Center-based Clustering under Perturbation Stability

Clustering under most popular objective functions is NP-hard, even to approximate well, and so unlikely to be efficiently solvable in the worst case. Recently, Bilu and Linial \cite{Bilu09} suggested an approach aimed at bypassing this computational barrier by using properties of instances one might hope to hold in practice. In particular, they argue that instances in practice should be stable to small perturbations in the metric space and give an efficient algorithm for clustering instances of the Max-Cut problem that are stable to perturbations of size $O(n^{1/2})$. In addition, they conjecture that instances stable to as little as O(1) perturbations should be solvable in polynomial time. In this paper we prove that this conjecture is true for any center-based clustering objective (such as $k$-median, $k$-means, and $k$-center). Specifically, we show we can efficiently find the optimal clustering assuming only stability to factor-3 perturbations of the underlying metric in spaces without Steiner points, and stability to factor $2+\sqrt{3}$ perturbations for general metrics. In particular, we show for such instances that the popular Single-Linkage algorithm combined with dynamic programming will find the optimal clustering. We also present NP-hardness results under a weaker but related condition

Further approximations for Aharoni's rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. With Pelik\'anov\'a and Pokorn\'a, we showed that this conjecture is true if each color class has size ${\Omega}(k\log k)$. In this paper, we present a proof of the conjecture if each color class has size ${\Omega}(k)$, which improved the previous result and is only a constant factor away from Aharoni's conjecture. We also consider what happens when the condition on the number of colors is relaxed

Minimum degree of minimal (\emph{n}-10)-factor-critical graphs

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is called minimal if for any edge $e\in E(G)$, $G-e$ is not $k$-factor-critical. In 1998, O. Favaron and M. Shi conjectured that every minimal $k$-factor-critical graph of order $n$ has the minimum degree $k+1$ and confirmed it for $k=1, n-2, n-4$ and $n-6$. By using a novel approach, we have confirmed it for $k = n - 8$ in a previous paper. Continuing this method, we prove the conjecture to be true for $k=n-10$ in this paper.Comment: 21 pages,14 figures

On matchings and factors of graphs /

In Section 1, we recall the historical sketch of matching and factor theory of graphs, and also introduce some necessary definitions and notation. In Section 2, we present a sufficient condition for the existence of a (g, f)-factor in graphs with the odd-cycle property, which is simpler than that of Lovasz\u27s (g, f)-Factor Theorem. From this, we derive some further results, and we show that (a) every r-regular graph G with the odd-cycle property has a k-factor, where 0 ≤ k ≤ r and k|V(G)| ≡ 0 (mod 2), (b) every graph G with the strong odd-cycle property with k|V(G)|≡ 0 (mod 2) is k-factorable if and only if G is a km-regular graph for some m ≥ 1, and (c) every regular graph of even order with the strong odd-cycle property is of the second class (i.e. the edge chromatic number is Δ). Chvátal [26] presented the following two conjectures that (1) a graph G has a 2-factor if tough(G) ≥ 3/2, and (2) a graph G has a k-factor if k|V(G)| ≡ 0 (mod 2) and tough(G) ≥ k. Enomoto et.al. [32] proved the second conjecture. They also proved the sharpness of the bound on tough(G) that guarantees the existence of a k-factor. This implies that the first conjecture is false. In Section 3, we show that the result of the second conjecture can be improved in some sense, and the first conjecture is also true if the graph considered has the odd-cycle property. Anderson [3] stated that a graph G of even order has a 1-factor if bind(G) ≥ 4/3, and Katerinis and Woodall [48] proved that a graph G of order n has a k-factor if bind(G) ˃ (2k -I)(n - 1)/(k(n - 2) + 3), where k ≥ 2, n ≥ 4k - 6 and kn ≡ 0 (mod 2). In Section 4, we shall present some similar conditions for the existence of [a, b]-factors. In Section 5, we study the existence of [a, b]-parity-factors in a graph, among which we extend some known theorems from 1-factors to {1, 3, ... , 2n - 1}-factors, or from k-factors to [a, b]-parity-factors. Also, extending Petersen\u27s 2-Factorization Theorem, we proved that a graph is [2a, 2b]-even-factorable if and only if it is a [2na, 2nb]-even-graph for some n ≥ 1. Plummer showed that (a) (in [58]) every graph G of even order is k-extendable if tough(G) ˃ k, and (b) (in [59]) every (2k+1)-connected graph G is k-extendable if G is K1,3-free, respectively. In Section 6, we give a counterpart of the former in terms of binding number, and extend the latter from K1,3-free graphs to K1,n-free graphs. Furthermore, we present a result toward the problem, posed by Saito [61] and Plummer [60], of characterizing the graphs that are maximal k-extendable

Maximum Edge-Disjoint Paths in kk-sums of Graphs

We consider the approximability of the maximum edge-disjoint paths problem (MEDP) in undirected graphs, and in particular, the integrality gap of the natural multicommodity flow based relaxation for it. The integrality gap is known to be $\Omega(\sqrt{n})$ even for planar graphs due to a simple topological obstruction and a major focus, following earlier work, has been understanding the gap if some constant congestion is allowed. In this context, it is natural to ask for which classes of graphs does a constant-factor constant-congestion property hold. It is easy to deduce that for given constant bounds on the approximation and congestion, the class of "nice" graphs is nor-closed. Is the converse true? Does every proper minor-closed family of graphs exhibit a constant factor, constant congestion bound relative to the LP relaxation? We conjecture that the answer is yes. One stumbling block has been that such bounds were not known for bounded treewidth graphs (or even treewidth 3). In this paper we give a polytime algorithm which takes a fractional routing solution in a graph of bounded treewidth and is able to integrally route a constant fraction of the LP solution's value. Note that we do not incur any edge congestion. Previously this was not known even for series parallel graphs which have treewidth 2. The algorithm is based on a more general argument that applies to $k$-sums of graphs in some graph family, as long as the graph family has a constant factor, constant congestion bound. We then use this to show that such bounds hold for the class of $k$-sums of bounded genus graphs