Increasing arc-connectivity by bounded- and fixed-size inversions

Abstract: For a digraph $D$ and some $X \subseteq V(D)$, the inversion of $X$ is the operation of flipping all arcs both of whose endvertices are in $X$. We initiate the study of establishing arc-connectivity properties by applying inversions of bounded or fixed size. For fixed-size inversions, the feasibility problem is interesting. For all integers $p \geq 2$ and $k \geq 1$, we give a characterization of the digraphs that can be made $k$-arc-strong by applying inversions of size exactly $p$, provided they are sufficiently large. For bounded-size inversions, the feasibility problem is easy, so we focus on minimising the number of inversions. We prove that for all integers $p\geq 3$ and $k \geq 1$ and any $ε>0$, there exists a polynomial-time $(4k-2+ε)$-approximation algorithm for computing the minimum number of inversions of size at most $p$ that make a given digraph $k$-arc-strong. This is in stark contrast to other results on inversion optimization problems. On the other hand, we show that for any $p\geq 3$ and $k \geq 1$ the problem is NP-hard, and, moreover, APX-hard. As a result on parameterized complexity, we show that for any $k \geq 2$, it is $W[1]$-hard with respect to $p$ to decide whether a given digraph can be made $k$-arc-strong by applying a single inversion of size at most $p$. We also prove that for a given multidigraph, it is $W[1]$-hard with respect to $\ell$ to decide whether it can be made 2-arc-strong by applying $\ell$ inversions of size 2.

• The paper presents new structural characterizations and algorithmic strategies that determine when fixed- and bounded-size inversions can achieve k-arc-strong connectivity in digraphs.
• It introduces efficient polynomial-time algorithms for p=2, proves NP-hardness for p≥3, and offers approximation algorithms with factors such as 4k-2+ε and 3/2+ε for specific cases.
• The work emphasizes parameterized hardness and ETH-based lower bounds, providing rigorous insights into the reconfiguration of digraphs via local inversion operations.

Increasing Arc-Connectivity by Bounded- and Fixed-Size Inversions: Technical Summary

Introduction and Problem Formulation

The paper "Increasing arc-connectivity by bounded- and fixed-size inversions" (2604.22584) rigorously investigates, for directed graphs (digraphs), the feasibility and complexity of increasing arc-connectivity via a constrained class of reconfiguration operations: inversions of vertex subsets of fixed or bounded size. Specifically, inverting a subset $X$ of vertices in a digraph $D$ denotes reversing all arcs with both ends in $X$. The focus is on two problem regimes:

1. Fixed-size inversions: only inversions of sets of exactly size $p$ are allowed.
2. Bounded-size inversions: inversions of any subset of size at most $p$ are permitted, aiming to minimize the number required.

The primary connectivity objective is transforming $D$ to a $k$-arc-strong digraph (i.e., deletion of up to $k-1$ arcs retains strong connectivity). The work delineates feasible instances, develops algorithms, and establishes hardness results, parameterized and approximability lower bounds, and approximation algorithms for both regimes.

Main Results: Feasibility of Achieving $k$-Arc-Strong Connectivity

Fixed-Size Inversions

For fixed inversion size $p\geq 2$ and target $D$0-arc-strongness:

• Even $D$1: For sufficiently large $D$2 compared to $D$3 and $D$4, a $D$5-edge-connected digraph can be made $D$6-arc-strong by inversions of size exactly $D$7. This generalizes Nash-Williams' result on strong orientations and shows that fixed-size operations with even $D$8 are algorithmically as powerful as unconstrained inversions in large, well-connected digraphs.
• Odd $D$9: The authors provide a complete structural characterization of digraphs that cannot be made $X$0-arc-strong by fixed-size inversions of odd $X$1, via the notion of $X$2-obstructions (specific vertex partitions with precise cut and parity conditions). This obstruction-based characterization is algorithmic: for fixed $X$3 and $X$4, and sufficiently large $X$5, existence of a $X$6-arc-strong outcome via inversions of size $X$7 is decidable in polynomial time.
• For both even and odd $X$8, when $X$9 is small (specifically $p$0), feasibility can be NP-hard, even for strong connectivity ($p$1) when $p$2.

