[Paper Review] A Combinatorial $ ilde{O}(m^{3/2})$-time Algorithm for the Min-Cost Flow Problem

Ruben Becker, Andreas Karrenbauer|arXiv (Cornell University)|Dec 13, 2013

Low-power high-performance VLSI design13 references3 citations

TL;DR

This paper presents a combinatorial $ ilde{O}(m^{3/2})$-time algorithm for the min-cost flow problem by constructing an auxiliary network with interior primal and dual solutions of potential $P_0 = ilde{O}(ar{m}^{1/2})$ in linear time, using a novel potential reduction method guided by electrical flow computations and a crossover procedure to compute optimal integral potentials in near-linear time, achieving the best-known time bound combinatorially for the first time.

ABSTRACT

We present a combinatorial method for the min-cost flow problem and prove that its expected running time is bounded by $ ilde O(m^{3/2})$. This matches the best known bounds, which previously have only been achieved by numerical algorithms or for special cases. Our contribution contains three parts that might be interesting in their own right: (1) We provide a construction of an equivalent auxiliary network and interior primal and dual points with potential $P_0= ilde{O}(\sqrt{m})$ in linear time. (2) We present a combinatorial potential reduction algorithm that transforms initial solutions of potential $P_0$ to ones with duality gap below $1$ in $ ilde O(P_0\cdot \mbox{CEF}(n,m,ε))$ time, where $ε^{-1}=O(m^2)$ and $\mbox{CEF}(n,m,ε)$ denotes the running time of any combinatorial algorithm that computes an $ε$-approximate electrical flow. (3) We show that solutions with duality gap less than $1$ suffice to compute optimal integral potentials in $O(m+n\log n)$ time with our novel crossover procedure. All in all, using a variant of a state-of-the-art $ε$-electrical flow solver, we obtain an algorithm for the min-cost flow problem running in $ ilde O(m^{3/2})$.

Motivation & Objective

• To close the gap between combinatorial and numerical algorithms for min-cost flow by achieving the $ ilde{O}(m^{3/2})$ time bound combinatorially.
• To construct an auxiliary network with low-initial-potential interior primal and dual solutions in linear time.
• To design a combinatorial potential reduction algorithm that reduces duality gap below 1 in $ ilde{O}(P_0 \cdot \text{CEF}(n,m,\epsilon))$ time using electrical flow subroutines.
• To develop a novel crossover procedure that computes optimal integral potentials in $O(m + n\log n)$ time from solutions with duality gap less than 1.
• To achieve the best-known time complexity for min-cost flow using only combinatorial operations, avoiding logarithms and square roots in computation.

Proposed method

• Construct an auxiliary network $G_1$ with $3m$ arcs and $n_1 = O(n)$ nodes, preserving the optimal solution of the original graph $G_0$.
• Compute a tree solution in $G_0$ and derive initial primal and dual solutions $x$, $y$, $s$ in $G_1$ such that $x_a s_a \in [t, t + \Gamma^2]$ for all arcs, with $t = m\Gamma^3$.
• Use a potential function $P(x,s) = q\ln(x^T s) - \sum_a \ln(x_a s_a) - m_1 \ln m_1$ with $q = m_1 + p$, $p = \min\{k \in \mathbb{Z} : k^2 \geq m_1\}$, to guide the search toward optimality.
• Perform primal steps by updating flow along cycles, guided by the gradient of the potential function projected onto the cycle space, and dual steps when the gradient is nearly orthogonal to the cycle space.
• Compute the required projections via electrical flow subroutines, where resistances are inversely proportional to primal variable values and current sources are derived from the gradient.
• Apply a novel crossover procedure to transform solutions with duality gap less than 1 into optimal integral potentials in $O(m + n\log n)$ time.

Experimental results

Research questions

• RQ1 Can a combinatorial algorithm achieve the $\tilde{O}(m^{3/2})$ time bound for min-cost flow, matching the best-known numerical methods?
• RQ2 Can a low-initial-potential interior solution be constructed in linear time for an auxiliary network with the same optimal solution as the original?
• RQ3 Can a combinatorial potential reduction method reduce the duality gap below 1 in $\tilde{O}(P_0 \cdot \text{CEF}(n,m,\epsilon))$ time using only combinatorial operations?
• RQ4 Can a near-linear time crossover procedure convert solutions with duality gap less than 1 into optimal integral potentials?
• RQ5 Can the entire algorithm be implemented using only basic arithmetic and comparisons, avoiding logarithms and square roots in computation?

Key findings

• The algorithm constructs an auxiliary network $G_1$ with $m_1 = 3m$ arcs and $n_1 = O(n)$ nodes in $O(m)$ time, preserving the optimal solution of the original graph.
• The initial primal and dual solutions in $G_1$ have potential $P_0 = \tilde{O}(\sqrt{m})$, achieved in linear time via Algorithm 3.
• The potential reduction algorithm reduces the duality gap below 1 in $\tilde{O}(P_0 \cdot \text{CEF}(n,m,\epsilon))$ time, where $\epsilon^{-1} = O(m^2)$.
• The crossover procedure computes optimal integral potentials in $O(m + n\log n)$ time from solutions with duality gap less than 1.
• Using a state-of-the-art combinatorial $\varepsilon$-electrical flow solver, the overall algorithm runs in $\tilde{O}(m^{3/2})$ expected time.
• The algorithm is fully combinatorial: it uses only basic arithmetic and comparisons, with logarithms and square roots appearing only in the analysis, not in computation.