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Faster Approximate Lossy Generalized Flow via Interior Point Algorithms
Samuel I. Daitch, Daniel A. Spielman
TL;DR
The paper targets the cost of repeatedly solving structured linear systems within interior-point methods for generalized flow problems. It exploits symmetric M-matrix structure and obtains additive approximations for generalized flows while exactly solving standard flow problems.
Problem
Interior-point complexity is dominated by repeatedly solving linear systems, but generalized flow formulations produce symmetric M-matrices that may admit faster specialized solvers.
Method
The algorithm uses an interior-point method with an M-matrix solver based on approximate linear-system solving in the matrix norm.
Results
The generalized max-flow algorithm produces a flow no worse than ǫ less than the maximum possible flow to the sink, while the standard flow algorithms are exact.
Takeaways & Limitations
The algorithms use arithmetic with bit-length O(log(nU/ǫ)); the standard minimum-cost flow algorithm is exact for any integer flow value.
Takeaways & Limitations
The interior-point algorithm cannot currently produce an exact generalized-flow solution because the M-matrix solver depends on the matrix condition number.
Abstract
from arXiv · showhide
We present faster approximation algorithms for generalized network flow problems. A generalized flow is one in which the flow out of an edge differs from the flow into the edge by a constant factor. We limit ourselves to the lossy case, when these factors are at most 1. Our algorithm uses a standard interior-point algorithm to solve a linear program formulation of the network flow problem. The system of linear equations that arises at each step of the interior-point algorithm takes the form of a symmetric M-matrix. We present an algorithm for solving such systems in nearly linear time. The algorithm relies on the Spielman-Teng nearly linear time algorithm for solving linear systems in diagonally-dominant matrices. For a graph with m edges, our algorithm obtains an additive epsilon approximation of the maximum generalized flow and minimum cost generalized flow in time tildeO(m^(3/2) * log(1/epsilon)). In many parameter ranges, this improves over previous algorithms by a factor of approximately m^(1/2). We also obtain a similar improvement for exactly solving the standard min-cost flow problem.
1 Introduction
The paper targets generalized flow problems whose interior-point linear systems have symmetric M-matrix structure, enabling faster specialized solving. Its algorithms provide additive approximations for generalized flows and exact solutions for standard flow problems, with reduced arithmetic precision requirements.
- Motivation: Interior-point methods for generalized flow problems produce symmetric M-matrices, unlike the symmetric diagonally-dominant systems arising in standard flow problems.The paper exploits this specialized matrix structure instead of applying generic worst-case linear-system bounds.
- Generalized flow problems: The generalized maximum-flow formulation allows edge-specific multipliers, with lossy instances restricting every multiplier to at most 1.Flow is measured by the amount entering each edge, subject to inward capacities.
- Results: The algorithms return additive ǫ approximations for generalized flow problems, while the compared alternatives provide multiplicative error (1 + ǫ).For standard flow problems, the paper's algorithms are exact.
- Precision: The algorithms use arithmetic with bit length O(log(nU/ǫ)), whereas the authors suspect multiplicative-approximation algorithms may require longer numbers.This precision observation is stated as a suspicion about the compared algorithms, not as an established bound.
- Results: The comparison chart reports running-time expressions including m2(m + n log n) log U and min(n3/2, m1/2)m log(n2/m) log U.The supplied passages present these expressions as chart entries without fully identifying their associated problem labels.
- M-matrix solver: The M-matrix solver first finds a positive diagonal scaling that makes the matrix diagonally dominant, then applies the Spielman-Teng solver.Finding the scaling uses the Spielman-Teng solver an expected O(log n) times.
2 Interior-Point Algorithm using an Approximate Solver
The paper modifies a Renegar-style dual path-following interior-point method to use approximate linear-system solves. Under stated matrix, eigenvalue, boundedness, initialization, and error assumptions, the algorithm returns an approximately feasible, approximately optimal solution.
- Iteration structure: Each interior-point iteration takes a step through the linear-program polytope by solving a linear system.The fastest cited interior-point methods require O(√n) iterations to approach the solution.
- Approximate interior-point method: The method analyzes Renegar's short-step dual path-following algorithm while replacing exact linear-system solves with an approximate solver.The analysis follows Ye's treatment and is modified to account for approximate solving.
- Assumptions: The algorithm assumes A is n × m, AAT is positive definite, and λmin is a positive lower bound on the eigenvalues of AAT.These are among the input conditions stated for InteriorPoint.
