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Minimum Cost Nowhere-zero Flows and Cut-balanced Orientations

Karthekeyan Chandrasekaran, Siyue Liu, R. Ravi

TL;DR

This work studies optimization versions of nowhere-zero flows and cut-balanced orientations on 2-edge-connected graphs, linking flow and orientation concepts via bidirected graphs and Jaeger’s equivalence. It establishes strong hardness results, proving that $ ext{WNZF}(k)$ and $ ext{WCBO}(k)$ admit no finite-factor approximations for any finite $k\ge3$, while delivering practical bicriteria approximations: a $(6,6)$-approximation for min-cost nowhere-zero $k$-flow and a $(k,6)$-approximation for min-cost $k$-cut-balanced orientation. For symmetric costs, the symmetric variant $ ext{SWNZF}(k)$ remains NP-hard, yet admits a 3-approximation when $k\ge6$ (and for $k=\infty$), and similarly a 3-approximation for the infinite-cost case. The methods combine LP relaxations with half-integrality, submodular-flow theory, and reductions between problems, yielding insights into both intractability and practical bicriteria strategies with connections to postman problems and well-balanced/orientation theory.

Abstract

Flows and colorings are disparate concepts in graph algorithms -- the former is tractable while the latter is intractable. Tutte introduced the concept of nowhere-zero flows to unify these two concepts. Jaeger showed that nowhere-zero flows are equivalent to cut-balanced orientations. Motivated by connections between nowhere-zero flows, cut-balanced orientations, Nash-Williams' well-balanced orientations, and postman problems, we study optimization versions of nowhere-zero flows and cut-balanced orientations. Given a bidirected graph with asymmetric costs on two orientations of each edge, we study the min cost nowhere-zero $k$-flow problem and min cost $k$-cut-balanced orientation problem. We show that both problems are NP-hard to approximate within any finite factor. Given the strong inapproximability result, we design bicriteria approximations for both problems: we obtain a $(6,6)$-approximation to the min cost nowhere-zero $k$-flow and a $(k,6)$-approximation to the min cost $k$-cut-balanced orientation. For the case of symmetric costs (where the costs of both orientations are the same for every edge), we show that the nowhere-zero $k$-flow problem remains NP-hard and admits a $3$-approximation.

Minimum Cost Nowhere-zero Flows and Cut-balanced Orientations

TL;DR

This work studies optimization versions of nowhere-zero flows and cut-balanced orientations on 2-edge-connected graphs, linking flow and orientation concepts via bidirected graphs and Jaeger’s equivalence. It establishes strong hardness results, proving that and admit no finite-factor approximations for any finite , while delivering practical bicriteria approximations: a -approximation for min-cost nowhere-zero -flow and a -approximation for min-cost -cut-balanced orientation. For symmetric costs, the symmetric variant remains NP-hard, yet admits a 3-approximation when (and for ), and similarly a 3-approximation for the infinite-cost case. The methods combine LP relaxations with half-integrality, submodular-flow theory, and reductions between problems, yielding insights into both intractability and practical bicriteria strategies with connections to postman problems and well-balanced/orientation theory.

Abstract

Flows and colorings are disparate concepts in graph algorithms -- the former is tractable while the latter is intractable. Tutte introduced the concept of nowhere-zero flows to unify these two concepts. Jaeger showed that nowhere-zero flows are equivalent to cut-balanced orientations. Motivated by connections between nowhere-zero flows, cut-balanced orientations, Nash-Williams' well-balanced orientations, and postman problems, we study optimization versions of nowhere-zero flows and cut-balanced orientations. Given a bidirected graph with asymmetric costs on two orientations of each edge, we study the min cost nowhere-zero -flow problem and min cost -cut-balanced orientation problem. We show that both problems are NP-hard to approximate within any finite factor. Given the strong inapproximability result, we design bicriteria approximations for both problems: we obtain a -approximation to the min cost nowhere-zero -flow and a -approximation to the min cost -cut-balanced orientation. For the case of symmetric costs (where the costs of both orientations are the same for every edge), we show that the nowhere-zero -flow problem remains NP-hard and admits a -approximation.

Paper Structure

This paper contains 16 sections, 20 theorems, 31 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background, Motivations, and Connections
  3. Our Results
  4. Techniques
  5. Preliminaries
  6. Reductions between $\operatorname{WNZF}(k)$ and $\operatorname{WCBO}(k)$
  7. Hardness of $\operatorname{WNZF}(k)$ and $\operatorname{WCBO}(k)$
  8. Bicriteria approximations for $\operatorname{WNZF}(k)$ and $\operatorname{WCBO}(k)$
  9. LP relaxation for $\operatorname{WNZF}(k)$ and its extreme point structure
  10. Bicriteria approximation for $\operatorname{WNZF}(k)$
  11. LP relaxation for $\operatorname{WCBO}(k)$ and its extreme point structure
  12. Bicriteria approximation algorithms for $\operatorname{WCBO}(k)$
  13. Min cost Symmetric Nowhere-Zero Flows
  14. Hardness of $\operatorname{SWNZF}(k)$
  15. Approximation algorithms
  16. ...and 1 more sections

Key Result

Theorem 1

For every finite integer $k\ge 3$, $\operatorname{WNZF}(k)$ and $\operatorname{WCBO}(k)$ are NP-hard to approximate within any finite factor.

Figures (5)

  • Figure 1: Left: part of graph $G=(V,E)$ and a partial orientation $\vec{F}$. Right: part of a nowhere-zero $k$-flow $(\vec{E},f)$ of $G$ such that $\vec{F}\subseteq \vec{E}$.
  • Figure 2: Left: the graph $G$ for restricted SAT instance $(x_1\vee x_2\vee x_3)\wedge (\bar{x}_1\vee \bar{x}_2\vee x_3)\wedge (x_1\vee \bar{x}_2\vee \bar{x}_3)$ and $k=4$. Right: an infeasible assignment $x_1=x_2=x_3=0$ yields a cut $X=\{u_1,u_2,u_3,v_1\}$ that violates the $k$-cut-balancedness condition. Here, arcs in $\delta^+(X)$ are colored red; arcs in $\delta^-(X)$ are colored blue.
  • Figure 3: (1) Cycle $R_i$ corresponding to variable $x_i$ with $d_i=3$. (2) Part of a nowhere-zero $(\vec{E},f)$ when $x_i=1$. (3) Part of a nowhere-zero $(\vec{E},f)$ when $x_i=0$.
  • Figure 4: The graph $G$ for NAE3SAT instance $(x_1\vee x_2\vee x_3)\wedge (\bar{x}_1\vee x_2\vee x_3)\wedge (\bar{x}_1\vee x_2\vee \bar{x}_3)$.
  • Figure 5: The nowhere-zero flow of total value $|E|+\sum_{i=1}^n d_i$ corresponding to a feasible assignment $x_1=1,x_2=1,x_3=0$ for NAE3SAT instance $(x_1\vee x_2\vee x_3)\wedge (\bar{x}_1\vee x_2\vee x_3)\wedge (\bar{x}_1\vee x_2\vee \bar{x}_3)$. The arcs of flow value $2$ are colored red, which are the even arcs of $R_1$, the even arcs of $R_2$, and the odd arcs of $R_3$.

Theorems & Definitions (40)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 7
  • proof
  • Lemma 8: Jaeger jaeger1976balanced
  • proof
  • ...and 30 more