Narrow sieves, representative sets and divide-and-color are three breakthrough techniques related to color coding, which led to the design of extremely fast parameterized algorithms. We present a novel family of strategies for applying mixtures of them. This includes: (a) a mix of representative sets and narrow sieves; (b) a faster computation of representative sets under certain separateness conditions, mixed with divide-and-color and a new technique, called “balanced cutting”; (c) two mixtures of representative sets and a new technique, called “unbalanced cutting”. We demonstrate our strategies by obtaining, among other results, significantly faster algorithms for k-Internal Out-Branching and Weighted 3-Set k-Packing, and a general framework for speeding-up the previous best deterministic algorithms for k-Path, k-Tree, r-Dimensional k-Matching, Graph Motif and Partial Cover.

Mixing Color Coding-Related Techniques

We consider the $\mathcal{NP}$-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form $\mathcal{O}^{*}(c^{n})$ with c≤2. For graphs with bounded degree, c<2. The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of $\mathcal{O}(1.8612^{n})$ when analyzed with respect to the number of vertices. We also show that its running time is $2.1364^{k} n^{\mathcal{O}(1)}$ when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.

/pdf/exact-and-parameterized-algorithms-for-max-internal-spanning-23mw9n46s3.pdf

Exact and Parameterized Algorithms for Max Internal Spanning Tree

The maximum internal spanning tree problem asks for a spanning tree of a given graph that has the maximum number of internal vertices among all spanning trees of this graph. In its parameterized version, we are interested in whether the graph has a spanning tree with at least k internal vertices. Fomin et al. (2013) 4 crafted a very ingenious reduction rule, and showed that a simple application of this rule is sufficient to yield a 3k-vertex kernel, implying an O * ( 8 k ) -time parameterized algorithm. Using depth-2 local search, Knauer and Spoerhase (2015) 9 developed a (5/3)-approximation algorithm for the optimization version. We try deeper local search: We conduct a thorough combinatorial analysis on the obtained spanning trees and explore their algorithmic consequences. We first observe that from the spanning tree obtained by depth-3 local search, one can easily find a reducible structure and apply the reduction rule of Fomin et al. This gives an improved kernel of 2k vertices, and as a by-product, a deterministic algorithm running in time O * ( 4 k ) . We then go even deeper by considering the spanning tree obtained by depth-5 local search. It is shown that the number of internal vertices of this spanning tree is at least 2/3 of the maximum number a spanning tree can have, thereby delivering an improved approximation algorithm with ratio 1.5 for the problem.

Deeper local search for parameterized and approximation algorithms for maximum internal spanning tree

Given an undirected connected graph G we consider the problem of finding a spanning tree of G which has a maximum number of internal (non-leaf) vertices among all spanning trees of G. This problem, called Maximum Internal Spanning Tree problem, is clearly NP-hard since it is a generalization of the Hamiltonian Path problem. From the optimization point of view the Maximum Internal Spanning Tree problem is equivalent to the Minimum Leaf Spanning Tree problem. However, the two problems have different approximability properties. Lu and Ravi proved that the latter has no constant factor approximation-unless P = NP-, while Salamon and Wiener gave a linear-time 2-approximation algorithm for the Maximum Internal Spanning Tree problem. In this paper, we improve this approximation ratio by giving an O(|V(G)|^4)-time7/4-approximation algorithm for graphs without pendant vertices. Our approach is based on the successive execution of local improvement steps. We use a linear programming formulation and a primal-dual technique to prove the approximation ratio. We also investigate the vertex-weighted case, that is to find a spanning tree of a vertex-weighted graph G in which the weight sum of internal vertices is maximal among all spanning trees of G. For this problem we present a ([email protected](G)-3)-approximation algorithm, where @D(G) is the maximum vertex-degree of G. A slight modification of this algorithm ensures a 2-approximation whenever the input graph is claw-free. Both algorithms run in O(|V(G)|^4) time for graphs with no pendant vertices.

Approximating the Maximum Internal Spanning Tree problem

We examine the problem of determining a spanning tree of a given graph such that the number of internal nodes is maximum The best approximation algorithm known so far for this problem is due to Prieto and Sloper and has a ratio of 2 For graphs without pendant nodes, Salamon has lowered this factor to $\frac74$ by means of local search However, the approximative behaviour of his algorithm on general graphs has remained open In this paper we show that a simplified and faster version of Salamon's algorithm yields a $\frac53$-approximation even on general graphs In addition to this, we investigate a node weighted variant of the problem for which Salamon achieved a ratio of 2·Δ(G ) *** 3 Modifying Salamon's approach we obtain a factor of 3 + *** for any *** > 0 We complement our results with worst case instances showing that our bounds are tight

Better Approximation Algorithms for the Maximum Internal Spanning Tree Problem

We consider the MaximumInternalSpanningTree problem which is to find a spanning tree of a given graph with a maximum number of non-leaf nodes. From an optimization point of view, this problem is equivalent to the MinimumLeafSpanningTree problem, and is NP-hard as being a generalization of the HamiltonianPath problem. Although there is no constant factor approximation for the MinimumLeafSpanningTree problem [1], MaximumInternalSpanningTree can be approximated within a factor of 2 [2].

In this paper we improve this factor by giving a 7/4 -approximation algorithm. We also investigate the node-weighted case, when the weighted sum of the internal nodes is to be maximized. For this problem, we give a (2Δ - 3)-approximation for general graphs, and a 2-approximation for claw-free graphs. All our algorithms are based on local improvement steps.

Approximation algorithms for the maximum internal spanning tree problem

A Survey on Algorithms for the Maximum Internal Spanning Tree and Related Problems

Hamiltonicity and vulnerability of graphs are in a strong connection. A basic necessary condition states that a graph containing a 2-leaf spanning tree (that is a Hamiltonian path) cannot be split into more than k + 1 components by deleting k of its vertices. In this paper we consider a more general approach and investigate the connection between the number of spanning tree leaves and two vulnerability parameters namely scattering number sc( G ) and cut-asymmetry ca( G ). We prove that any spanning tree of a graph G has at least sc( G ) + 1 leaves. We also show that if X $\subset$ V is a maximum cardinality independent set of G = ( V , E ) such that the elements of X are all leaves of a particular spanning tree then | X | = ca( G ) + 1 = | V | − cvc( G ), where cvc( G ) is the size of a minimum connected vertex cover of G . As a consequence we obtain a new proof for the following results: any spanning tree with independent leaves provides a 2-approximation for both MaximumInternalSpanningTree and MinimumConnectedVertexCover problems. We also consider the opposite point of view by fixing the number of leaves to q and looking for a q -leaf subtree of G that spans a maximum number of vertices. Bermond proved that a 2-connected graph on n vertices always contains a path (a 2-leaf subtree) of length min {n,δ 2 }, where δ 2 is the minimum degree-sum of a 2-element independent set. We generalize this result to obtain a sufficient condition for the existence of a large q -leaf subtree.

/pdf/vulnerability-bounds-on-the-number-of-spanning-tree-leaves-3s0rgzrrc7.pdf

Vulnerability bounds on the number of spanning tree leaves

On finding spanning trees with few leaves

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