where the sought TSP tour can hit each region at any point. We also give (iii) a simple O(log n)-approximation algorithm for continuous non-fat neighborhoods. The most distinguishing features of these algorithms are their simplicity and low running-time complexities
Rectilinear Shortest Path and Rectilinear Minimum Spanning Tree with Neighborhoods
We consider a setting where we are given a graph \mathcal {g}=(\mathcal {r},e)\mathcal {g}=(\mathcal {r},e), where \mathcal {r}=\{r_1,\ldots ,r_n\}\mathcal {r}=\{r_1,\ldots ,r_n\} is a set of polygonal regions in the plane. Placing a point p_ip_i inside each region r_ir_i turns gg into an edge-weighted graph g_{\varvec{p}}g_{\varvec{p}}, {\varvec{p}}=\{p_1,\ldots ,p_n\}{\varvec{p}}=\{p_1,\ldots ,p_n\}, where the cost of (r_i,r_j)\in e(r_i,r_j)\in e is the distance between p_ip_i and p_jp_j. The shortest path problem with neighborhoods asks, for given r_sr_s and r_tr_t, to find a placement \varvec{p}\varvec{p} such that the cost of a resulting shortest stst-path in \mathcal {g}_{\varvec{p}}\mathcal {g}_{\varvec{p}} is minimum among all graphs \mathcal {g}_{\varvec{p}}\mathcal {g}_{\varvec{p}}. The minimum spanning tree problem with neighborhoods asks to find a placement \varvec{p}\varvec{p} such that the cost of a resulting minimum spanning tree is minimum among all graphs \mathcal {g}_{\varvec{p}}\mathcal {g}_{\varvec{p}}. We study these problems in the l_1l_1 metric, and show that the shortest path problem with neighborhoods is solvable in polynomial time, whereas the minimum spanning tree problem with neighborhoods is \mathsf {apx}\mathsf {apx}-hard, even if the neighborhood regions are segments
Dualization in lattices given by implicational bases
International audienc