Shape formation is one of the most thoroughly studied problems in programmable matter and swarm robotics. However, in many models, the class of shapes that can be formed is highly restricted due to the particles’ limited memory. In the hybrid model, an active agent with the computational power of a deterministic finite automaton can form shapes by lifting and placing passive tiles on the triangular lattice. We study the shape reconfiguration problem where the agent additionally has the ability to distinguish so-called target nodes from non-target nodes and needs to form a target shape from the initial tile configuration. We present a worst-case optimal 𝒪(mn) algorithm for simply connected target shapes, where m is the initial number of unoccupied target nodes and n is the total number of tiles. Furthermore, we show how an agent can reconfigure a large class of target shapes with holes in 𝒪(n^4) steps.
We study the query complexity of finding tournament solutions in near-transitive tournaments, which are tournaments obtained by flipping a small number of edges in a transitive tournament. While general tournaments require (n^2) queries for many solution concepts such as Copeland, Top Cycle, and Uncovered Set, we show that we can do better for near-transitive tournaments in several scenarios. In particular, we introduce three query models: the standard model, where queries return orientations in the input tournament; the partial-flip model, where queries return orientations from the underlying transitive tournament and we have access to edge-flip queries; and the full-flip model, which additionally provides vertex-flip queries that indicate whether a vertex is incident to a flipped edge. Our main result shows that in the full-flip model, if at most ℓ edges are flipped from a transitive tournament where ℓ⩽√(n log n) , then the set of Copeland winners, top cycle, and uncovered set can all be determined in O(n log n) queries. This represents an improvement from the (n^2) lower bound in general tournaments. We also show the fine-grained query complexity of obtaining Copeland winners parameterized by the number of flipped edges. Finally, we study pseudo-transitive tournaments—tournaments that are at most a single flip away from a transitive tournament—at length, and consider the query complexity of finding the Copeland winners, Top cycle, and Uncovered set in all three models. We also study the query complexity of determining if a pseudo-transitive tournament has a Condorcet winner in these models. Surprisingly, in the standard model, we need 2n - ⌈log n ⌉ - 2 queries to determine if a pseudo-transitive tournament has a Condorcet winner: and these many queries are in fact sufficient to address the question on general tournaments, so the restriction to pseudo-transitive tournaments is not helpful. This situation persists in the partial-flip model, although we show an improved bound of n + ⌈log n ⌉ queries in the full-flip model.
The graph coloring problem is a well-known combinatorial problem that seeks a coloring pattern such that no two adjacent vertices share the same color. In this paper, we propose an efficient method not only for finding one solution but also for efficiently enumerating all solutions that satisfy the graph coloring constraints. Our method uses ZDDs (Zero-suppressed Binary Decision Diagrams) for efficiently indexing sets of graph coloring patterns. In addition, we propose an algorithm to generate all coloring patterns that do not split any cell of a given set partition. Our experimental results demonstrate that the proposed method significantly outperforms existing solvers. Compared to the leading ASP (Answer Set Programming) solver, our approach generates all coloring patterns more than 1,000 times faster. Our method is applicable to many practical graph instances with up to around 25 vertices, in a feasible computation time.
Given an outerplane drawing 𝒟 with n vertices, a 𝒟 -constrained maximum outerplane drawing is an outerplane drawing that contains 𝒟 and has the maximum number of edges among all such drawings. We show that (1) any such 𝒟 -constrained maximum outerplane drawing has at least n+1 edges, and (2) this bound is best possible. If 𝒟 is a plane spanning path, we present an O(n^3) -time algorithm to compute a 𝒟 -constrained maximum outerplane drawing that minimizes its weight, that is, the sum of the Euclidean length of the edges of its drawing. For the unweighted setting, our results imply the following: It can be decided in O(n^3) time whether a given plane spanning path admits a polygon such that every edge of the path is on the polygon or an internal edge. Further, we present for several minimum-weight structures, like minimum spanning trees, points sets such that they are not subdrawings of the minimum-weight outerplane Laman graph of that point set.
We investigate the computational complexity of Minimum Coverage in Multi-Interface Networks (Min CMI), which has applications in the field of Internet of Things (IoT). In this problem, we are given an undirected graph where each vertex represents a device (e.g., smartphones, environmental sensors, smart speakers) capable of activating multiple interfaces (e.g., Bluetooth, Wi-Fi, 4G/5G) to establish connections with one another. The objective is to ensure the connectivity requirement of the network while minimizing its total energy consumption by activating only the necessary interfaces. Contributing to the computational complexity landscape of Min CMI, we unify and extend previously known algorithmic results and demonstrate their limits. In particular, we analyze two scenarios based on the number of different interfaces across the network. When this number is unbounded, we show that the problem remains fixed-parameter intractable with respect to the number p of active interfaces per node within highly restricted graph classes such as stars and cliques. Additionally, we show that even in the case that p is a small constant, strong structural parameters like vertex cover number, feedback edge number, or distance to clique do not help to obtain fixed-parameter tractability. On the positive side, we provide a polynomial-time algorithm for cographs when p=2 . When the number of different interfaces is bounded by a constant, the problem, while still NP-hard, allows for more tractable special cases, including sparse graphs with small separators (i.e., low treewidth) and dense graphs.
