An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In addition to being a natural combinatorial problem, the rainbow connectivity problem is motivated by applications in cellular networks. In this paper we give the first proof that computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every $ε$ > 0, a connected graph with minimum degree at least $εn$ has bounded rainbow connectivity, where the bound depends only on $ε$, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also pre sented.
Let $G$ be a nontrivial connected, edge-colored graph. An edge-cut $S$ of $G$ is called a rainbow cut if no two edges in $S$ are colored with a same color. An edge-coloring of $G$ is a rainbow disconnection coloring if for every two distinct vertices $s$ and $t$ of $G$, there exists a rainbow cut $S$ in $G$ such that $s$ and $t$ belong to different components of $G\setminus S$. For a connected graph $G$, the {\it rainbow disconnection number} of $G$, denoted by $rd(G)$, is defined as the smallest number of colors such that $G$ has a rainbow disconnection coloring by using this number of colors…
The concept of rainbow connection number of a graph was introduced by Chartrand et al. in 2008. Inspired by this concept, other concepts on colored version of connectivity in graphs were introduced, such as the monochromatic connection number by Caro and Yuster in 2011, the proper connection number by Borozan et al. in 2012, and the conflict-free connection number by Czap et al. in 2018, as well as some other variants of connection numbers later on. Chakraborty et al. proved that to compute the rainbow connection number of a graph is NP-hard. For a long time, it has been tried to fix the compu…
This note concerns the trade-off between the degree of the constraint graph and the gap in hardness of approximating the Min-Rep variant of Label Cover (aka Projection Game). We make a very simple observation that, for NP-hardness with gap $g$, the degree can be made as small as $O(g \log g)$, which improves upon the previous $\tilde{O}(g^{1/2})$ bound from a work of Laekhanukit (SODA'14). Note that our bound is optimal up to a logarithmic factor since there is a trivial $Δ$-approximation for Min-Rep where $Δ$ is the maximum degree of the constraint graph. Thanks to known reductions, this impr…
We study the problem #IndSub(P) of counting all induced subgraphs of size k in a graph G that satisfy the property P. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties P including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(P), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph m…
Bobkov, Houdré, and the last author [2000] introduced a Poincaré-type functional parameter, $λ_\infty$, of a graph and related it to connectivity of the graph via Cheeger-type inequalities. A work by the second author, Raghavendra, and Vempala [2013] related the complexity of $λ_\infty$ to the so-called small-set expansion (SSE) problem and further set forth the desiderata for NP-hardness of this optimization problem. We confirm the conjecture that computing $λ_\infty$ is NP-hard for weighted trees. Beyond measuring connectivity in many applications we want to optimize it. This, via convex dua…
Given a set of obstacles and two points, is there a path between the two points that does not cross more than $k$ different obstacles? This is a fundamental problem that has undergone a tremendous amount of work. It is known to be NP-hard, even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). The problem can be generalized into the following graph problem: Given a planar graph $G$ whose vertices are colored by color sets, two designated vertices $s, t \in V(G)$, and $k \in \mathbb{N}$, is there an $s$-$t$ path in $G$ that uses at most $k$ colors? If each o…
Duke University（杜克大学）；Tandon School of Engineering, New York University（纽约大学坦登工程学院）；Tel-Aviv University（特拉维夫大学）
Assembly planning is a fundamental problem in robotics and automation, which involves designing a sequence of motions to bring the separate constituent parts of a product into their final placement in the product. Assembly planning is naturally cast as a disassembly problem, giving rise to the assembly partitioning problem: Given a set $A$ of parts, find a subset $S\subset A$, referred to as a subassembly, such that $S$ can be rigidly translated to infinity along a prescribed direction without colliding with $A\setminus S$. While assembly partitioning is efficiently solvable, it is further des…
Carleton University（卡尔顿大学）；Massachusetts Institute of Technology（麻省理工学院）；Tufts University（塔夫茨大学）；Harvard University（哈佛大学）；Columbia University（哥伦比亚大学）；University of Waterloo（滑铁卢大学）；TU Eindhoven（埃因霍温理工大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
We give both efficient algorithms and hardness results for reconfiguring between two connected configurations of modules in the hexagonal grid. The reconfiguration moves that we consider are "pivots", where a hexagonal module rotates around a vertex shared with another module. Following prior work on modular robots, we define two natural sets of hexagon pivoting moves of increasing power: restricted and monkey moves. When we allow both moves, we present the first universal reconfiguration algorithm, which transforms between any two connected configurations using $O(n^3)$ monkey moves. This res…
Query Complexity of Densest Sub(hyper)graph Problems
Pacific Northwest National Laboratory（太平洋西北国家实验室）；School of Computing, University of Utah（犹他大学计算机学院）；Department of Computer Science and Engineering, Texas A&M University（德克萨斯农工大学计算机科学工程系）
