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Solution Discovery for Vertex Cover, Independent Set, Dominating Set, and Feedback Vertex Set

Rin Saito, Anouk Sommer, Tatsuhiro Suga, Takahiro Suzuki, Yuma Tamura

TL;DR

This work analyzes solution discovery (Pi-Discovery) for four fundamental vertex-subset problems under a token-sliding model. It delivers XP algorithms parameterized by clique-width and an FPT algorithm for FVS-D when parameterized by the number of tokens $k$, while mapping out computational boundaries by graph class: NP-hard on chordal and diameter-2 graphs, yet polynomial on split graphs. The key methodological innovations include dynamic programming on $w$-expressions with a $v^*$ gadget and the use of $k$-compact representations to enable a single-exponential FPT approach via minimum-weight matchings. Collectively, the results delineate the tractability landscape for Pi-Discovery across graph structure and parameter regimes, with practical implications for reconfiguration tasks and related optimization problems.

Abstract

In the solution discovery problem for a search problem on graphs, we are given an initial placement of $k$ tokens on the vertices of a graph and asked whether this placement can be transformed into a feasible solution by applying a small number of modifications. In this paper, we study the computational complexity of solution discovery for several fundamental vertex-subset problems on graphs, namely Vertex Cover Discovery, Independent Set Discovery, Dominating Set Discovery, and Feedback Vertex Set Discovery. We first present XP algorithms for all four problems parameterized by clique-width. We then prove that Vertex Cover Discovery, Independent Set Discovery, and Feedback Vertex Set Discovery are NP-complete for chordal graphs and graphs of diameter 2, which have unbounded clique-width. In contrast to these hardness results, we show that all three problems can be solved in polynomial time on split graphs. Furthermore, we design an FPT algorithm for Feedback Vertex Set Discovery parameterized by the number of tokens.

Solution Discovery for Vertex Cover, Independent Set, Dominating Set, and Feedback Vertex Set

TL;DR

This work analyzes solution discovery (Pi-Discovery) for four fundamental vertex-subset problems under a token-sliding model. It delivers XP algorithms parameterized by clique-width and an FPT algorithm for FVS-D when parameterized by the number of tokens , while mapping out computational boundaries by graph class: NP-hard on chordal and diameter-2 graphs, yet polynomial on split graphs. The key methodological innovations include dynamic programming on -expressions with a gadget and the use of -compact representations to enable a single-exponential FPT approach via minimum-weight matchings. Collectively, the results delineate the tractability landscape for Pi-Discovery across graph structure and parameter regimes, with practical implications for reconfiguration tasks and related optimization problems.

Abstract

In the solution discovery problem for a search problem on graphs, we are given an initial placement of tokens on the vertices of a graph and asked whether this placement can be transformed into a feasible solution by applying a small number of modifications. In this paper, we study the computational complexity of solution discovery for several fundamental vertex-subset problems on graphs, namely Vertex Cover Discovery, Independent Set Discovery, Dominating Set Discovery, and Feedback Vertex Set Discovery. We first present XP algorithms for all four problems parameterized by clique-width. We then prove that Vertex Cover Discovery, Independent Set Discovery, and Feedback Vertex Set Discovery are NP-complete for chordal graphs and graphs of diameter 2, which have unbounded clique-width. In contrast to these hardness results, we show that all three problems can be solved in polynomial time on split graphs. Furthermore, we design an FPT algorithm for Feedback Vertex Set Discovery parameterized by the number of tokens.

Paper Structure

This paper contains 19 sections, 11 theorems, 1 equation, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. Our problems
  6. Parameterized Algorithms
  7. XP algorithm parameterized by clique-width
  8. Dynamic Programming
  9. FPT algorithm parameterized by the number of tokens
  10. Graphs of Unbounded Clique-width
  11. $\NP$-completeness
  12. Polynomial-time Algorithms
  13. Conclusion
  14. Omitted Proofs in \ref{['sec:parameterized']}
  15. Proof of \ref{['lem:solsizeFPT']}
  16. ...and 4 more sections

Key Result

theorem thmcountertheorem

Given a $w$-expression tree of the input graph, the problems VC-D, FVS-D, IS-D, and DS-D can be solved in $n^{O(w)}$ time.

Figures (2)

  • Figure 1: A rough illustration for a join node. If $f$ tokens directly move from the $i$-vertices to the $j$-vertices, we decrease both $\operatorname{\texttt{P}}(i)$ and $\operatorname{\texttt{A}}(j)$ by $f$, and increase the total number of moves $\ell$ by $f$.
  • Figure 2: Construction of an instance $(G,S,b)$ of VC-D from an instance $(U,\mathcal{X})$ of Exact Cover by 3-Set, where $U=\{1,\ldots,9\}$ and $\mathcal{X}=\{X_1=\{1,2,3\}, X_2=\{2,3,4\}, X_3=\{1,5,6\}, X_4=\{5,6,7\}, X_5=\{6,7,8\}, X_6=\{7,8,9\}\}$. The vertices $q_0,\ldots,q_9$ inside the dotted box form a clique of $G$. Each double line between a vertex $q_i$ and a solid box labeled $A_j$ represents all edges between $q_i$ and the vertices in $A_j$. The vertices marked in gray constitute the initial configuration $S$, and $b=9+3=12$.

Theorems & Definitions (11)

  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem: Theorem 3 of solsize:Misra12
  • lemma thmcounterlemma: $\ast$
  • theorem thmcountertheorem
  • lemma thmcounterlemma: $\ast$
  • corollary thmcountercorollary
  • lemma thmcounterlemma: $\ast$
  • corollary thmcountercorollary
  • theorem thmcountertheorem: $\ast$
  • ...and 1 more