Directed Capacity-Preserving Subgraphs: Hardness and Exact Polynomial Algorithms

TL;DR

This work studies MCPS, the problem of finding the smallest subgraph that preserves at least $α$ times the maximum $u$-$v$-flow for every pair $(u,v)$ in a directed graph, motivating practical network power savings. It proves NP-hardness even on restricted DAGs and establishes para-NP-hardness w.r.t. several graph-structure parameters, then introduces Laminar Series-Parallel Graphs (LSPs) to extend the class of graphs for which strong results can be obtained. The authors provide two DSP-based algorithms—linear-time for MCPS1 and cubic-time for MCPS on DSP—and lift these techniques to LSPs, yielding a polynomial-time MCPS on LSPs and a quadratic-time MCPS1 on LSPs, by decomposing LSPs into DSP subproblems. They further connect MCPS to related problems like MED and DHC, showing how the LSP framework enables efficient solutions and broad applicability, while outlining open questions about undirected variants, tighter approximations, and linear-time possibilities on LSPs.

Abstract

We introduce and discuss the Minimum Capacity-Preserving Subgraph (MCPS) problem: given a directed graph and a retention ratio $α\in (0,1)$, find the smallest subgraph that, for each pair of vertices $(u,v)$, preserves at least a fraction $α$ of a maximum $u$-$v$-flow's value. This problem originates from the practical setting of reducing the power consumption in a computer network: it models turning off as many links as possible while retaining the ability to transmit at least $α$ times the traffic compared to the original network. First we prove that MCPS is NP-hard already on a restricted set of directed acyclic graphs (DAGs) with unit edge capacities. Our reduction also shows that a closely related problem (which only considers the arguably most complicated core of the problem in the objective function) is NP-hard to approximate within a sublogarithmic factor already on DAGs. In terms of positive results, we present two algorithms that solve MCPS optimally on directed series-parallel graphs (DSPs): a simple linear-time algorithm for the special case of unit edge capacities and a cubic-time dynamic programming algorithm for the general case of non-uniform edge capacities. Further, we introduce the family of laminar series-parallel graphs (LSPs), a generalization of DSPs that also includes cyclic and very dense graphs. Their properties allow us to solve MCPS on LSPs by employing our DSP-algorithms as subroutines. In addition, we give a separate quadratic-time algorithm for MCPS on LSPs with unit edge capacities that also yields straightforward quadratic time algorithms for several related problems such as Minimum Equivalent Digraph and Directed Hamiltonian Cycle on LSPs.

Figures (7)

• Figure 1: Two examples of subgraphs (in red) whose stretch differs greatly from their retention ratio. All edge lengths and edge capacities are 1, making clear that using the reciprocals of the edge lengths as edge capacities does not lead to a direct relation between stretch and capacity either.
• Figure 2: pr:MCPS*1 instance constructed from the pr:SC instance $(U,\mathcal{S})$ with $U = \{a,b,c,d\}$, $\mathcal{S}=\{\{a,b,c\},\newline\ \{c,d\},\ \{b,c\}\}$. An optimal solution contains the MED (shown as solid black lines) as well as one corresponding red (dashed) or green (dotted) edge for each $u \in U$. Edges are directed from upper to lower vertices.
• Figure 3: The digraph $W$, whose subdivisions cannot be contained in DSP.
• Figure 4: The digraph $W$ with two added paths of length 2, see \ref{['th:obs_cps_spshadows']}. Assume unit edge capacities. Then, the $x$-$y$-capacity is 3. For $\alpha = \frac{1}{2}$, all edges of the original graph are covered by the MED (solid black edges) but the non-adjacent vertex pair $(x,y)$ is not. Observe that the graph is not a DSP but its shadow is $z_1$-$z_2$-series-parallel ($(x,y) \neq (z_1,z_2)$).
• Figure 5: Examples of LSP. Every edge represents a DSP of arbitrary size.

Theorems & Definitions (27)

• proof
• proof
• Theorem 4
• proof
• Corollary 5
• Corollary 6
• proof
• Theorem 7: see DBLP:journals/siamcomp/ValdesTL82
• Definition 1: Laminar Series-Parallel Graph
• Theorem 8