Linear-time Algorithms for Eliminating Claws in Graphs

Since many NP-complete graph problems have been shown polynomial-time solvable when restricted to claw-free graphs, we study the problem of determining the distance of a given graph to a claw-free graph, considering vertex elimination as measure. CLAW-FREE VERTEX DELETION (CFVD) consists of determining the minimum number of vertices to be removed from a graph such that the resulting graph is claw-free. Although CFVD is NP-complete in general and recognizing claw-free graphs is still a challenge, where the current best algorithm for a graph $G$ has the same running time of the best algorithm for matrix multiplication, we present linear-time algorithms for CFVD on weighted block graphs and weighted graphs with bounded treewidth. Furthermore, we show that this problem can be solved in linear time by a simpler algorithm on forests, and we determine the exact values for full $k$-ary trees. On the other hand, we show that CLAW-FREE VERTEX DELETION is NP-complete even when the input graph is a split graph. We also show that the problem is hard to approximate within any constant factor better than $2$, assuming the Unique Games Conjecture.Comment: 20 page

On minimum tt-claw deletion in split graphs

For $t\geq 3$, $K_{1, t}$ is called $t$-claw. In minimum $t$-claw deletion problem (\texttt{Min-$t$-Claw-Del}), given a graph $G=(V, E)$, it is required to find a vertex set $S$ of minimum size such that $G[V\setminus S]$ is $t$-claw free. In a split graph, the vertex set is partitioned into two sets such that one forms a clique and the other forms an independent set. Every $t$-claw in a split graph has a center vertex in the clique partition. This observation motivates us to consider the minimum one-sided bipartite $t$-claw deletion problem (\texttt{Min-$t$-OSBCD}). Given a bipartite graph $G=(A \cup B, E)$, in \texttt{Min-$t$-OSBCD} it is asked to find a vertex set $S$ of minimum size such that $G[V \setminus S]$ has no $t$-claw with the center vertex in $A$. A primal-dual algorithm approximates \texttt{Min-$t$-OSBCD} within a factor of $t$. We prove that it is \UGC-hard to approximate with a factor better than $t$. We also prove it is approximable within a factor of 2 for dense bipartite graphs. By using these results on \texttt{Min-$t$-OSBCD}, we prove that \texttt{Min-$t$-Claw-Del} is \UGC-hard to approximate within a factor better than $t$, for split graphs. We also consider their complementary maximization problems and prove that they are \APX-complete.Comment: 11 pages and 1 figur

Calculating the error percentage of an automated part-of-speech tagger when analyzing Estonian learner English: an empirical analysis

Teksti sõnaliikideks jaotamine sündis koos lingvistikaga, kuid selle protsessi automatiseerimine on muutunud võimalikuks alles viimastel kümnenditel ning seda tänu arvutite võimsuse kasvule. Tekstitöötluse algoritmid on alates sellest ajast iga aastaga üha paranenud. Selle magistritöö raames pannakse üks selle valdkonna lipulaevadest proovile korpuse peal, mis hõlmab eesti keelt emakeelena kõnelevate inglise keele õppijate tekste (TCELE korpus). Korpuse suurus on antud hetkel ca. 25 000 sõna (127 kirjalikku esseed) ning 11 transkribeeritud intervjuud (~100 minutit). Eesmärk on hinnata TCELE ja muude sarnaste korpuste veaprotsenti. Töö esimeses osas tutvustatakse lugejale korpuse kokkupanemist, annoteerimist ja väljavõtet (ingl. ​retrieval​ ) ning antakse ülevaade sõnaliikide määramisest ja veaprotsendist. Pärast seda antakse ülevaade varasematest uuringutest ning vastatakse muuhulgas, järgnevatele küsimustele: mida on eelnevalt tehtud? Mis olid uuringute leiud? Millised automaatsed märgendajad (ingl. ​taggers) ja sõnaliikide loendeid (ingl. ​tagset​ ) kasutati?http://www.ester.ee/record=b5142572*es

Free-hand sketch recognition by multi-kernel feature learning

Abstract Free-hand sketch recognition has become increasingly popular due to the recent expansion of portable touchscreen devices. However, the problem is non-trivial due to the complexity of internal structures that leads to intra-class variations, coupled with the sparsity in visual cues that results in inter-class ambiguities. In order to address the structural complexity, a novel structured representation for sketches is proposed to capture the holistic structure of a sketch. Moreover, to overcome the visual cue sparsity problem and therefore achieve state-of-the-art recognition performance, we propose a Multiple Kernel Learning (MKL) framework for sketch recognition, fusing several features common to sketches. We evaluate the performance of all the proposed techniques on the most diverse sketch dataset to date (Mathias et al., 2012), and offer detailed and systematic analyses of the performance of different features and representations, including a breakdown by sketch-super-category. Finally, we investigate the use of attributes as a high-level feature for sketches and show how this complements low-level features for improving recognition performance under the MKL framework, and consequently explore novel applications such as attribute-based retrieval

Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs

For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ? V(G) such that G? S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion can be solved in time f_H(t) ? n^{?(1)} assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively. We show that f_H(t) = 2^{?(t^{h-2})} for every graph H on h ? 3 vertices, and that f_H(t) = 2^{?(t)} if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function f_H(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then f_H(t) = 2^{?(t^{h-2})}. We also show that f_H(t) = 2^{?(t^{h})} when H = K_{h,h}, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function f_{C?}(t) for the subgraph version. Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V(H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function f_H(t) for every connected graph H on h vertices: if h ? 2 the problem can be solved in polynomial time; if h ? 3, f_H(t) = 2^{?(t)} if H is a clique, and f_H(t) = 2^{?(t^{h-2})} otherwise

Traffic Management and Congestion Control in the ATM Network Model.