- Assumptions: It also assumes a positive dual bound T, an initial point y0 satisfying AT y0 < c, and an error parameter 0 < ǫ < 1.The smallest initial slack s0_min and the largest absolute input entry U are also used in the stated conditions.
- Guarantee: The output satisfies approximate feasibility, ∥Ax − b∥ ≤ ǫ, and approximate objective quality, cT x < z∗ + ǫ.The output is specified as a positive vector x.
- Complexity: The interior-point algorithm makes O calls to the approximate solver, with the supplied passage leaving the full bound's expression incomplete.The theorem identifies the call count as a complexity component but does not provide the complete expression in the supplied text.
3 Solving Generalized Flow
The paper formulates lossy generalized flow problems as linear programs, uses interior-point outputs to construct approximate flows, and then rounds them to exact flows when needed. The resulting algorithms provide additive approximations for generalized flows and an exact algorithm for standard minimum-cost flow.
- Problem formulation: The network uses edge capacities, loss multipliers below 1, and costs, with a single sink in-edge assumed without loss of generality.Lossy flow means each edge multiplier γ(e) is less than 1; the sink has no out-edges and one in-edge.
- Approximate generalized flows: The generalized maximum-flow algorithm loses no more than ǫ flow relative to the maximum possible sink flow.Its output is an additive approximation rather than a multiplicative one.
- Approximate generalized flows: The generalized minimum-cost algorithm is within ǫ of maximum flow and costs no more than a minimum-cost maximum flow.The cost guarantee is preserved while approximating the attainable flow value.
- Rounding approximate flows: Interior-point solutions are approximate flows, which are balanced and scaled to obtain exact flows while preserving the relevant value and cost guarantees.The resulting exact generalized flow is within ǫ of the original maximum flow and costs no more than a minimum-cost maximum flow.
- Preprocessing: A least-lossy-paths tree supports preprocessing, after which vertices with excessive loss can be deleted with bounded impact on maximum-flow value.The tree is computed using shortest paths with edge weights −log γ(e); deleting such vertices reduces maximum flow by at most the stated ǫ-dependent amount.
- Linear-program formulation: The linear program enforces generalized flow conservation through Ax_1 = 0 and capacity feasibility through x_1 + x_2 = c.x_1 contains edge flows and x_2 contains unused inward capacities.
- Standard minimum-cost flow: The standard minimum-cost flow problem is solved exactly by combining interior-point output with a unique minimum-cost perturbation and rounding.The Isolation Lemma gives a unique minimum-cost vertex with probability at least 1/2, and the rounded flow is minimum-cost.
4 Solving linear systems in symmetric M-Matrices
The paper solves symmetric M-matrix systems by iteratively scaling them into diagonally-dominant systems, then applying the Spielman–Teng approximate solver. Random scaling reduces nondominant rows by a constant factor, yielding expected logarithmic iterations.
- A symmetric M-matrix is positive definite with non-positive off-diagonals and admits a factorization M = AA^T with at most two nonzeros per column of A.
- The algorithm constructs a positive diagonal matrix D so that DMD becomes diagonally dominant, enabling use of the Spielman–Teng solver.It iteratively scales the system while preserving the M-matrix structure.
- Each iteration solves O(log n) diagonally-dominant systems, and the algorithm terminates after O(log n) iterations in expectation.
- Random diagonal scaling makes a constant fraction of previously nondominant rows diagonally dominant with constant probability.The scaled matrix has no positive off-diagonals, so diagonal dominance can be checked through nonnegative row sums.
- Approximate diagonal Schur-complement values are computed with Johnson–Lindenstrauss random projections, avoiding construction of the full Schur complement.Only diagonal approximations are needed for the scaling lemma.
- The exact solver calls are replaced by Spielman–Teng approximate solves, whose expected time is eO(m log(1/ε)) for diagonally-dominant systems.
5 Final Remarks
The approach cannot currently produce exact generalized-flow solutions because the M-matrix solver depends on matrix conditioning. Extending the result to networks with gains is also limited by potentially ill-conditioned linear programs.
- Condition-number dependence prevents the interior-point algorithm from producing exact solutions to generalized flow problems.
- Networks with gains remain outside the result because the resulting linear programs may be ill-conditioned.
A Proofs for Section 4
The proofs establish that random scaling progressively improves diagonal dominance while randomized projections and concentration bounds control the approximations used in the process.
- Johnson–Lindenstrauss projections preserve the relevant vector norms with probability at least p, enabling approximate Schur-complement diagonal estimates.
- Chi-square concentration bounds control the number of projected norms that fall below a threshold, supporting the scaling analysis.
