Complexity and Approximation Algorithms for Fixed Charge Transportation Problems

TL;DR

This work initiates a systematic study of the Fixed Charge Transportation (FCT) problem and its variants, including PFCT, PFCT-S, PFCT-U, FCT, FCT-S, and FCT-U, and establishes a comprehensive map of their approximability and computational complexity. It introduces tight constant-factor algorithms for several variants (notably a 2-approximation for PFCT-S and FCT-U, and a (6/5+ε)-approximation for PFCT-U), while proving strong hardness results that link PFCT and related problems to DST, Set Cover, and 3DM-B. The paper also provides an Efficient Parameterized Approximation Scheme (EPAS) for PFCT parameterized by the number of sources $n$, and a bicriteria approximation for FCT, enabling feasible but approximately optimal solutions when exact feasibility is relaxed. The core techniques combine LP-based greedy schemes, balanced-set and $k$-set packing reductions, and carefully crafted reductions from DST, Set Cover, and 3DM-B, offering both practical algorithms and deep insights into the intrinsic difficulty of these network-design problems. Overall, the results lay the groundwork for a richer theory of fixed-charge transportation and related fixed-cost network problems, with implications for routing, pipelines, and infrastructure design where fixed setup costs dominate variable costs.

Abstract

The Fixed Charge Transportation (FCT) problem models transportation scenarios where we need to send a commodity from $n$ sources to $m$ sinks, and the cost of sending a commodity from a source to a sink consists of a linear component and a fixed component. Despite extensive research on exponential time exact algorithms and heuristic algorithms for FCT and its variants, their approximability and computational complexity are not well understood. In this work, we initiate a systematic study of the approximability and complexity of these problems. When there are no linear costs, we call the problem the Pure Fixed Charge Transportation (PFCT) problem. We also distinguish between cases with general, sink-independent, and uniform fixed costs; we use the suffixes ``-S'' and ``-U'' to denote the latter two cases, respectively. This gives us six variants of the FCT problem. We give a complete characterization of the existence of $O(1)$-approximation algorithms for these variants. In particular, we give $2$-approximation algorithms for FCT-U and PFCT-S, and a $(6/5 + ε)$-approximation for PFCT-U. On the negative side, we prove that FCT and PFCT are NP-hard to approximate within a factor of $O(\log^{2-ε} (\max\{n, m\}))$ for any constant $ε> 0$, FCT-S is NP-hard to approximate within a factor of $c\log (\max\{n, m\})$ for some constant $c> 0$, and PFCT-U is APX-hard. Additionally, we design an Efficient Parameterized Approximation Scheme (EPAS) for PFCT when parameterized by the number $n$ of sources, and an $O(1/ε)$-bicriteria approximation for the FCT problem, when we are allowed to violate the demand constraints for sinks by a factor of $1\pm ε$.

Figures (4)

• Figure 1: Reductions used in our hardness results.
• Figure 2: Splitting operation in reduction from PFCT-Digraph to PFCT.
• Figure 3: Notations used in proof of Theorem \ref{['thm:reduce-candidate-set']}. In the three sub-figures, the $n$ sources are denoted by squares, the $m$ sinks are denoted by dots, the sinks $[m']$ are those in the yellow rectangle, and sinks $T'$ are represented by red dots. The lines connecting the sources and the sinks in sub-figure (a) denote $\mathrm{supp}(x^*)$. The lines in sub-figures (b) and (c) denote the pre-selected edges for the first and second PFCT-S instances respectively. The blue thick lines show the difference between the two instances.
• Figure 4: Reduction from Set Cover to FCT-S.

Theorems & Definitions (36)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10: BK