A path following algorithm for the graph matching problem

We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We therefore construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore to perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four datasets: simulated graphs, QAPLib, retina vessel images and handwritten chinese characters. In all cases, the results are competitive with the state-of-the-art.Comment: 23 pages, 13 figures,typo correction, new results in sections 4,5,

Minimum Weight Perfect Matching via Blossom Belief Propagation

Max-product Belief Propagation (BP) is a popular message-passing algorithm for computing a Maximum-A-Posteriori (MAP) assignment over a distribution represented by a Graphical Model (GM). It has been shown that BP can solve a number of combinatorial optimization problems including minimum weight matching, shortest path, network flow and vertex cover under the following common assumption: the respective Linear Programming (LP) relaxation is tight, i.e., no integrality gap is present. However, when LP shows an integrality gap, no model has been known which can be solved systematically via sequential applications of BP. In this paper, we develop the first such algorithm, coined Blossom-BP, for solving the minimum weight matching problem over arbitrary graphs. Each step of the sequential algorithm requires applying BP over a modified graph constructed by contractions and expansions of blossoms, i.e., odd sets of vertices. Our scheme guarantees termination in O(n^2) of BP runs, where n is the number of vertices in the original graph. In essence, the Blossom-BP offers a distributed version of the celebrated Edmonds' Blossom algorithm by jumping at once over many sub-steps with a single BP. Moreover, our result provides an interpretation of the Edmonds' algorithm as a sequence of LPs

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

Additive Spanners: A Simple Construction

We consider additive spanners of unweighted undirected graphs. Let $G$ be a graph and $H$ a subgraph of $G$. The most na\"ive way to construct an additive $k$-spanner of $G$ is the following: As long as $H$ is not an additive $k$-spanner repeat: Find a pair $(u,v) \in H$ that violates the spanner-condition and a shortest path from $u$ to $v$ in $G$. Add the edges of this path to $H$. We show that, with a very simple initial graph $H$, this na\"ive method gives additive $6$- and $2$-spanners of sizes matching the best known upper bounds. For additive $2$-spanners we start with $H=\emptyset$ and end with $O(n^{3/2})$ edges in the spanner. For additive $6$-spanners we start with $H$ containing $\lfloor n^{1/3} \rfloor$ arbitrary edges incident to each node and end with a spanner of size $O(n^{4/3})$.Comment: To appear at proceedings of the 14th Scandinavian Symposium and Workshop on Algorithm Theory (SWAT 2014

Exact Weight Subgraphs and the k-Sum Conjecture

We consider the Exact-Weight-H problem of finding a (not necessarily induced) subgraph H of weight 0 in an edge-weighted graph G. We show that for every H, the complexity of this problem is strongly related to that of the infamous k-Sum problem. In particular, we show that under the k-Sum Conjecture, we can achieve tight upper and lower bounds for the Exact-Weight-H problem for various subgraphs H such as matching, star, path, and cycle. One interesting consequence is that improving on the O(n^3) upper bound for Exact-Weight-4-Path or Exact-Weight-5-Path will imply improved algorithms for 3-Sum, 5-Sum, All-Pairs Shortest Paths and other fundamental problems. This is in sharp contrast to the minimum-weight and (unweighted) detection versions, which can be solved easily in time O(n^2). We also show that a faster algorithm for any of the following three problems would yield faster algorithms for the others: 3-Sum, Exact-Weight-3-Matching, and Exact-Weight-3-Star

A Tight Algorithm for Strongly Connected Steiner Subgraph On Two Terminals With Demands

