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Approximation algorithms for non-sequential star packing problems

Mengyuan Hu, An Zhang, Yong Chen, Mingyang Gong, Guohui Lin

TL;DR

A local search algorithms and a $\frac 32$-approximation algorithm that improves the best known approximation algorithms for the problems, respectively.

Abstract

For a positive integer $k \ge 1$, a $k$-star ($k^+$-star, $k^-$-star, respectively) is a connected graph containing a degree-$\ell$ vertex and $\ell$ degree-$1$ vertices, where $\ell = k$ ($\ell \ge k$, $1 \le \ell \le k$, respectively). The $k^+$-star packing problem is to cover as many vertices of an input graph $G$ as possible using vertex-disjoint $k^+$-stars in $G$; and given $k > t \ge 1$, the $k^-/t$-star packing problem is to cover as many vertices of $G$ as possible using vertex-disjoint $k^-$-stars but no $t$-stars in $G$. Both problems are NP-hard for any fixed $k \ge 2$. We present a $(1 + \frac {k^2}{2k+1})$- and a $\frac 32$-approximation algorithms for the $k^+$-star packing problem when $k \ge 3$ and $k = 2$, respectively, and a $(1 + \frac 1{t + 1 + 1/k})$-approximation algorithm for the $k^-/t$-star packing problem when $k > t \ge 2$. They are all local search algorithms and they improve the best known approximation algorithms for the problems, respectively.

Approximation algorithms for non-sequential star packing problems

TL;DR

A local search algorithms and a -approximation algorithm that improves the best known approximation algorithms for the problems, respectively.

Abstract

For a positive integer , a -star (-star, -star, respectively) is a connected graph containing a degree- vertex and degree- vertices, where (, , respectively). The -star packing problem is to cover as many vertices of an input graph as possible using vertex-disjoint -stars in ; and given , the -star packing problem is to cover as many vertices of as possible using vertex-disjoint -stars but no -stars in . Both problems are NP-hard for any fixed . We present a - and a -approximation algorithms for the -star packing problem when and , respectively, and a -approximation algorithm for the -star packing problem when . They are all local search algorithms and they improve the best known approximation algorithms for the problems, respectively.

Paper Structure

This paper contains 10 sections, 16 theorems, 8 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Approximating the $k^+$-star packing problem
  3. The algorithm for $k \ge 2$
  4. Performance analysis
  5. Case 1 where $k \ge 2$
  6. Case 2 where $k = 2$
  7. Approximating the $k^-/t$-star packing problem
  8. The algorithm
  9. Performance analysis
  10. Conclusion

Key Result

Lemma 1

When operation Collect is not applicable, the maximum degree of the remainder graph with respect to the feasible solution $\mathcal{P}$ is at most $k-1$ and no uncovered vertex is adjacent to the center of any internal star.

Figures (11)

  • Figure 1: An illustration to apply operation Pull-by-$(k, (k+1)^+)$ on two internal $k$-star centered at $w$, which is a $k_{v\hbox{-}u}$-star, and $(k+1)^+$-star centered at $y$, which is a $(k+1)^+_x$-star. The filled vertices are covered, the empty vertices are uncovered, the edges in the internal stars are solid while the dashed edges are in the input graph $G$. In this case, $k = 4$, a vertex of the $k_{v\hbox{-}u}$-star other than $v$ is adjacent to $x$. After removing the $k_{v\hbox{-}u}$-star from $\mathcal{P}$ and the satellite $x$ from the $(k+1)^+_x$-star, two vertex-disjoint $k$-stars centered at $u$ and $x$, respectively, are extracted.
  • Figure 2: An illustration to apply operation Pull-by-$(k, k)$ on two internal $k$-stars centered at $w$ and $y$, which are a $k_{v\hbox{-}u}$-star and a $k_x$-star, respectively. The filled vertices are covered, the empty vertices are uncovered, the edges in the internal stars are solid while the dashed edges are in the input graph $G$. In this case, $k = 4$, a vertex of the $k_{v\hbox{-}u}$-star other than $v$ is adjacent to $x$. After removing the two $k$-stars from $\mathcal{P}$, two vertex-disjoint $k$-star centered at $u$ and $(k+1)$-star centered at $x$, respectively, are extracted.
  • Figure 3: A high-level description of LocalSearch-$k^+$ for $k^+$-star packing.
  • Figure 4: An illustration to apply operation Pull-by-$k$ when the center vertex of the Type-$2$ optimal star is uncovered and contains two $a$-vertices. In this case, $k = 4$; at the top these two $a$-vertices are associated with the same $c$-vertex; at the bottom these two $a$-vertices are associated with different $c$-vertices. The edges in Type-$2$ optimal star are dotted.
  • Figure 5: An illustration to apply operation Pull-by-$k$ when the center vertex of the Type-$2$ optimal star is uncovered and contains an $a$-vertex. The edges in Type-$2$ optimal star are dotted.
  • ...and 6 more figures

Theorems & Definitions (25)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Definition 3
  • Lemma 2
  • Definition 4
  • Definition 5
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 15 more