A graph G is a support for a hypergraph H = (V, S) if the vertices of G correspond to the vertices of H such that for each hyperedge S i ∈ S the subgraph of G induced by S i is connected. G is a planar support if it is a support and planar. Johnson and Pollak [9] proved that it is NPcomplete to decide if a given hypergraph has a planar support. In contrast, there are polynomial time algorithms to test whether a given hypergraph has a planar support that is a path, cycle, or tree. In this paper we present an algorithm which tests in polynomial time if a given hypergraph has a planar support that is a tree where the maximal degree of each vertex is bounded. Our algorithm is constructive and computes a support if it exists. Furthermore, we prove that it is already NP-hard to decide if a hypergraph has a 2-outerplanar support.

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We give a systematic study of Hamiltonicity of grids-the graphs induced by finite subsets of vertices of the tilings of the plane with congruent regular convex polygons (triangles, squares, or hexagons). Summarizing and extending existing classification of the usual, "square", grids, we give a comprehensive taxonomy of the grid graphs. For many classes of grid graphs we resolve the computational complexity of the Hamiltonian cycle problem. For graphs for which there exists a polynomial-time algorithm we give efficient algorithms to find a Hamiltonian cycle.We also establish, for any g ≥ 6, a one-to-one correspondence between Hamiltonian cycles in planar bipartite maximum-degree-3 graphs and Hamiltonian cycles in the class C g of girthg planar maximum-degree-3 graphs. As applications of the correspondence, we show that for graphs in C g the Hamiltonian cycle problem is NP-complete and that for any N ≥ 5 there exist graphs in C g that have exactly N Hamiltonian cycles. We also prove that for the graphs in C g , a Chinese Postman tour gives a (1 + 8 g )-approximation to TSP, improving thereby the Christofides ratio when g > 16. We show further that, on any graph, the tour obtained by Christofides' algorithm is not longer than a Chinese Postman tour.

We present a number of breakthroughs for coordinated motion planning, in which the objective is to reconfigure a swarm of labeled convex objects by a combination of parallel, continuous, collision-free translations into a given target arrangement. Problems of this type can be traced back to the classic work of Schwartz and Sharir (1983), who gave a method for deciding the existence of a coordinated motion for a set of disks between obstacles; their approach is polynomial in the complexity of the obstacles, but exponential in the number of disks. Other previous work has largely focused on sequential schedules, in which one robot moves at a time.We provide constant-factor approximation algorithms for minimizing the execution time of a coordinated, parallel motion plan for a swarm of robots in the absence of obstacles, provided some amount of separability.Our algorithm achieves constant stretch factor: If all robots are at most d units from their respective starting positions, the total duration of the overall schedule is O(d). Extensions include unlabeled robots and different classes of robots. We also prove that finding a plan with minimal execution time is NP-hard, even for a grid arrangement without any stationary obstacles. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor Ω(N 1/4 ) may be required. On the positive side, we establish a stretch factor of O(N 1/2 ) even in this case.

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In this paper, we study rectangle of influence drawings, i.e., drawings of graphs such that for any edge the axis-parallel rectangle defined by the two endpoints of the edge is empty. Specifically, we show that if G is a planar graph without filled 3-cycles, i.e., a planar graph that can be drawn such that the interior of every 3-cycle is empty, then G has a rectangle of influence drawing.

An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding-preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at most k reflex corners. We present polynomial time algorithms that test whether G has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of G are related to its number of crossings per edge and to its connectivity. More precisely, we prove that if G has at most one crossing per edge (i.e., G is a 1-plane graph), an OPVR of G always exists while this may not be the case if two crossings per edge are allowed. Also, if G is a 3-connected 1-plane graph, we can compute an OPVR of G whose vertex complexity is bounded by a constant in O(n) time. However, if G is a 2-connected 1-plane graph, the vertex complexity of any OPVR of G may be Ω(n). In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed in O(n) time. Finally, we present the results of an experimental study on the vertex complexity of ortho-polygon visibility representations of 1-plane graphs.

Abstract. We present a new algorithm for upper bounding the maximum average linear hull probability for SPNs, a value required to determine provable security against linear cryptanalysis. The best previous result (Hong et al. [9]) applies only when the linear transformation branch number (B) is M or (M + 1) (maximal case), where M is the number of s-boxes per round. In contrast, our upper bound can be computed for any value of B. Moreover, the new upper bound is a function of the number of rounds (other upper bounds known to the authors are not). When B = M , our upper bound is consistently superior to [9]. When B = (M + 1), our upper bound does not appear to improve on [9]. On application to Rijndael (128-bit block size, 10 rounds), we obtain the upper bound UB = 2 −75 , corresponding to a lower bound on the data complexity of 8 UB = 2 78 (for 96.7% success rate). Note that this does not demonstrate the existence of a such an attack, but is, to our knowledge, the first such lower bound.