Parameterized Algorithms for Graph Partitioning Problems

In parameterized complexity, a problem instance (I, k) consists of an input I and an extra parameter k. The parameter k usually a positive integer indicating the size of the solution or the structure of the input. A computational problem is called fixed-parameter tractable (FPT) if there is an algorithm for the problem with time complexity O(f(k).nc ), where f(k) is a function dependent only on the input parameter k, n is the size of the input and c is a constant. The existence of such an algorithm means that the problem is tractable for fixed values of the parameter. In this thesis, we provide parameterized algorithms for the following NP-hard graph partitioning problems: (i) Matching Cut Problem: In an undirected graph, a matching cut is a partition of vertices into two non-empty sets such that the edges across the sets induce a matching. The matching cut problem is the problem of deciding whether a given graph has a matching cut. The Matching Cut problem is expressible in monadic second-order logic (MSOL). The MSOL formulation, together with Courcelle’s theorem implies linear time solvability on graphs with bounded tree-width. However, this approach leads to a running time of f(||ϕ||, t) · n, where ||ϕ|| is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. The dependency of f(||ϕ||, t) on ||ϕ|| can be as bad as a tower of exponentials. In this thesis we give a single exponential algorithm for the Matching Cut problem with tree-width alone as the parameter. The running time of the algorithm is 2O(t) · n. This answers an open question posed by Kratsch and Le [Theoretical Computer Science, 2016]. We also show the fixed parameter tractability of the Matching Cut problem when parameterized by neighborhood diversity or other structural parameters. (ii) H-Free Coloring Problems: In an undirected graph G for a fixed graph H, the H-Free q-Coloring problem asks to color the vertices of the graph G using at most q colors such that none of the color classes contain H as an induced subgraph. That is every color class is H-free. This is a generalization of the classical q-Coloring problem, which is to color the vertices of the graph using at most q colors such that no pair of adjacent vertices are of the same color. The H-Free Chromatic Number is the minimum number of colors required to H-free color the graph. For a fixed q, the H-Free q-Coloring problem is expressible in monadic secondorder logic (MSOL). The MSOL formulation leads to an algorithm with time complexity f(||ϕ||, t) · n, where ||ϕ|| is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. In this thesis we present the following explicit combinatorial algorithms for H-Free Coloring problems: • An O(q O(t r ) · n) time algorithm for the general H-Free q-Coloring problem, where r = |V (H)|. • An O(2t+r log t · n) time algorithm for Kr-Free 2-Coloring problem, where Kr is a complete graph on r vertices. The above implies an O(t O(t r ) · n log t) time algorithm to compute the H-Free Chromatic Number for graphs with tree-width at most t. Therefore H-Free Chromatic Number is FPT with respect to tree-width. We also address a variant of H-Free q-Coloring problem which we call H-(Subgraph)Free q-Coloring problem, which is to color the vertices of the graph such that none of the color classes contain H as a subgraph (need not be induced). We present the following algorithms for H-(Subgraph)Free q-Coloring problems. • An O(q O(t r ) · n) time algorithm for the general H-(Subgraph)Free q-Coloring problem, which leads to an O(t O(t r ) · n log t) time algorithm to compute the H- (Subgraph)Free Chromatic Number for graphs with tree-width at most t. • An O(2O(t 2 ) · n) time algorithm for C4-(Subgraph)Free 2-Coloring, where C4 is a cycle on 4 vertices. • An O(2O(t r−2 ) · n) time algorithm for {Kr\e}-(Subgraph)Free 2-Coloring, where Kr\e is a graph obtained by removing an edge from Kr. • An O(2O((tr2 ) r−2 ) · n) time algorithm for Cr-(Subgraph)Free 2-Coloring problem, where Cr is a cycle of length r. (iii) Happy Coloring Problems: In a vertex-colored graph, an edge is happy if its endpoints have the same color. Similarly, a vertex is happy if all its incident edges are happy. we consider the algorithmic aspects of the following Maximum Happy Edges (k-MHE) problem: given a partially k-colored graph G, find an extended full k-coloring of G such that the number of happy edges are maximized. When we want to maximize the number of happy vertices, the problem is known as Maximum Happy Vertices (k-MHV). We show that both k-MHE and k-MHV admit polynomial-time algorithms for trees. We show that k-MHE admits a kernel of size k + `, where ` is the natural parameter, the number of happy edges. We show the hardness of k-MHE and k-MHV for some special graphs such as split graphs and bipartite graphs. We show that both k-MHE and k-MHV are tractable for graphs with bounded tree-width and graphs with bounded neighborhood diversity. vii In the last part of the thesis we present an algorithm for the Replacement Paths Problem which is defined as follows: Let G (|V (G)| = n and |E(G)| = m) be an undirected graph with positive edge weights. Let PG(s, t) be a shortest s − t path in G. Let l be the number of edges in PG(s, t). The Edge Replacement Path problem is to compute a shortest s − t path in G\{e}, for every edge e in PG(s, t). The Node Replacement Path problem is to compute a shortest s−t path in G\{v}, for every vertex v in PG(s, t). We present an O(TSP T (G) + m + l 2 ) time and O(m + l 2 ) space algorithm for both the problems, where TSP T (G) is the asymptotic time to compute a single source shortest path tree in G. The proposed algorithm is simple and easy to implement