Asynchronous Transfer Mode (ATM) networking technology has been chosen by the International Telegraph and Telephony Consultative Committee (CCITT) for use on future local as well as wide area networks to handle traffic types of a wide range. It is a cell based network architecture that resembles circuit switched networks, providing Quality of Service (QoS) guarantees not normally found on data networks. Although the specifications for the architecture have been continuously evolving, traffic congestion management techniques for ATM networks have not been very well defined yet. This thesis studies the traffic management problem in detail, provides some theoretical understanding and presents a collection of techniques to handle the problem under various operating conditions. A detailed simulation of various ATM traffic types is carried out and the collected data is analyzed to gain an insight into congestion formation patterns. Problems that may arise during migration planning from legacy LANs to ATM technology are also considered. We present an algorithm to identify certain portions of the network that should be upgraded to ATM first. The concept of adaptive burn-in is introduced to help ease the computational costs involved in virtual circuit setup and tear down operations

Hitting forbidden induced subgraphs on bounded treewidth graphs

For a fixed graph $H$, the $H$-IS-Deletion problem asks, given a graph $G$, for the minimum size of a set $S \subseteq V(G)$ such that $G\setminus S$ does not contain $H$ as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph $H$, the smallest function $f_H(t)$ such that $H$-IS-Deletion can be solved in time $f_H(t) \cdot n^{O(1)}$ assuming the Exponential Time Hypothesis (ETH), where $t$ and $n$ denote the treewidth and the number of vertices of the input graph, respectively. We show that $f_H(t) = 2^{O(t^{h-2})}$ for every graph $H$ on $h \geq 3$ vertices, and that $f_H(t) = 2^{O(t)}$ if $H$ is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when $H$ deviates slightly from a clique, the function $f_H(t)$ suffers a sharp jump: if $H$ is obtained from a clique of size $h$ by removing one edge, then $f_H(t) = 2^{\Theta(t^{h-2})}$. We also show that $f_H(t) = 2^{\Omega(t^{h})}$ when $H=K_{h,h}$, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function $f_{C_4}(t)$ for the subgraph version. Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of $G$ is colored with some color from $V(H)$ and we require to hit only induced copies of $H$ with matching colors. In this case, we determine, under the ETH, the function $f_H(t)$ for every connected graph $H$ on $h$ vertices: if $h\leq 2$ the problem can be solved in polynomial time; if $h\geq 3$, $f_H(t) = 2^{\Theta(t)}$ if $H$ is a clique, and $f_H(t) = 2^{\Theta(t^{h-2})}$ otherwise.Comment: 24 pages, 3 figure

Unified almost linear kernels for generalized covering and packing problems on nowhere dense classes

Let $\mathcal{F}$ be a family of graphs, and let $p,r$ be nonnegative integers. The \textsc{$(p,r,\mathcal{F})$-Covering} problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such that $G^p\setminus N_G^r[D]$ has no induced subgraph isomorphic to a graph in $\mathcal{F}$, where $G^p$ is the $p$-th power of $G$. The \textsc{$(p,r,\mathcal{F})$-Packing} problem asks whether for a graph $G$ and an integer $k$, $G^p$ has $k$ induced subgraphs $H_1,\ldots,H_k$ such that each $H_i$ is isomorphic to a graph in $\mathcal{F}$, and for distinct $i,j\in \{1, \ldots, k\}$, the distance between $V(H_i)$ and $V(H_j)$ in $G$ is larger than $r$. We show that for every fixed nonnegative integers $p,r$ and every fixed nonempty finite family $\mathcal{F}$ of connected graphs, the \textsc{$(p,r,\mathcal{F})$-Covering} problem with $p\leq2r+1$ and the \textsc{$(p,r,\mathcal{F})$-Packing} problem with $p\leq2\lfloor r/2\rfloor+1$ admit almost linear kernels on every nowhere dense class of graphs, and admit linear kernels on every class of graphs with bounded expansion, parameterized by the solution size $k$. We obtain the same kernels for their annotated variants. As corollaries, we prove that \textsc{Distance-$r$ Vertex Cover}, \textsc{Distance-$r$ Matching}, \textsc{$\mathcal{F}$-Free Vertex Deletion}, and \textsc{Induced-$\mathcal{F}$-Packing} for any fixed finite family $\mathcal{F}$ of connected graphs admit almost linear kernels on every nowhere dense class of graphs and linear kernels on every class of graphs with bounded expansion. Our results extend the results for \textsc{Distance-$r$ Dominating Set} by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and the result for \textsc{Distance-$r$ Independent Set} by Pilipczuk and Siebertz (EJC 2021).Comment: 38 page