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Polynomial-Size Enumeration Kernelizations for Long Path Enumeration

Christian Komusiewicz, Diptapriyo Majumdar, Frank Sommer

TL;DR

The paper tackles the problem of enumerating all $k$-paths in an undirected graph under structural parameters, introducing polynomial-delay polynomial-size (pd-ps) enumeration kernels for the vertex cover ${\sf vc}$, the dissociation number ${\sf diss}$, and the distance to clique ${\sf dtc}$. The authors develop kernelization schemes based on a marking strategy and the Expansion Lemma, coupled with sophisticated solution-lifting procedures that preserve a one-to-one correspondence between kernel solutions and global solutions while ensuring output without duplicates. Key innovations include the new expansion lemma applications to enumeration, a signature-based equivalence framework for $k$-paths, and two-phase lifting algorithms that guarantee polynomial-delay enumeration; these yield explicit delay bounds like $\mathcal{O}(n\cdot m\cdot k^2)$ for vc, $\mathcal{O}({\sf diss}\cdot n)$ for diss, and $\mathcal{O}(n\cdot {\sf dtc}(G))$ for dtc, with kernels of sizes $\mathcal{O}({\sf vc}^2)$, $\mathcal{O}({\sf diss}^3)$, and $\mathcal{O}({\sf dtc}^3)$ respectively. The framework generalizes to variants such as Long-Cycle and to broader parameters like $r$-${\sf coc}$, and to the dtc setting extending beyond a single parameter, underscoring the practical impact for design of enumeration algorithms with guaranteed delay and compact data reduction. These results advance the understanding of when pd-ps kernels exist for hard enumeration problems and illustrate how structural graph parameters can enable tractable, output-sensitive preprocessing and lifting techniques.

Abstract

Enumeration kernelization for parameterized enumeration problems was defined by Creignou et al. [Theory Comput. Syst. 2017] and was later refined by Golovach et al. [J. Comput. Syst. Sci. 2022, STACS 2021] to polynomial-delay enumeration kernelization. We consider ENUM LONG-PATH, the enumeration variant of the Long-Path problem, from the perspective of enumeration kernelization. Formally, given an undirected graph G and an integer k, the objective of ENUM LONG-PATH is to enumerate all paths of G having exactly k vertices. We consider the structural parameters vertex cover number, dissociation number, and distance to clique and provide polynomial-delay enumeration kernels of polynomial size for each of these parameters.

Polynomial-Size Enumeration Kernelizations for Long Path Enumeration

TL;DR

The paper tackles the problem of enumerating all -paths in an undirected graph under structural parameters, introducing polynomial-delay polynomial-size (pd-ps) enumeration kernels for the vertex cover , the dissociation number , and the distance to clique . The authors develop kernelization schemes based on a marking strategy and the Expansion Lemma, coupled with sophisticated solution-lifting procedures that preserve a one-to-one correspondence between kernel solutions and global solutions while ensuring output without duplicates. Key innovations include the new expansion lemma applications to enumeration, a signature-based equivalence framework for -paths, and two-phase lifting algorithms that guarantee polynomial-delay enumeration; these yield explicit delay bounds like for vc, for diss, and for dtc, with kernels of sizes , , and respectively. The framework generalizes to variants such as Long-Cycle and to broader parameters like -, and to the dtc setting extending beyond a single parameter, underscoring the practical impact for design of enumeration algorithms with guaranteed delay and compact data reduction. These results advance the understanding of when pd-ps kernels exist for hard enumeration problems and illustrate how structural graph parameters can enable tractable, output-sensitive preprocessing and lifting techniques.

Abstract

Enumeration kernelization for parameterized enumeration problems was defined by Creignou et al. [Theory Comput. Syst. 2017] and was later refined by Golovach et al. [J. Comput. Syst. Sci. 2022, STACS 2021] to polynomial-delay enumeration kernelization. We consider ENUM LONG-PATH, the enumeration variant of the Long-Path problem, from the perspective of enumeration kernelization. Formally, given an undirected graph G and an integer k, the objective of ENUM LONG-PATH is to enumerate all paths of G having exactly k vertices. We consider the structural parameters vertex cover number, dissociation number, and distance to clique and provide polynomial-delay enumeration kernels of polynomial size for each of these parameters.

