Constrained Flows in Networks

TL;DR

This work analyzes flows in networks under structural restrictions on the support, revealing broad NP-hardness even in acyclic digraphs for several degree- and path-based constraints. It develops both hardness reductions (notably from linkage problems) and approximation algorithms, including a $1/H(p)$-approximation for $p$-decomposable flows and a $1/\!H(p)$-bound that scales with the number of allowed paths. The paper also identifies tractable regimes, such as polynomial-time solvability for maximum $p$-path-flows in acyclic networks and XP-time algorithms for certain acyclic, vertex-disjoint decompositions, while highlighting limits for higher connectivity and persistence under arc/vertex deletions. These results illuminate the nuanced boundary between tractable and intractable constrained-flow problems, with implications for network design and resilience analysis.

Abstract

The support of a flow $x$ in a network is the subdigraph induced by the arcs $uv$ for which $x(uv)>0$. We discuss a number of results on flows in networks where we put certain restrictions on structure of the support of the flow. Many of these problems are NP-hard because they generalize linkage problems for digraphs. For example deciding whether a network ${\cal N}=(D,s,t,c)$ has a maximum flow $x$ such that the maximum out-degree of the support $D_x$ of $x$ is at most 2 is NP-complete as it contains the 2-linkage problem as a very special case. Another problem which is NP-complete for the same reason is that of computing the maximum flow we can send from $s$ to $t$ along $p$ paths (called a maximum {\bf $p$-path-flow}) in ${\cal N}$. Baier et al. (2005) gave a polynomial time algorithm which finds a $p$-path-flow $x$ whose value is at least $\frac{2}{3}$ of the value of a optimum $p$-path-flow when $p\in \{2,3\}$, and at least $\frac{1}{2}$ when $p\geq 4$. When $p=2$, they show that this is best possible unless P=NP. We show for each $p\geq 2$ that the value of a maximum $p$-path-flow cannot be approximated by any ratio larger than $\frac{9}{11}$, unless P=NP. We also consider a variant of the problem where the $p$ paths must be disjoint. For this problem, we give an algorithm which gets within a factor $\frac{1}{H(p)}$ of the optimum solution, where $H(p)$ is the $p$'th harmonic number ($H(p) \sim \ln(p)$). We show that in the case where the network is acyclic, we can find such a maximum $p$-path-flow in polynomial time for every $p$. We determine the complexity of a number of related problems concerning the structure of flows. For the special case of acyclic digraphs, some of the results we obtain are in some sense best possible.

Figures (8)

• Figure 1: The network $\mathcal{N}_\mathcal{F}$ when $\mathcal{F} = \textcolor{orange}{(x_1 \vee x_2 \vee \neg x_3)} \wedge \textcolor{g-green}{(\neg x_1 \vee x_2 \vee x_3)} \wedge \textcolor{g-blue}{(\neg x_1 \vee \neg x_2 \vee \neg x_3)}$.
• Figure 2: The structure of the network $\mathcal{N}$.
• Figure 3: The variable-gadget $W$.
• Figure 4: An illustration of the network $\mathcal{N}_{\mathcal{F}}$. Three undirected edges between the same pair of vertices represent a copy of $W$. Arcs in a copy of $W$ have the same capacities as in Figure \ref{['fig:vertex_gadget_npc_dplus']}. Arcs incident to vertices in $\{q_i,r_i \mid 1 \leq i \leq m\}$ have capacity $1$.
• Figure 5: A gadget that ensures $d^-(t) \leq 2$ in the support of every flow.

Theorems & Definitions (41)

• Lemma 1
• Lemma 2
• proof
• Theorem 3: Fortune, Hopcroft and Wyllie fortuneTCS10
• Theorem 4: Fortune, Hopcroft and Wyllie fortuneTCS10
• Theorem 5: Slivkins slivkinsSJDM24
• Theorem 6: berman2004approximation
• Theorem 7
• proof
• Remark 8