A linear-time algorithm for finding a complete graph minor in a dense graph

Let g(t) be the minimum number such that every graph G with average degree d(G) \geq g(t) contains a K_{t}-minor. Such a function is known to exist, as originally shown by Mader. Kostochka and Thomason independently proved that g(t) \in \Theta(t*sqrt{log t}). This article shows that for all fixed \epsilon > 0 and fixed sufficiently large t \geq t(\epsilon), if d(G) \geq (2+\epsilon)g(t) then we can find this K_{t}-minor in linear time. This improves a previous result by Reed and Wood who gave a linear-time algorithm when d(G) \geq 2^{t-2}.Comment: 6 pages, 0 figures; Clarification added in several places, no change to arguments or result

On Adaptive Algorithms for Maximum Matching

In the fundamental Maximum Matching problem the task is to find a maximum cardinality set of pairwise disjoint edges in a given undirected graph. The fastest algorithm for this problem, due to Micali and Vazirani, runs in time O(sqrt{n}m) and stands unbeaten since 1980. It is complemented by faster, often linear-time, algorithms for various special graph classes. Moreover, there are fast parameterized algorithms, e.g., time O(km log n) relative to tree-width k, which outperform O(sqrt{n}m) when the parameter is sufficiently small. We show that the Micali-Vazirani algorithm, and in fact any algorithm following the phase framework of Hopcroft and Karp, is adaptive to beneficial input structure. We exhibit several graph classes for which such algorithms run in linear time O(n+m). More strongly, we show that they run in time O(sqrt{k}m) for graphs that are k vertex deletions away from any of several such classes, without explicitly computing an optimal or approximate deletion set; before, most such bounds were at least Omega(km). Thus, any phase-based matching algorithm with linear-time phases obliviously interpolates between linear time for k=O(1) and the worst case of O(sqrt{n}m) when k=Theta(n). We complement our findings by proving that the phase framework by itself still allows Omega(sqrt{n}) phases, and hence time Omega(sqrt{n}m), even on paths, cographs, and bipartite chain graphs

Separator Theorems for Minor-Free and Shallow Minor-Free Graphs with Applications

Alon, Seymour, and Thomas generalized Lipton and Tarjan's planar separator theorem and showed that a $K_h$-minor free graph with $n$ vertices has a separator of size at most $h^{3/2}\sqrt n$. They gave an algorithm that, given a graph $G$ with $m$ edges and $n$ vertices and given an integer $h\geq 1$, outputs in $O(\sqrt{hn}m)$ time such a separator or a $K_h$-minor of $G$. Plotkin, Rao, and Smith gave an $O(hm\sqrt{n\log n})$ time algorithm to find a separator of size $O(h\sqrt{n\log n})$. Kawarabayashi and Reed improved the bound on the size of the separator to $h\sqrt n$ and gave an algorithm that finds such a separator in $O(n^{1 + \epsilon})$ time for any constant $\epsilon > 0$, assuming $h$ is constant. This algorithm has an extremely large dependency on $h$ in the running time (some power tower of $h$ whose height is itself a function of $h$), making it impractical even for small $h$. We are interested in a small polynomial time dependency on $h$ and we show how to find an $O(h\sqrt{n\log n})$-size separator or report that $G$ has a $K_h$-minor in O(\poly(h)n^{5/4 + \epsilon}) time for any constant $\epsilon > 0$. We also present the first O(\poly(h)n) time algorithm to find a separator of size $O(n^c)$ for a constant $c < 1$. As corollaries of our results, we get improved algorithms for shortest paths and maximum matching. Furthermore, for integers $\ell$ and $h$, we give an $O(m + n^{2 + \epsilon}/\ell)$ time algorithm that either produces a $K_h$-minor of depth $O(\ell\log n)$ or a separator of size at most $O(n/\ell + \ell h^2\log n)$. This improves the shallow minor algorithm of Plotkin, Rao, and Smith when $m = \Omega(n^{1 + \epsilon})$. We get a similar running time improvement for an approximation algorithm for the problem of finding a largest $K_h$-minor in a given graph.Comment: To appear at FOCS 201

Maximum Matchings in Geometric Intersection Graphs

Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O(ρ3ω/2nω/2) time with high probability, where ρ is the density of the geometric objects and ω>2 is a constant such that n×n matrices can be multiplied in O(nω) time. The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O(nω/2) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [1,Ψ] can be found in O(Ψ6log11n+Ψ12ωnω/2) time with high probability