Paper Structure

This paper contains 52 sections, 32 theorems, 4 figures.

Table of Contents

  1. Introduction
  2. Our Contributions.
  3. Further Related Work.
  4. Preliminaries
  5. Sets, Numbers, and Graph Theory.
  6. Expansion Lemma.
  7. Graph Parameters.
  8. Parameterized Complexity and Kernelization.
  9. Parameterized Enumeration and Enumeration Kernelization.
  10. Parameterization by the Vertex Cover Number
  11. Marking Scheme and Kernelization Algorithm
  12. Signatures and Equivalence among $k$-Paths
  13. A Challenge: Avoiding Duplicate Enumeration
  14. The Solution-Lifting Algorithm
  15. Extension to Other Path and Cycle Variants
  16. ...and 37 more sections

Key Result

theorem thmcountertheorem

Enum Long-Path parameterized by ${{\sf vc}}$ admits an $\mathcal{O}(n\cdot m\cdot k^2)$-delay enumeration kernel with $\mathcal{O}({{\sf vc}}^2)$ vertices.

Figures (4)

  • Figure 1: A hierarchy of parameters. A cyan box indicates polynomial sized enumeration kernels. A pink box indicates the non-existence of polynomial sized enumeration kernels. A yellow box indicates open status. An edge from a parameter $\alpha$ to a parameter $\beta$ below $\alpha$ means that there is a function $f$ such that $\beta \le f(\alpha)$ in every graph.
  • Figure 2: Part $a)$ shows a graph $G$ and its decomposition into the modulator $X$ and the independent set $I$. Here, the vertices in $I_1$ are not shown. The two paths $P_1=(x_1,x_2,i_1,x_3,i_4,x_4,x_5,i_6,x_6)$ (in red) and $P_1=(x_1,x_2,i_4,x_3,i_5,x_4,x_5,i_6,x_6)$ (in blue) have the same signature ${\textsf{sig}}(x_1)=1,{\textsf{sig}}(x_2)=2,{\textsf{sig}}(x_3)=4,{\textsf{sig}}(x_4)=6,{\textsf{sig}}(x_5)=7,{\textsf{sig}}(i_6)=8,{\textsf{sig}}(x_6)=9$. Part $b)$ shows the corresponding auxiliary bipartite graph and the highlighted edges show the edges occupied (see paragraph above \ref{['lemma:edge-occupation-vc']} for a definition) of both paths $P_1$ and $P_2$.
  • Figure 3: A graph $G$ with modulator $X$ such that each connected component in $G[I]$ has at most 2 vertices. The two paths $P_1=(v_1,v_2,x_2,v_5,x_4,v_8,x_5,v_9,v_{10})$ (in red) and $P_2=(v_1,v_2,x_2,v_6,x_4,v_8,x_5,v_{13},v_{14})$ (in blue) have the same signature. Furthermore, $P_1$ is entirely contained in $G'$, while $P_2$ contains the two non-kernel vertices $v_{13}$ and $v_{14}$.
  • Figure 4: A graph $G$ with clique $C$ and modulator $X$ where $c_1$ and $c_4$ are rare, and all other clique vertices are frequent. The two paths $P_1=(c_8,c_1,x_1,x_2,c_2,x_4,c_4,c_{12},c_{13},c_6,x_6)$ (in red) and $P_2=(c_1,x_1,x_2,c_3,x_4,c_4,c_{10},c_{11},c_5,c_{13},x_6)$ (in blue) have the same signature $(c_1,x_1,x_2,g_C,x_4,c_4,g_N,x_6)$.

Theorems & Definitions (71)

  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • proposition thmcounterproposition
  • proof
  • theorem thmcountertheorem
  • proposition thmcounterproposition: New $q$-Expansion Lemma - Lemma 3.2 of FominLLSTZ19
  • proposition thmcounterproposition: KobayashiKW22, KobayashiKW21
  • definition thmcounterdefinition: GolovachKKL22
  • proposition thmcounterproposition
  • ...and 61 more