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Solving Co-Path/Cycle Packing and Co-Path Packing Faster Than $3^k$

Yuxi Liu, Mingyu Xiao

TL;DR

This work studies Co-Path/Cycle Packing (CPC) and Co-Path Packing (CPP), asking whether at most $k$ vertices can be deleted to obtain a graph whose components are induced paths (and cycles in CPC).The authors develop a two-pronged approach that combines a branch-and-search phase to reduce hard instances to structured graphs with bounded pathwidth and a dynamic programming phase on path decompositions; for CPP they additionally integrate a Cut & Count based randomized DP to handle global connectivity.Their main results are a deterministic $O^*(2.8192^k)$-time algorithm for CPC with $O^*(2.5199^k)$ space, and a randomized $O^*(2.9241^k)$-time algorithm for CPP with failure probability at most $1/3$; CPP also admits a $5^{p}$-time DP if a path decomposition of width $p$ is given.The paper demonstrates that breaking the $3^k$ barrier is achievable by pruning bottlenecks via a pathwidth-aware branch-and-search strategy and by leveraging Cut & Count for connectivity-type problems.These methods and insights may extend to other parameterized, connectivity-type graph problems and motivate future work toward deterministic CPP algorithms faster than the $3^k$ threshold.

Abstract

The \textsc{Co-Path/Cycle Packing} problem (resp. The \textsc{Co-Path Packing} problem) asks whether we can delete at most $k$ vertices from the input graph such that the remaining graph is a collection of induced paths and cycles (resp. induced paths). These two problems are fundamental graph problems that have important applications in bioinformatics. Although these two problems have been extensively studied in parameterized algorithms, it seems hard to break the running time bound $3^k$. In 2015, Feng et al. provided an $O^*(3^k)$-time randomized algorithms for both of them. Recently, Tsur showed that they can be solved in $O^*(3^k)$ time deterministically. In this paper, by combining several techniques such as path decomposition, dynamic programming, cut \& count, and branch-and-search methods, we show that \textsc{Co-Path/Cycle Packing} can be solved in $O^*(2.8192^k)$ time deterministically and \textsc{Co-Path Packing} can be solved in $O^*(2.9241^{k})$ time with failure probability $\leq 1/3$. As a by-product, we also show that the \textsc{Co-Path Packing} problem can be solved in $O^*(5^p)$ time with probability at least 2/3 if a path decomposition of width $p$ is given.

Solving Co-Path/Cycle Packing and Co-Path Packing Faster Than $3^k$

TL;DR

This work studies Co-Path/Cycle Packing (CPC) and Co-Path Packing (CPP), asking whether at most $k$ vertices can be deleted to obtain a graph whose components are induced paths (and cycles in CPC).The authors develop a two-pronged approach that combines a branch-and-search phase to reduce hard instances to structured graphs with bounded pathwidth and a dynamic programming phase on path decompositions; for CPP they additionally integrate a Cut & Count based randomized DP to handle global connectivity.Their main results are a deterministic $O^*(2.8192^k)$-time algorithm for CPC with $O^*(2.5199^k)$ space, and a randomized $O^*(2.9241^k)$-time algorithm for CPP with failure probability at most $1/3$; CPP also admits a $5^{p}$-time DP if a path decomposition of width $p$ is given.The paper demonstrates that breaking the $3^k$ barrier is achievable by pruning bottlenecks via a pathwidth-aware branch-and-search strategy and by leveraging Cut & Count for connectivity-type problems.These methods and insights may extend to other parameterized, connectivity-type graph problems and motivate future work toward deterministic CPP algorithms faster than the $3^k$ threshold.

Abstract

The \textsc{Co-Path/Cycle Packing} problem (resp. The \textsc{Co-Path Packing} problem) asks whether we can delete at most vertices from the input graph such that the remaining graph is a collection of induced paths and cycles (resp. induced paths). These two problems are fundamental graph problems that have important applications in bioinformatics. Although these two problems have been extensively studied in parameterized algorithms, it seems hard to break the running time bound . In 2015, Feng et al. provided an -time randomized algorithms for both of them. Recently, Tsur showed that they can be solved in time deterministically. In this paper, by combining several techniques such as path decomposition, dynamic programming, cut \& count, and branch-and-search methods, we show that \textsc{Co-Path/Cycle Packing} can be solved in time deterministically and \textsc{Co-Path Packing} can be solved in time with failure probability . As a by-product, we also show that the \textsc{Co-Path Packing} problem can be solved in time with probability at least 2/3 if a path decomposition of width is given.
Paper Structure (14 sections, 19 theorems, 21 equations, 4 figures, 3 tables)

This paper contains 14 sections, 19 theorems, 21 equations, 4 figures, 3 tables.

Table of Contents

  1. Preliminaries
  2. A Parameterized Algorithm for Co-Path/Cycle Packing
  3. Proper Graphs with Small Pathwidth
  4. A Branch-and-Search Algorithm
  5. Reduction and Branching Rules
  6. Steps
  7. A Parameterized Algorithm for Co-Path Packing
  8. A DP Algorithm via Cut & Count for Co-Path Packing
  9. Cutting
  10. Counting
  11. The Whole Algorithm
  12. Proper Graphs with Small Pathwidth
  13. A Branch-and-Search Algorithm
  14. Conclusion

Key Result

Lemma 2

If a graph $G$ admits a path decomposition of width at most $p$, then it also admits a nice path decomposition of width at most $p$. Moreover, given a path decomposition $P = (X_1, X_2, \cdots, X_r)$ of $G$ of width at most $p$, one can in time $O(p^2\cdot \max(r, |V(G)|))$ compute a nice path decom

Figures (4)

  • Figure 2: Degree-4 vertex $v$ dominates a degree-3 vertex $u_1$.
  • Figure 3: Degree-4 vertex $v$ is in a heavy triangle $\{v, u_1, u_2\}$.
  • Figure 4: In the Case 2 of Step 4, degree-4 vertex $v$ is in a triangle $\{v, u_1, u_2\}$ and both of $u_1$ and $u_2$ are degree-3 vertices.
  • Figure 5: Vertices in the deletion set are denoted by black vertices, and vertices not allowed to be deleted are denoted by grey vertices.

Theorems & Definitions (24)

  • Definition 1: cygan2015parameterized
  • Lemma 2: cygan2015parameterized
  • Theorem 3: fomin2009two
  • Lemma 4
  • Lemma 5
  • Theorem 6: $\clubsuit$
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 14 more