Bounded-Size Inversions

When considering arbitrary inversions up to size $p$3 (bounded-size):

• Feasibility is tractable once $p$4 and $p$5 are fixed, and the minimum number of inversions can be efficiently computed for $p$6 via reduction to arc flip problems and orientations (using Frank's theorem).
• For $p$7, feasibility remains easy, but the minimization version (fewest inversions to achieve $p$8-arc-strongness) is NP-hard and APX-hard, even for $p$9.

Algorithmic Results: Exact and Approximate Computation

Exact Algorithms

• For $p$0 (arc flips): Exact minimization is polynomial-time solvable via a reduction to weighted $p$1-arc-strong orientation problems (Frank [FRANK198297]).
• Parameterizable Obstruction Detection: For fixed $p$2, a kernel of size $p$3 is achievable for deciding feasibility.

Hardness and Inapproximability

• For $p$4: The number of inversions required for $p$5-arc-strongness cannot be polynomial-time approximated within a $p$6 factor, for some universal constant $p$7 depending on $p$8, unless $p$9. This is shown via reductions from hard combinatorial optimization problems such as $D$0-packing and hypergraph matching, preserving approximation gaps.

Approximation Algorithms

• Approximation Ratio: For all $D$1, $D$2, and any $D$3, there is a $D$4-approximation for minimizing bounded-size inversions to achieve $D$5-arc-strongness. For $D$6 and strong connectivity, this is improved to $D$7-approximation.
• Method: The approximation leverages a reduction from $D$8-inversions to the $D$9 (arc flip) case, bounding the growth of the optimum via combinatorial compression and Ramsey-theoretic arguments. This demonstrates atypical approximability compared to related orientation/inversion optimization problems which resist even constant-factor approximations.

Parameterized Complexity

• $k$0-hardness: Deciding whether a digraph can be made $k$1-arc-strong via a single inversion of size at most $k$2 is $k$3-hard parameterized by $k$4 for any $k$5. This shows no fpt algorithms in $k$6 unless $k$7.
• For Multidigraphs: It is $k$8-hard parameterized by $k$9 (number of inversions allowed) to achieve strong connectivity by $k-1$0 inversions of size 2.
• ETH barriers: The exponent in $k-1$1 or $k-1$2 brute-force approaches is essentially optimal under the Exponential Time Hypothesis.

Theoretical and Practical Implications

The research elucidates the landscape of structural and algorithmic properties of digraphs under fixed-size and bounded-size reconfiguration. Major implications are:

• Structural: The link between edge-connectivity and achievable arc-connectivity by fixed-size inversions formalizes the limitations and power of local reconfiguration, especially the role of parity and large enough host graphs.
• Algorithmic: The existence of efficient constant-factor approximation algorithms for $k-1$3 contrasts sharply with prior strongly inapproximable inversion or orientation modification problems, opening new algorithmic avenues for digraph resilience design.
• Parameterized hardness: The $k-1$4-hardness results signal that no significant efficiency gains are feasible for parameterized minimization by inversion size or count, limiting the scope of fpt approaches in practice.

Future Directions

The work raises several open questions:

• Tightening Approximation Ratios: Can the $k-1$5 factor be improved for general $k-1$6? Is a constant-factor approximation always attainable independent of $k-1$7?
• Small-graph Cases: Whether the $k-1$8 size lower bound in obstruction characterizations can be reduced remains unclear.
• Stronger Parameterizations: Complexity for strong connectivity ($k-1$9) under single bounded-size inversions is not fully resolved.
• Generalization to other connectivity augmentations: Adapting methods to vertex-connectivity or weighted variants of arc-strongness may yield further insights.

Conclusion

This work offers a comprehensive, technical treatment of the interplay between local digraph inversion operations and global connectivity, with broad algorithmic and structural complexity contributions. The clarity of feasible/infeasible regimes, hardness lower bounds, and new approximation algorithms together provide a foundational toolkit for further study of digraph reconfiguration under locality constraints.