Given an edge-weighted directed graph $G=(V,E)$ on $n$ vertices and a set $T=\{t_1, t_2, \ldots, t_p\}$ of $p$ terminals, the objective of the \scss ($p$-SCSS) problem is to find an edge set $H\subseteq E$ of minimum weight such that $G[H]$ contains an $t_{i}\rightarrow t_j$ path for each $1\leq i\neq j\leq p$. In this paper, we investigate the computational complexity of a variant of $2$-SCSS where we have demands for the number of paths between each terminal pair. Formally, the \sharinggeneral problem is defined as follows: given an edge-weighted directed graph $G=(V,E)$ with weight function $\omega: E\rightarrow \mathbb{R}^{\geq 0}$, two terminal vertices $s, t$, and integers $k_1, k_2$ ; the objective is to find a set of $k_1$ paths $F_1, F_2, \ldots, F_{k_1}$ from $s\leadsto t$ and $k_2$ paths $B_1, B_2, \ldots, B_{k_2}$ from $t\leadsto s$ such that $\sum_{e\in E} \omega(e)\cdot \phi(e)$ is minimized, where $\phi(e)= \max \Big\{|\{i\in [k_1] : e\in F_i\}|\ ,\ |\{j\in [k_2] : e\in B_j\}|\Big\}$. For each $k\geq 1$, we show the following: The \sharing problem can be solved in $n^{O(k)}$ time. A matching lower bound for our algorithm: the \sharing problem does not have an $f(k)\cdot n^{o(k)}$ algorithm for any computable function $f$, unless the Exponential Time Hypothesis (ETH) fails. Our algorithm for \sharing relies on a structural result regarding an optimal solution followed by using the idea of a "token game" similar to that of Feldman and Ruhl. We show with an example that the structural result does not hold for the \sharinggeneral problem if $\min\{k_1, k_2\}\geq 2$. Therefore \sharing is the most general problem one can attempt to solve with our techniques.Comment: To appear in Algorithmica. An extended abstract appeared in IPEC '1

Forbidden Substructures in Graphs and Trigraphs, and Related Coloring Problems

Given a graph G, χ(G) denotes the chromatic number of G, and ω(G) denotes the clique number of G (i.e. the maximum number of pairwise adjacent vertices in G). A graph G is perfect provided that for every induced subgraph H of G, χ(H) = ω(H). This thesis addresses several problems from the theory of perfect graphs and generalizations of perfect graphs. The bull is a five-vertex graph consisting of a triangle and two vertex-disjoint pendant edges; a graph is said to be bull-free provided that no induced subgraph of it is a bull. The first result of this thesis is a structure theorem for bull-free perfect graphs. This is joint work with Chudnovsky, and it first appeared in [12]. The second result of this thesis is a decomposition theorem for bull-free perfect graphs, which we then use to give a polynomial time combinatorial coloring algorithm for bull-free perfect graphs. We remark that de Figueiredo and Maffray [33] previously solved this same problem, however, the algorithm presented in this thesis is faster than the algorithm from [33]. We note that a decomposition theorem that is very similar (but slightly weaker) than the one from this thesis was originally proven in [52], however, the proof in this thesis is significantly different from the one in [52]. The algorithm from this thesis is very similar to the one from [52]. A class G of graphs is said to be χ-bounded provided that there exists a function f such that for all G in G, and all induced subgraphs H of G, we have that χ(H) ≤ f(ω(H)). χ-bounded classes were introduced by Gyarfas [41] as a generalization of the class of perfect graphs (clearly, the class of perfect graphs is χ-bounded by the identity function). Given a graph H, we denote by Forb*(H) the class of all graphs that do not contain any subdivision of H as an induced subgraph. In [57], Scott proved that Forb*(T) is χ-bounded for every tree T, and he conjectured that Forb*(H) is χ-bounded for every graph H. Recently, a group of authors constructed a counterexample to Scott's conjecture [51]. This raises the following question: for which graphs H is Scott's conjecture true? In this thesis, we present the proof of Scott's conjecture for the cases when H is the paw (i.e. a four-vertex graph consisting of a triangle and a pendant edge), the bull, and a necklace (i.e. a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge). This is joint work with Chudnovsky, Scott, and Trotignon, and it originally appeared in [13]. Finally, we consider several operations (namely, "substitution," "gluing along a clique," and "gluing along a bounded number of vertices"), and we show that the closure of a χ-bounded class under any one of them, as well as under certain combinations of these three operations (in particular, the combination of substitution and gluing along a clique, as well as the combination of gluing along a clique and gluing along a bounded number of vertices) is again χ-bounded. This is joint work with Chudnovsky, Scott, and Trotignon, and it originally appeared in [14]