Approximating Independent Set and Dominating Set on VPG graphs

Approximating MIS over equilateral $B_1$-VPG graphs

We present an approximation algorithm for the maximum independent set (MIS) problem over the class of equilateral $B_1$-VPG graphs. These are intersection graphs of $L$-shaped planar objects % (and their rotations by multiples of $90^o$) with both arms of each object being equal. We obtain a $36(\log 2d)$-approximate algorithm running in $O(n(\log n)^2)$ time for this problem, where $d$ is the ratio $d_{max}/d_{min}$ and $d_{max}$ and $d_{min}$ denote respectively the maximum and minimum length of any arm in the input equilateral $L$-representation of the graph. In particular, we obtain $O(1)$…

Approximating $k$-connected $m$-dominating sets

A subset $S$ of nodes in a graph $G$ is a $k$-connected $m$-dominating set ($(k,m)$-cds) if the subgraph $G[S]$ induced by $S$ is $k$-connected and every $v \in V \setminus S$ has at least $m$ neighbors in $S$. In the $k$-Connected $m$-Dominating Set ($(k,m)$-CDS) problem the goal is to find a minimum weight $(k,m)$-cds in a node-weighted graph. For $m \geq k$ we obtain the following approximation ratios. For general graphs our ratio $O(k \ln n)$ improves the previous best ratio $O(k^2 \ln n)$ and matches the best known ratio for unit weights. For unit disc graphs we improve the ratio $O(k \ln…

Upper and Lower Bounds on Approximating Weighted Mixed Domination

A mixed dominating set of a graph $G = (V, E)$ is a mixed set $D$ of vertices and edges, such that for every edge or vertex, if it is not in $D$, then it is adjacent or incident to at least one vertex or edge in $D$. The mixed domination problem is to find a mixed dominating set with a minimum cardinality. It has applications in system control and some other scenarios and it is $NP$-hard to compute an optimal solution. This paper studies approximation algorithms and hardness of the weighted mixed dominating set problem. The weighted version is a generalization of the unweighted version, where …

Approximating Highly Inapproximable Problems on Graphs of Bounded Twin-Width

For any $\varepsilon > 0$, we give a polynomial-time $n^\varepsilon$-approximation algorithm for Max Independent Set in graphs of bounded twin-width given with an $O(1)$-sequence. This result is derived from the following time-approximation trade-off: We establish an $O(1)^{2^q-1}$-approximation algorithm running in time $\exp(O_q(n^{2^{-q}}))$, for every integer $q \geqslant 0$. Guided by the same framework, we obtain similar approximation algorithms for Min Coloring and Max Induced Matching. In general graphs, all these problems are known to be highly inapproximable: for any $\varepsilon > 0…

近似集合覆盖重配置的最优PSPACE难度

Optimal PSPACE-hardness of Approximating Set Cover Reconfiguration

本文证明了Minmax集合覆盖重配置和支配集重配置在2−1/polyloglog N因子内PSPACE难近似，首次揭示重配置问题近似因子的尖锐阈值，并基于FGLSS归约的重配置类比给出证明。

Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class

We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to $\textit{edit}$ a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then $\textit{lift}$ the solution to apply to the original graph. We give a general characterization of when an optimi…