Finding Induced Subgraphs via Minimal Triangulations

Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulations problems including Minimum Fill-in and Treewidth. We discover unexpected applications of these notions to the field of moderate exponential algorithms. In particular, we show that given an n-vertex graph G together with its set of potential maximal cliques Pi_G, and an integer t, it is possible in time |Pi_G| * n^(O(t)) to find a maximum induced subgraph of treewidth t in G; and for a given graph F of treewidth t, to decide if G contains an induced subgraph isomorphic to F. Combined with an improved algorithm enumerating all potential maximal cliques in time O(1.734601^n), this yields that both problems are solvable in time 1.734601^n * n^(O(t)).Comment: 14 page

Large induced subgraphs via triangulations and CMSO

We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models \phi. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from \Gamma\ as a minor. 3) "Independent \Pi-packing", where for a fixed finite set of connected graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of connected components, such that each connected component of G[F] is isomorphic to some graph from \Pi. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and \phi\ only. We also show how a similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with polynomial number of minimal separators

Long Circuits and Large Euler Subgraphs

An undirected graph is Eulerian if it is connected and all its vertices are of even degree. Similarly, a directed graph is Eulerian, if for each vertex its in-degree is equal to its out-degree. It is well known that Eulerian graphs can be recognized in polynomial time while the problems of finding a maximum Eulerian subgraph or a maximum induced Eulerian subgraph are NP-hard. In this paper, we study the parameterized complexity of the following Euler subgraph problems: - Large Euler Subgraph: For a given graph G and integer parameter k, does G contain an induced Eulerian subgraph with at least k vertices? - Long Circuit: For a given graph G and integer parameter k, does G contain an Eulerian subgraph with at least k edges? Our main algorithmic result is that Large Euler Subgraph is fixed parameter tractable (FPT) on undirected graphs. We find this a bit surprising because the problem of finding an induced Eulerian subgraph with exactly k vertices is known to be W[1]-hard. The complexity of the problem changes drastically on directed graphs. On directed graphs we obtained the following complexity dichotomy: Large Euler Subgraph is NP-hard for every fixed k>3 and is solvable in polynomial time for k<=3. For Long Circuit, we prove that the problem is FPT on directed and undirected graphs

Bidimensionality and Geometric Graphs

In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. We proceed to show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on fail for disk graphs and that therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed d

Preventing Unraveling in Social Networks Gets Harder

The behavior of users in social networks is often observed to be affected by the actions of their friends. Bhawalkar et al. \cite{bhawalkar-icalp} introduced a formal mathematical model for user engagement in social networks where each individual derives a benefit proportional to the number of its friends which are engaged. Given a threshold degree $k$ the equilibrium for this model is a maximal subgraph whose minimum degree is $\geq k$. However the dropping out of individuals with degrees less than $k$ might lead to a cascading effect of iterated withdrawals such that the size of equilibrium subgraph becomes very small. To overcome this some special vertices called "anchors" are introduced: these vertices need not have large degree. Bhawalkar et al. \cite{bhawalkar-icalp} considered the \textsc{Anchored $k$-Core} problem: Given a graph $G$ and integers $b, k$ and $p$ do there exist a set of vertices $B\subseteq H\subseteq V(G)$ such that $|B|\leq b, |H|\geq p$ and every vertex $v\in H\setminus B$ has degree at least $k$ is the induced subgraph $G[H]$. They showed that the problem is NP-hard for $k\geq 2$ and gave some inapproximability and fixed-parameter intractability results. In this paper we give improved hardness results for this problem. In particular we show that the \textsc{Anchored $k$-Core} problem is W[1]-hard parameterized by $p$, even for $k=3$. This improves the result of Bhawalkar et al. \cite{bhawalkar-icalp} (who show W[2]-hardness parameterized by $b$) as our parameter is always bigger since $p\geq b$. Then we answer a question of Bhawalkar et al. \cite{bhawalkar-icalp} by showing that the \textsc{Anchored $k$-Core} problem remains NP-hard on planar graphs for all $k\geq 3$, even if the maximum degree of the graph is $k+2$. Finally we show that the problem is FPT on planar graphs parameterized by $b$ for all $k\geq 7$.Comment: To appear in AAAI 201