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Maximizing Minimum Cycle Bases Intersection

Dimitri Watel, Marc-Antoine Weisser, Dominique Barth, Ylène Aboulfath, Thierry Mautor

TL;DR

This work studies maximizing the intersection of minimum cycle bases across $k$ graphs sharing the same vertex set, formulating max-MCBI as a structured instance of matroid intersection on the cycle space $C$ with independence determined by minimum cycle bases. A key method is restricting attention to a polynomial-size candidate cycle set $L$ and leveraging a polynomial-time independence oracle (via a Horton-based modification), which allows efficient handling of the otherwise exponential ground set. The authors establish a complete complexity partition in terms of four parameters $k$, $\gamma$ (maximum cycle size in a basis), $\Delta$ (maximum degree), and $K$ (intersection target), showing polynomial-time solvability for $k=2$, polynomial-time solvability in the cases $\gamma=3$ or $(\gamma=4, \Delta=3)$, and NP-hardness/W[1]-hardness with tight inapproximability results for larger regimes. They also provide approximation and XP results, linking the cycle-space structure to practical chemoinformatics questions about conserved molecular substructures across trajectories. The findings elucidate the complexity landscape of MI in this specialized setting and suggest avenues where small graph edits could yield tractable algorithms for real-world molecular data.

Abstract

We address a specific case of the matroid intersection problem: given a set of graphs sharing the same set of vertices, select a minimum cycle basis for each graph to maximize the size of their intersection. We provide a comprehensive complexity analysis of this problem, which finds applications in chemoinformatics. We establish a complete partition of subcases based on intrinsic parameters: the number of graphs, the maximum degree of the graphs, and the size of the longest cycle in the minimum cycle bases. Additionally, we present results concerning the approximability and parameterized complexity of the problem.

Maximizing Minimum Cycle Bases Intersection

TL;DR

This work studies maximizing the intersection of minimum cycle bases across graphs sharing the same vertex set, formulating max-MCBI as a structured instance of matroid intersection on the cycle space with independence determined by minimum cycle bases. A key method is restricting attention to a polynomial-size candidate cycle set and leveraging a polynomial-time independence oracle (via a Horton-based modification), which allows efficient handling of the otherwise exponential ground set. The authors establish a complete complexity partition in terms of four parameters , (maximum cycle size in a basis), (maximum degree), and (intersection target), showing polynomial-time solvability for , polynomial-time solvability in the cases or , and NP-hardness/W[1]-hardness with tight inapproximability results for larger regimes. They also provide approximation and XP results, linking the cycle-space structure to practical chemoinformatics questions about conserved molecular substructures across trajectories. The findings elucidate the complexity landscape of MI in this specialized setting and suggest avenues where small graph edits could yield tractable algorithms for real-world molecular data.

Abstract

We address a specific case of the matroid intersection problem: given a set of graphs sharing the same set of vertices, select a minimum cycle basis for each graph to maximize the size of their intersection. We provide a comprehensive complexity analysis of this problem, which finds applications in chemoinformatics. We establish a complete partition of subcases based on intrinsic parameters: the number of graphs, the maximum degree of the graphs, and the size of the longest cycle in the minimum cycle bases. Additionally, we present results concerning the approximability and parameterized complexity of the problem.
Paper Structure (7 sections, 21 theorems, 2 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 7 sections, 21 theorems, 2 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Formal Definitions
  3. Case with $k = 2$, Approximability and Parameterized Complexity
  4. Hardness of MCBI
  5. Cases $\gamma = 3$, and $\gamma = 4, \Delta = 3$
  6. Concluding remarks
  7. Proofs of Lemmas \ref{['lem:exchangecyclesinbasis']}, \ref{['lem:generatecyclebylowerweight']}, \ref{['lem:basisexchangeaxiom']} and \ref{['lem:circuit']}

Key Result

lemma 1

Given $B$ a cycle basis of a graph $G$ with two cycles $c_1 \in B$ and $c_2 \notin B$, if $\lambda_B(c_1, c_2)~=~1$ then $(B \setminus \{c_1\}) \cup \{c_2\}$ is a cycle basis of $G$.

Figures (4)

  • Figure 1: Example of cycles such that $c_1 \oplus c' \oplus c_2 = c$.
  • Figure 2: Illustration of the procedure $CONN(c_1, c_2)$
  • Figure 3: Possible cases of intersection of squares and of triangles.
  • Figure 4: Possible circuits containing squares in a graph $G$ with degree at most 3. Each square contains four node $(a_i, b_i, c_i, d_i)$.

Theorems & Definitions (38)

  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • proof
  • lemma 5
  • proof
  • theorem 1
  • proof
  • theorem 2
  • ...and 28 more