Average Sensitivity of Graph Algorithms

In modern applications of graphs algorithms, where the graphs of interest are large and dynamic, it is unrealistic to assume that an input representation contains the full information of a graph being studied. Hence, it is desirable to use algorithms that, even when only a (large) subgraph is available, output solutions that are close to the solutions output when the whole graph is available. We formalize this idea by introducing the notion of average sensitivity of graph algorithms, which is the average earth mover's distance between the output distributions of an algorithm on a graph and its subgraph obtained by removing an edge, where the average is over the edges removed and the distance between two outputs is the Hamming distance. In this work, we initiate a systematic study of average sensitivity. After deriving basic properties of average sensitivity such as composition, we provide efficient approximation algorithms with low average sensitivities for concrete graph problems, including the minimum spanning forest problem, the global minimum cut problem, the minimum $s$-$t$ cut problem, and the maximum matching problem. In addition, we prove that the average sensitivity of our global minimum cut algorithm is almost optimal, by showing a nearly matching lower bound. We also show that every algorithm for the 2-coloring problem has average sensitivity linear in the number of vertices. One of the main ideas involved in designing our algorithms with low average sensitivity is the following fact; if the presence of a vertex or an edge in the solution output by an algorithm can be decided locally, then the algorithm has a low average sensitivity, allowing us to reuse the analyses of known sublinear-time algorithms and local computation algorithms (LCAs). Using this connection, we show that every LCA for 2-coloring has linear query complexity, thereby answering an open question.Comment: 39 pages, 1 figur

Semi-algebraic colorings of complete graphs

We consider $m$-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case $m = 2$ was first studied by Alon et al., who applied this framework to obtain surprisingly strong Ramsey-type results for intersection graphs of geometric objects and for other graphs arising in computational geometry. Considering larger values of $m$ is relevant, e.g., to problems concerning the number of distinct distances determined by a point set. For $p\ge 3$ and $m\ge 2$, the classical Ramsey number $R(p;m)$ is the smallest positive integer $n$ such that any $m$-coloring of the edges of $K_n$, the complete graph on $n$ vertices, contains a monochromatic $K_p$. It is a longstanding open problem that goes back to Schur (1916) to decide whether $R(p;m)=2^{O(m)}$, for a fixed $p$. We prove that this is true if each color class is defined semi-algebraically with bounded complexity. The order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle {\em et al.}, and on a Szemer\'edi-type regularity lemma for multicolored semi-algebraic graphs, which is of independent interest. The same technique is used to address the semi-algebraic variant of a more general Ramsey-type problem of Erd\H{o}s and Shelah

