Approximation Algorithms for Packing Cycles and Paths in Complete Graphs

TL;DR

This work studies approximation algorithms for metric and general $k$-cycle/$k$-path packing on complete graphs, focusing on maximizing total weight. It leverages reductions to MAX TSP and path-patching techniques, together with matching-based strategies, to derive tighter approximation ratios across several regimes: metric $k$CP for odd and even constant $k\ge5$, metric $k$PP for even $k$ in $6\le k\le10$, and the special case $k=4$ for both metric and general weights. Notable contributions include the new ratios $\alpha\cdot(1-0.5/k)(1-1/k)$ for odd $k$ and $\alpha\cdot(1-0.5/k)(1-1/k+1/(k(k-1)))$ for even $k$ in metric $k$CP, a $(27k^2-48k+16)/(32k^2-36k-24)$-approximation for metric $k$PP with even $k\in[6,10]$, and a $5/6$-approximation for metric $4$CP (plus $7/8$ on {1,2}-weighted graphs) and a $14/17$-approximation for metric $4$PP. It also shows that a $\rho$-approximation on {0,1}-weighted graphs can be transformed into a $(1+\rho)/2$-approximation on {1,2}-weighted graphs, enabling a $9/11$-approximation for $3$CP on {1,2}-weighted graphs. These results substantially narrow the gap to optimal packings and provide a unified framework combining TSP techniques, path patching, and matching-based constructions. The paper also lays out a precise organization of techniques and proofs across multiple cases and weight models, highlighting both improvements and intrinsic limits.

Abstract

Given an edge-weighted (metric/general) complete graph with $n$ vertices, the maximum weight (metric/general) $k$-cycle/path packing problem is to find a set of $\frac{n}{k}$ vertex-disjoint $k$-cycles/paths such that the total weight is maximized. In this paper, we consider approximation algorithms. For metric $k$-cycle packing, we improve the previous approximation ratio from $3/5$ to $7/10$ for $k=5$, and from $7/8\cdot(1-1/k)^2$ for $k>5$ to $(7/8-0.125/k)(1-1/k)$ for constant odd $k>5$ and to $7/8\cdot (1-1/k+\frac{1}{k(k-1)})$ for even $k>5$. For metric $k$-path packing, we improve the approximation ratio from $7/8\cdot (1-1/k)$ to $\frac{27k^2-48k+16}{32k^2-36k-24}$ for even $10\geq k\geq 6$. For the case of $k=4$, we improve the approximation ratio from $3/4$ to $5/6$ for metric 4-cycle packing, from $2/3$ to $3/4$ for general 4-cycle packing, and from $3/4$ to $14/17$ for metric 4-path packing.

Figures (5)

• Figure 1: An illustration of the $k$-cycle $C_{ij}$ obtained from $P_i$, where $j\in\{1,2,\dots,k-1\}$
• Figure 2: A tight example of the $7/10$-approximation algorithm for metric 5CP
• Figure 3: A tight example of the $3/4$-approximation algorithm for general 4CP, where each solid edge has a weight of 1 and each omitted edge has a weight of 0
• Figure 4: A tight example of the $3/4$-approximation algorithm for general 4PP, where each solid edge has a weight of 1 and each omitted edge has a weight of 0
• Figure 5: A tight example of the $5/6$-approximation algorithm for metric 4CP, where each gray edge has a weight of $1$, each red edge has a weight of $2$, each blue edge has a weight of $3$, and each orange edge has a weight of $4$

Theorems & Definitions (56)

• Lemma 1: DBLP:journals/ipl/HassinR02kostochka1985polynomial
• Lemma 2
• Theorem 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Lemma 6: DBLP:journals/tcs/KowalikM09
• Lemma 7