Maximum Volume Subset Selection for Anchored Boxes

Let B be a set of n axis-parallel boxes in d-dimensions such that each box has a corner at the origin and the other corner in the positive quadrant, and let k be a positive integer. We study the problem of selecting k boxes in B that maximize the volume of the union of the selected boxes. The research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known algorithms in any dimension d>2 enumerate all size-k subsets. We show that: * The problem is NP-hard already in 3 dimensions. * In 3 dimensions, we break the enumeration of all size-k subsets, by providing an n^O(sqrt(k)) algorithm. * For any constant dimension d, we give an efficient polynomial-time approximation scheme

Parallel Transitive Closure and Point Location in Planar Structures

AMS(MOS) subject classifications. 68E05, 68C05, 68C25Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar st-graphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(logn) running time using n/logn processors in the EREW PRAM model, n being the number of vertices

A Provably Good Linear Algorithm for Embedding Graphs in the Rectilinear Grid

Coordinated Science Laboratory was formerly known as Control Systems LaboratorySemiconductor Research Corporation / RSCH 84-06-049-

The DFS-heuristic for orthogonal graph drawing☆☆Some of these result were published in the author's PhD thesis at Rutgers University; the author would like to thank her advisor, Prof. Endre Boros, for much helpful input. The results in Section 5 have been presented at the 8th Canadian Conference on Computational Geometry, Ottawa, 1996, see [1].

AbstractIn this paper, we present a new heuristic for orthogonal graph drawings, which creates drawings by performing a depth-first search and placing the nodes in the order they are encountered. This DFS-heuristic works for graphs with arbitrarily high degrees, and particularly well for graphs with maximum degree 3. It yields drawings with at most one bend per edge, and a total number of m−n+1 bends for a graph with n nodes and m edges; this improves significantly on the best previous bound of m−2 bends

Maximum Volume Subset Selection for Anchored Boxes

Let $B$ be a set of $n$ axis-parallel boxes in $\mathbb{R}^d$ such that each box has a corner at the origin and the other corner in the positive quadrant of $\mathbb{R}^d$, and let $k$ be a positive integer. We study the problem of selecting $k$ boxes in $B$ that maximize the volume of the union of the selected boxes. This research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known running time in any dimension $d \ge 3$ is $\Omega\big(\binom{n}{k}\big)$. We show that: - The problem is NP-hard already in 3 dimensions. - In 3 dimensions, we break the bound $\Omega\big(\binom{n}{k}\big)$, by providing an $n^{O(\sqrt{k})}$ algorithm. - For any constant dimension $d$, we present an efficient polynomial-time approximation scheme

An alternative proof for the NP-completeness of the Grid Subgraph problem

Στον τομέα της Γραφικής Αναπαράστασης Γραφημάτων υπάρχει μεγάλο ενδιαφέρον για αποτελέσματα σχετικά με την εμβάπτιση ενός δοθέντος γραφήματος πάνω σε μία σχάρα, κυρίως λόγω των εφαρμογών στο σχεδιασμό κυκλωμάτων VLSI. Πιο συγκεκριμένα, το ερώτημα αν ένα γράφημα επιδέχεται εμβάπτιση μοναδιαίου μήκους, δηλαδή μια αντιστοίχιση των κορυφών και των ακμών του γραφήματος σε κορυφές και ακμές μιας αρκετά μεγάλης σχάρας, ταυτίζεται με το ερώτημα αν το γράφημα είναι υπογράφημα της συγκεκριμένης σχάρας. Θεωρούμε το πρόβλημα Υπογράφημα Σχάρας, στο οποίο δοθέντος ενός επίπεδου (όχι απαραίτητα συνεκτικού) γραφήματος G, καλούμαστε να αποφανθούμε αν το G είναι ισόμορφο με κάποιο υπογράφημα μιας αρκετά μεγάλης σχάρας. Αποδεικνύουμε ότι το πρόβλημα αυτό είναι NP-πλήρες χρησιμοποιώντας απλά και διαισθητικά για να ανάγουμε σε αυτό μία παραλλαγή του προβλήματος SAT (ικανοποίησης λογικής φόρμουλας). Προς αυτό αποδεικνύουμε ότι και η ειδική περίπτωση του προβλήματος στην οποία το μέγεθος της σχάρας είναι προκαθορισμένο, γνωστό και ως το πρόβλημα Υπογράφημα (k*k)-Σχάρας, είναι επίσης NP-πλήρες.In the field of Graph Drawing, there is great interest for results regarding the embedding of a given graph on a grid, mainly due to the applications on the VLSI circuit design. Moreover, determining whether a graph accepts a unit-length embedding, i.e., a matching of its vertices and edges to vertices and edges of a large enough grid, is the same as asking whether the graph is a subgraph of that grid. We consider the Grid Subgraph problem, in which given a planar (not necessarily connected) graph G, we need to determine if G is isomorphic to a subgraph of a large enough grid. We prove that this problem is NP-complete by employing simple and intuitive gadgets to perform a reduction from a SAT-variant. In addition we prove that a special case of that problem, the (k*k)-Grid Subgraph problem, in which the size of the grid is given in the input, is also NP-complete