Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMaximum Clique in Disk-Like Intersection Graphs
182
0
0.0
(
0
)
Added by
\\'Edouard Bonnet
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark 90, Raghavan and Spinrad 03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. 18, Bonamy et al. 18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard. We observe that, so far, all the hardness proofs for Maximum Clique in intersection graph classes $mathcal I$ follow the same road. They show that, for every graph $G$ of a large-enough class $mathcal C$, the complement of an even subdivision of $G$ belongs to the intersection class $mathcal I$. Then they conclude invoking the hardness of Maximum Independent Set on the class $mathcal C$, and the fact that the even subdivision preserves that hardness. However there is a strong evidence that this approach cannot work for disk graphs [Bonnet et al. 18]. We suggest a new approach, based on a problem that we dub Max Interval Permutation Avoidance, which we prove unlikely to have a subexponential-time approximation scheme. We transfer that hardness to Maximum Clique in intersection graphs of objects which can be either half-planes (or unit disks) or axis-parallel rectangles. That problem is not amenable to the previous approach. We hope that a scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance could help making progress on the disk case, for instance by showing the NP-hardness for (convex) pseudo-disks.
rate research
Read More
Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for $t$-interval graphs when $tgeq 3$ and polynomial-time solvable when $t=1$. The problem is also known to be NP-complete in $t$-track graphs when $tgeq 4$ and polynomial-time solvable when $tleq 2$. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APX-complete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called $t$-circular interval graphs and $t$-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time $t$-approximation algorithm for WEIGHTED MAXIMUM CLIQUE on $t$-interval graphs, improving earlier work with approximation ratio $4t$.
We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a {em weakly robust} polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) that it computes a clique partition, we show that it is guaranteed to be within $(1+eps)$ ratio of the optimum if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDGs is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an $O(frac{log^* n}{eps^{O(1)}})$ time distributed PTAS. We consider a weighted version of the clique partition problem on vertex weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet, surprisingly, it admits a $(2+eps)$-approximation algorithm for the weighted case where the graph is expressed, say, as an adjacency matrix. This improves on the best known 8-approximation for the {em unweighted} case for UDGs expressed in standard form.
We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of $n$ unit disks in the plane there exists a line $ell$ such that $ell$ intersects at most $O(sqrt{(m+n)log{n}})$ disks and each of the halfplanes determined by $ell$ contains at most $2n/3$ unit disks from the set, where $m$ is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting $O(sqrt{m+n})$ disks exists, but each halfplane may contain up to $4n/5$ disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in $n$ exists when we look at disks of arbitrary radii, even when $m=0$. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size $O(sqrt{m})$).
Let $mathcal{D}$ be a set of $n$ disks in the plane. The disk graph $G_mathcal{D}$ for $mathcal{D}$ is the undirected graph with vertex set $mathcal{D}$ in which two disks are joined by an edge if and only if they intersect. The directed transmission graph $G^{rightarrow}_mathcal{D}$ for $mathcal{D}$ is the directed graph with vertex set $mathcal{D}$ in which there is an edge from a disk $D_1 in mathcal{D}$ to a disk $D_2 in mathcal{D}$ if and only if $D_1$ contains the center of $D_2$. Given $mathcal{D}$ and two non-intersecting disks $s, t in mathcal{D}$, we show that a minimum $s$-$t$ vertex cut in $G_mathcal{D}$ or in $G^{rightarrow}_mathcal{D}$ can be found in $O(n^{3/2}text{polylog} n)$ expected time. To obtain our result, we combine an algorithm for the maximum flow problem in general graphs with dynamic geometric data structures to manipulate the disks. As an application, we consider the barrier resilience problem in a rectangular domain. In this problem, we have a vertical strip $S$ bounded by two vertical lines, $L_ell$ and $L_r$, and a collection $mathcal{D}$ of disks. Let $a$ be a point in $S$ above all disks of $mathcal{D}$, and let $b$ a point in $S$ below all disks of $mathcal{D}$. The task is to find a curve from $a$ to $b$ that lies in $S$ and that intersects as few disks of $mathcal{D}$ as possible. Using our improved algorithm for minimum cuts in disk graphs, we can solve the barrier resilience problem in $O(n^{3/2}text{polylog} n)$ expected time.
Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
