Parameterized Complexity of Edge Interdiction Problems

We study the parameterized complexity of interdiction problems in graphs. For an optimization problem on graphs, one can formulate an interdiction problem as a game consisting of two players, namely, an interdictor and an evader, who compete on an objective with opposing interests. In edge interdiction problems, every edge of the input graph has an interdiction cost associated with it and the interdictor interdicts the graph by modifying the edges in the graph, and the number of such modifications is constrained by the interdictor's budget. The evader then solves the given optimization problem on the modified graph. The action of the interdictor must impede the evader as much as possible. We focus on edge interdiction problems related to minimum spanning tree, maximum matching and shortest paths. These problems arise in different real world scenarios. We derive several fixed-parameter tractability and W[1]-hardness results for these interdiction problems with respect to various parameters. Next, we show close relation between interdiction problems and partial cover problems on bipartite graphs where the goal is not to cover all elements but to minimize/maximize the number of covered elements with specific number of sets. Hereby, we investigate the parameterized complexity of several partial cover problems on bipartite graphs

Vertex and edge covers with clustering properties: complexity and algorithms

We consider the concepts of a t-total vertex cover and a t-total edge cover (t≥1), which generalise the notions of a vertex cover and an edge cover, respectively. A t-total vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has at least t vertices (edges). These definitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present NP-completeness and approximability results (both upper and lower bounds) and FTP algorithms for problems concerned with finding the minimum size of a t-total vertex cover, t-total edge cover and connected vertex cover, in particular improving on a previous FTP algorithm for the latter problem

Two Approaches to Sidorenko's Conjecture

Sidorenko's conjecture states that for every bipartite graph $H$ on $\{1,\cdots,k\}$, $\int \prod_{(i,j)\in E(H)} h(x_i, y_j) d\mu^{|V(H)|} \ge \left( \int h(x,y) \,d\mu^2 \right)^{|E(H)|}$ holds, where $\mu$ is the Lebesgue measure on $[0,1]$ and $h$ is a bounded, non-negative, symmetric, measurable function on $[0,1]^2$. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph $H$ to a graph $G$ is asymptotically at least the expected number of homomorphisms from $H$ to the Erd\H{o}s-R\'{e}nyi random graph with the same expected edge density as $G$. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph $H$ with bipartition $A \cup B$ is tree-arrangeable if neighborhoods of vertices in $A$ have a certain tree-like structure. We show that Sidorenko's conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko's conjecture holds if there are two vertices $a_1, a_2$ in $A$ such that each vertex $a \in A$ satisfies $N(a) \subseteq N(a_1)$ or $N(a) \subseteq N(a_2)$, and also implies a recent result of Conlon, Fox, and Sudakov \cite{CoFoSu}. Second, if $T$ is a tree and $H$ is a bipartite graph satisfying Sidorenko's conjecture, then it is shown that the Cartesian product $T \Box H$ of $T$ and $H$ also satisfies Sidorenko's conjecture. This result implies that, for all $d \ge 2$, the $d$-dimensional grid with arbitrary side lengths satisfies Sidorenko's conjecture.Comment: 20 pages, 2 figure

On the decomposition threshold of a given graph

We study the $F$-decomposition threshold $\delta_F$ for a given graph $F$. Here an $F$-decomposition of a graph $G$ is a collection of edge-disjoint copies of $F$ in $G$ which together cover every edge of $G$. (Such an $F$-decomposition can only exist if $G$ is $F$-divisible, i.e. if $e(F)\mid e(G)$ and each vertex degree of $G$ can be expressed as a linear combination of the vertex degrees of $F$.) The $F$-decomposition threshold $\delta_F$ is the smallest value ensuring that an $F$-divisible graph $G$ on $n$ vertices with $\delta(G)\ge(\delta_F+o(1))n$ has an $F$-decomposition. Our main results imply the following for a given graph $F$, where $\delta_F^\ast$ is the fractional version of $\delta_F$ and $\chi:=\chi(F)$: (i) $\delta_F\le \max\{\delta_F^\ast,1-1/(\chi+1)\}$; (ii) if $\chi\ge 5$, then $\delta_F\in\{\delta_F^{\ast},1-1/\chi,1-1/(\chi+1)\}$; (iii) we determine $\delta_F$ if $F$ is bipartite. In particular, (i) implies that $\delta_{K_r}=\delta^\ast_{K_r}$. Our proof involves further developments of the recent `iterative' absorbing approach.Comment: Final version, to appear in the Journal of Combinatorial Theory, Series