Complexity of Computing the Anti-Ramsey Numbers for Paths

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erd\" os, Simonovits and S\' os. For given graphs $G$ and $H$ the \emph{anti-Ramsey number} $\textrm{ar}(G,H)$ is defined to be the maximum number $k$ such that there exists an assignment of $k$ colors to the edges of $G$ in which every copy of $H$ in $G$ has at least two edges with the same color. There are works on the computational complexity of the problem when $H$ is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number $\textrm{ar}(G,P_k)$, where $P_k$ is a path of length $k$. First, we observe that when $k = \Omega(n)$, the problem is hard; hence, the challenging part is the computational complexity of the problem when $k$ is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing $\textrm{ar}(G,P_k)$ for every integer $k>2$ is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating $\textrm{ar}(G,P_3)$ to a factor of $n^{-1/2 - \epsilon}$ is hard already in $3$-partite graphs, unless P=NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant $k$. Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. We introduce the notion of color connected coloring and employing this structural property. We obtain a linear time algorithm to compute $\textrm{ar}(G,P_k)$, for every integer $k$, when the host graph, $G$, is a tree

Synchronizace automatů

V této práci studujeme Trahtmanův důkaz Problému barvení cesty a sou- visející algoritmus. Pro každý silne souvislý orientovaný multigraf výstupně d s periodou 1 existuje synchronizující barvení. Béal a Perrin dokázali, že Trahtma- nův důkaz lze jednoduše zobecnit pro každou periodu a k-synchronizující barvení. Ukážeme dané zobecnění. Trahtmanův důkaz je konstruktivní a je založený na hledání barvení s netriviální stabilní dvojicí. Dokážeme, že pokud v Pα je právě jeden maximální strom, potom barvení má netriviální stabilní dvojici. Podgraf Pα obsahuje všechny hrany se stejnou barvou. Ukážeme jak také barvení na- lézt. Potom popíšeme algoritmy na nalezení k-synchronizujícího barvení. První je přímočarou aplikací tvrzení z Trahtmanova důkazu se složitostí O((n − k)dn2 ). Potom ukážeme Trahtmanovu redukci a Béalin a Perrinův algoritmus založený na Trahtmanově důkazu, ale se složitostí O((n − k)dn), kde n je počet vrcholů. 1In this thesis we study Trahtman's proof of Road coloring problem and related algorithm. For every strongly connected directed multigraph with outdegree d and period 1, there exists synchronizing coloring. Béal and Perrin prove that Trahtman's proof can be simply generalized for every period and k-synchronizing coloring. We show generalized proof too. Trahtman's proof is constructive and is based on finding coloring with nontrivial stable states. We prove if there is only one maximal tree in Pα then the coloring is with nontrivial stable states. Subgraph Pα contains all edges with same color. We show how to find such coloring. Then we describe algorithms for finding k-synchronizing coloring. First algorithm uses proposition from Trahtman's proof with time complexity O((n−k)dn2 ). Then we show Trahtman's reduction and Béal and Perrin's algorithm based on Trahtman's proof but time complexity is O((n − k)dn) where n is the number of vertices. 1Department of AlgebraKatedra algebryFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Parameterized Complexity of Equitable Coloring

A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex coloring, being $\mathsf{NP\text{-}Complete}$ even for cographs and interval graphs. In this work, we prove that it is $\mathsf{W[1]\text{-}Hard}$ for block graphs and for disjoint union of split graphs when parameterized by the number of colors; and $\mathsf{W[1]\text{-}Hard}$ for $K_{1,4}$-free interval graphs when parameterized by treewidth, number of colors and maximum degree, generalizing a result by Fellows et al. (2014) through a much simpler reduction. Using a previous result due to Dominique de Werra (1985), we establish a dichotomy for the complexity of equitable coloring of chordal graphs based on the size of the largest induced star. Finally, we show that \textsc{equitable coloring} is $\mathsf{FPT}$ when parameterized by the treewidth of the complement graph

Complexity of C_k-Coloring in Hereditary Classes of Graphs

For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f:V(G) -> V(H) such that for every edge uv in E(G) it holds that f(u)f(v)in E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of P_t-free graphs. We show that for every odd k >= 5 the C_k-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P_9-free graphs. On the other hand, we prove that the extension version of C_k-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw