(1,1)-Cluster Editing is Polynomial-time Solvable

• URL: http://arxiv.org/abs/2210.07722v2
• Date: Tue, 4 Jul 2023 08:02:59 GMT
• Title: (1,1)-Cluster Editing is Polynomial-time Solvable
• Authors: Gregory Gutin and Anders Yeo
• Abstract summary: Abu-Khzam introduced the $(a,d)$- Editing problem, where for fixed natural numbers $a,d$, given a graph $G and affirmative-weights $a*: V(G)rightarrow 0,1,dots, a and $d*: V(G)rightarrow 0,1,dots, a and $d*: V(G)rightarrow 0,1,dots, a and $d*: V(
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques. Abu-Khzam (2017) introduced the $(a,d)$-{Cluster Editing} problem, where for fixed natural numbers $a,d$, given a graph $G$ and vertex-weights $a^*:\ V(G)\rightarrow \{0,1,\dots, a\}$ and $d^*{}:\ V(G)\rightarrow \{0,1,\dots, d\}$, we are to decide whether $G$ can be turned into a cluster graph by deleting at most $d^*(v)$ edges incident to every $v\in V(G)$ and adding at most $a^*(v)$ edges incident to every $v\in V(G)$. Results by Komusiewicz and Uhlmann (2012) and Abu-Khzam (2017) provided a dichotomy of complexity (in P or NP-complete) of $(a,d)$-{Cluster Editing} for all pairs $a,d$ apart from $a=d=1.$ Abu-Khzam (2017) conjectured that $(1,1)$-{Cluster Editing} is in P. We resolve Abu-Khzam's conjecture in affirmative by (i) providing a serious of five polynomial-time reductions to $C_3$-free and $C_4$-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving $(1,1)$-{Cluster Editing} on $C_3$-free and $C_4$-free graphs of maximum degree at most 3.

Related papers

• A Polynomial-Time Approximation for Pairwise Fair $k$-Median Clustering [10.697784653113095]
  We study pairwise fair clustering with $ell ge 2$ groups, where for every cluster $C$ and every group $i in [ell]$, the number of points in $C$ from group $i$ must be at most $t times the number of points in $C$ from any other group $j in [ell]$. We show that our problem even when $ell=2$ is almost as hard as the popular uniform capacitated $k$-median, for which no-time algorithm with an approximation factor of $o
  arXiv  Detail & Related papers  (2024-05-16T18:17:44Z)
• $\ell_p$-Regression in the Arbitrary Partition Model of Communication [59.89387020011663]
  We consider the randomized communication complexity of the distributed $ell_p$-regression problem in the coordinator model. For $p = 2$, i.e., least squares regression, we give the first optimal bound of $tildeTheta(sd2 + sd/epsilon)$ bits. For $p in (1,2)$,we obtain an $tildeO(sd2/epsilon + sd/mathrmpoly(epsilon)$ upper bound.
  arXiv  Detail & Related papers  (2023-07-11T08:51:53Z)
• Detection of Dense Subhypergraphs by Low-Degree Polynomials [72.4451045270967]
  Detection of a planted dense subgraph in a random graph is a fundamental statistical and computational problem. We consider detecting the presence of a planted $Gr(ngamma, n-alpha)$ subhypergraph in a $Gr(n, n-beta) hypergraph. Our results are already new in the graph case $r=2$, as we consider the subtle log-density regime where hardness based on average-case reductions is not known.
  arXiv  Detail & Related papers  (2023-04-17T10:38:08Z)
• Repeated Averages on Graphs [2.363388546004777]
  We prove that $frac(1-epsilon)2log2nlog n-O(n)$ is a general lower bound for all connected graphs on $n$ nodes. We also get sharp magnitude of $t_epsilon,1$ for several important families of graphs, including star, expander, dumbbell, and cycle.
  arXiv  Detail & Related papers  (2022-05-09T20:18:31Z)
• Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
  We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
  arXiv  Detail & Related papers  (2022-02-10T16:10:41Z)
• Fast Graph Sampling for Short Video Summarization using Gershgorin Disc Alignment [52.577757919003844]
  We study the problem of efficiently summarizing a short video into several paragraphs, leveraging recent progress in fast graph sampling. Experimental results show that our algorithm achieves comparable video summarization as state-of-the-art methods, at a substantially reduced complexity.
  arXiv  Detail & Related papers  (2021-10-21T18:43:00Z)
• Exact Matching of Random Graphs with Constant Correlation [2.578242050187029]
  This paper deals with the problem of graph matching or network alignment for ErdHos--R'enyi graphs. It can be viewed as a noisy average-case version of the graph isomorphism problem.
  arXiv  Detail & Related papers  (2021-10-11T05:07:50Z)
• Local Correlation Clustering with Asymmetric Classification Errors [12.277755088736864]
  In the Correlation Clustering problem, we are given a complete weighted graph $G$ with its edges labeled as "similar" and "dissimilar" For a clustering $mathcalC$ of graph $G$, a similar edge is in disagreement with $mathcalC$, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with $mathcalC$ if its endpoints belong to the same cluster. We produce a clustering that minimizes the $ell_p$ norm of the disagreements vector for $pgeq 1$
  arXiv  Detail & Related papers  (2021-08-11T12:31:48Z)
• A common variable minimax theorem for graphs [3.0079490585515343]
  We study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in $mathcalG$, simultaneously, and how to find it if it exists.
  arXiv  Detail & Related papers  (2021-07-30T16:47:25Z)
• Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Models [56.98280399449707]
  We show that there exists an $epsilon$-cover for $S$ of cardinality $M = (k/epsilon)O_d(k1/d)$. Building on our structural result, we obtain significantly improved learning algorithms for several fundamental high-dimensional probabilistic models hidden variables.
  arXiv  Detail & Related papers  (2020-12-14T18:14:08Z)
• Agnostic Q-learning with Function Approximation in Deterministic Systems: Tight Bounds on Approximation Error and Sample Complexity [94.37110094442136]
  We study the problem of agnostic $Q$-learning with function approximation in deterministic systems. We show that if $delta = Oleft(rho/sqrtdim_Eright)$, then one can find the optimal policy using $Oleft(dim_Eright)$.
  arXiv  Detail & Related papers  (2020-02-17T18:41:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.