Efficient Approximation Schemes for Uniform-Cost Clustering Problems in Planar Graphs

We consider the k-Median problem on planar graphs: given an edge-weighted planar graph G, a set of clients C subseteq V(G), a set of facilities F subseteq V(G), and an integer parameter k, the task is to find a set of at most k facilities whose opening minimizes the total connection cost of clients, where each client contributes to the cost with the distance to the closest open facility. We give two new approximation schemes for this problem: - FPT Approximation Scheme: for any epsilon>0, in time 2^{O(k epsilon^{-3} log (k epsilon^{-1}))}* n^O(1) we can compute a solution that has connection cost at most (1+epsilon) times the optimum, with high probability. - Efficient Bicriteria Approximation Scheme: for any epsilon>0, in time 2^{O(epsilon^{-5} log (epsilon^{-1}))}* n^O(1) we can compute a set of at most (1+epsilon)k facilities whose opening yields connection cost at most (1+epsilon) times the optimum connection cost for opening at most k facilities, with high probability. As a direct corollary of the second result we obtain an EPTAS for Uniform Facility Location on planar graphs, with same running time. Our main technical tool is a new construction of a "coreset for facilities" for k-Median in planar graphs: we show that in polynomial time one can compute a subset of facilities F_0 subseteq F of size k * (log n/epsilon)^O(epsilon^{-3}) with a guarantee that there is a (1+epsilon)-approximate solution contained in F_0

Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth

We give a fixed-parameter tractable algorithm that, given a parameter $k$ and two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running time of the algorithm on an $n$-vertex graph is $2^{O(k^5\log k)}\cdot n^5$, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in $2^{O(k^5\log k)}\cdot n^5$ time that, for a given graph $G$ on $n$ vertices, either concludes that the treewidth of $G$ is at least $k$, or: * finds in an isomorphic-invariant way a graph $\mathfrak{c}(G)$ that is isomorphic to $G$; * finds an isomorphism-invariant construction term --- an algebraic expression that encodes $G$ together with a tree decomposition of $G$ of width $O(k^4)$. Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for $G_1$ and $G_2$ are equal.Comment: Full version of a paper presented at FOCS 201

Polynomial-time algorithm for Maximum Weight Independent Set on P6P_6-free graphs

In the classic Maximum Weight Independent Set problem we are given a graph $G$ with a nonnegative weight function on vertices, and the goal is to find an independent set in $G$ of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any $P_6$-free graph, that is, a graph that has no path on $6$ vertices as an induced subgraph. This improves the polynomial-time algorithm on $P_5$-free graphs of Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on $P_6$-free graphs of Lokshtanov et al (SODA 2016). The main technical contribution leading to our main result is enumeration of a polynomial-size family $\mathcal{F}$ of vertex subsets with the following property: for every maximal independent set $I$ in the graph, $\mathcal{F}$ contains all maximal cliques of some minimal chordal completion of $G$ that does not add any edge incident to a vertex of $I$

Edge Bipartization Faster Than 2^k

In the Edge Bipartization problem one is given an undirected graph $G$ and an integer $k$, and the question is whether $k$ edges can be deleted from $G$ so that it becomes bipartite. In 2006, Guo et al. [J. Comput. Syst. Sci., 72(8):1386-1396, 2006] proposed an algorithm solving this problem in time $O(2^k m^2)$; today, this algorithm is a textbook example of an application of the iterative compression technique. Despite extensive progress in the understanding of the parameterized complexity of graph separation problems in the recent years, no significant improvement upon this result has been yet reported. We present an algorithm for Edge Bipartization that works in time $O(1.977^k nm)$, which is the first algorithm with the running time dependence on the parameter better than $2^k$. To this end, we combine the general iterative compression strategy of Guo et al. [J. Comput. Syst. Sci., 72(8):1386-1396, 2006], the technique proposed by Wahlstrom [SODA 2014, 1762-1781] of using a polynomial-time solvable relaxation in the form of a Valued Constraint Satisfaction Problem to guide a bounded-depth branching algorithm, and an involved Measure & Conquer analysis of the recursion tree