An approximation algorithm for maximum packing of 3-edge paths

Given a digraph $G=(V,E)$, the k-path partition problem aims to find a minimum collection of vertex-disjoint directed paths, of order at most k, to cover all the vertices. The problem has various applications. Its special case on undirected graphs is NP-hard when $k\ge 3$, and has received much study recently from the approximation algorithm perspective. However, the general problem on digraphs is seemingly untouched in the literature. We fill the gap with the first $k/2$-approximation algorithm, based on a novel concept of enlarging walk to minimize the number of singletons. Secondly, for $k=3$, we define a second novel kind of enlarging walks to greedily reduce the number of 2-paths in the 3-path partition and propose an improved 13/9-approximation algorithm. Lastly, for any $k\ge 7$, we present an improved $(k+2)/3$-approximation algorithm built on the maximum path-cycle cover followed by a careful 2-cycle elimination process.

Given a graph, we wish to find a maximum number of vertex-disjoint paths of length 2. We propose a series of local improvement algorithms for this problem, and present a linear-programming based method for analyzing their performance.

We investigate two $\mathsf{NP}$-complete vertex partition problems on edge-weighted complete graphs with $3k$ vertices. The first problem asks to partition the graph into $k$ vertex disjoint paths of length 2 (referred to as 2-paths) such that the total weight of the paths is maximized. We present a cubic time approximation algorithm that computes a 2-path partition whose total weight is at least $.5833$ of the weight of an optimal partition, improving upon the $(.5265-ϵ)$-approximation algorithm of Tanahashi and Chen (2010). Restricting the input to graphs with edge weights in $\{0,1\}$, we present a $.75$ approximation algorithm improving upon the $.55$-approximation algorithm of Hassin and Schneider (2013).

Combining this algorithm with a previously known approximation algorithm for the 3-Set Packing problem, we obtain a $.6$-approximation algorithm for the problem of partitioning a $\{0,1\}$-edge-weighted graph into $k$ vertex disjoint triangles of maximum total weight. The best known approximation algorithm for general weights is due to Chen and Tanahashi (2009) and achieves an approximation ratio of $.5257$.

Let $G$ be a complete (undirected) graph with $3l$ vertices. Given a binary weight function on the edges of $G$, the binary maximum 2-path partitioning problem is to compute a set of $l$ vertex-disjoint simple 2-edge paths with maximum total edge weight. The problem is NP-hard (Garey and Johnson 1979) [1].

In this paper we propose a simple local search algorithm with polynomial running time for the problem and analyze its performance for several search depths. For depth 2, we show that the algorithm is a 0.3333-approximation, and that the bound is tight. For depth 3, we show that the algorithm is a 0.4-approximation. For depth 9, we show that the algorithm is a 0.55-approximation, improving on the best-known 0.5265 bound for the problem.

We also consider the special case where $G$ is subcubic, that is, the maximum degree in its subgraph induced by the unit-weight edges is 3. In this case we show that the algorithm is a 0.375-approximation for depth 2 and a 0.5-approximation for depth 3. In addition, we show that depth 7 is sufficient for the 0.55 bound guarantee.

Finally we give, by means of bad instances, upper bounds on the performance guarantees of the algorithm. For depth 2 we show a 0.4 upper bound in the subcubic case. For depth 3 we show a 0.6 upper bound, as well as a 0.7 upper bound in the subcubic case. For the general (non-negative) weight problem we show a 0.5556 upper bound for depth 3 (for depth 2, the tight 0.3333 ratio holds for this problem as well).

We consider packing problems in graphs under a parameterized perspective. Starting from a maximal path packing $\mathcal{P}$ of size $j$ we use extremal arguments for determining how many vertices of $\mathcal{P}$ appear in a path packing of size $j+1$. Generally, one can re-use $2j$ vertices of $j$ paths of length three, four and five. This is improved to $3j$, $2.5j$ and $3j$ vertices.

We present several new standard and differential approximation results for the ${\mathrm{P}}_4$-partition problem using the Hassin and Rubinstein algorithm [Information Processing Letters 63 (1997) 63–67]. Those results concern both minimization and maximization versions of the problem. However, the main point of this paper lies in the establishment of the robustness of this algorithm, in the sense that it provides good quality solutions for a variety of versions of the problem, under both standard and differential approximation ratios.