- Random diagonal scaling preserves positive diagonals and uses row-sum behavior to make a constant fraction of rows diagonally dominant.
- The proof defines near-positive rows through bounds on off-diagonal sums, then shows these rows enter the favorable set with substantial probability.
- The constructed matrix D′ is positive diagonal, preserving the M-matrix-compatible scaling required by the algorithm.
B Solving Matrices from the Interior-Point Method
Interior-point systems are reduced to M-matrix systems and solved using the paper’s MMatrixSolve procedure, with Schur complements and Sherman–Morrison handling structured updates.
- The interior-point method requires solving systems involving an M-matrix and structured diagonal or rank-one modifications.
- Schur complements reduce the structured systems to M-matrix systems whose eigenvalues can be bounded from the diagonal scaling parameters.
- A solver for the Schur complement can be lifted to a solver for the original matrix through a standard Schur-complement identity.
- Systems of the form M + vv^T are handled with the Sherman–Morrison formula and approximate M-matrix solves.
- The approximation guarantee is analyzed in the M-induced inner product using Cauchy–Schwarz-type bounds.
C Interior-Point Method using an Approximate Solver
This section formulates an approximate interior-point solver for a canonical primal–dual linear program and states its accuracy and solver-call complexity under explicit assumptions.
- Linear-program formulation: The primal and dual linear programs have the same solution, with z∗ defined as the maximum dual objective over the dual polytope.The dual objective is expressed as b^T y over feasible (y, s) pairs.
- Linear-program formulation: The linear program uses an n × m matrix A, length-m primal variables x, s, c, and length-n dual variables y, b, with m ≥ n.The paper notes that its n and m convention follows graph-theory usage rather than standard linear-programming notation.
- Algorithmic framework: The algorithm is based on Renegar’s interior-point method and uses an approximate solver in place of exact linear-system solves.The analysis follows Ye’s treatment of interior-point methods.
- Assumptions: The solver assumes positive definiteness of AAT, a lower eigenvalue bound λmin, bounded dual coordinates T, and a strictly feasible initial point y0.It also assumes bounded matrix and vector entries through U and defines s0_min from the initial slacks.
- Complexity: The interior-point algorithm makes O calls to the approximate solver under these assumptions.The supplied passage states the asymptotic form without preserving the omitted expression.
C.1 The Analytic Center
The analytic center is characterized through slack variables and a proximity measure, enabling Newton-type approximate-solver steps that remain feasible while moving closer to the center.
- Definition: The analytic center maximizes the product of the dual slacks and serves as the reference point for interior-point steps.The paper also uses an equivalent definition for its analysis.
- Proximity measure: For any interior dual point, ηA measures distance from the analytic center through the scaled slack vector.The listed properties include a lower bound involving all x satisfying Ax = 0.
- Newton step: The analytic-center slack bounds support feasibility and conditioning arguments for the NewtonStep procedure.The section explicitly establishes that the center’s slacks cannot be arbitrarily small.
- Proximity measure: If ηA(s) < 1, the point is close enough to the analytic center that its slacks remain within a constant ratio of the center’s slacks.This provides the neighborhood condition used by subsequent Newton steps.
- Newton step: NewtonStep uses one approximate-solver call to produce a point closer to the analytic center from a sufficiently close interior point.The resulting point remains inside the dual polytope because its new slacks are positive.
C.2 The Path-Following Algorithm
The path-following algorithm advances toward the optimum by shifting a dual-polytope constraint, then uses Newton correction and an approximate solver to maintain central-path proximity.
- Central path: The central path is the set of analytic centers of polytopes obtained by adding the constraint b^T y ≥ z, with z approaching z∗.As z approaches z∗, the polytope center approaches the dual linear-program solution.
- Shift procedure: Shift increases z toward z∗, updates the gap slack, and uses one solver call to return a point sufficiently close to the central path.The intermediate shifted point may move slightly farther from the path before NewtonStep restores proximity.
- InteriorPoint algorithm: The complete InteriorPoint algorithm starts from a point near the central path and repeatedly applies Shift until the gap slack is sufficiently small.The supplied passages identify this procedure as the dual path-following algorithm using an approximate solver.
- Correctness: The returned primal solution is positive, has bounded feasibility residual, and satisfies c^T x < z∗+ǫ.The objective guarantee is stated directly by Lemma C.9(iii).
- Iteration complexity: Each Shift iteration decreases the potential B(z) by Θ(√m), and this decrease implies progress through a smaller gap slack.The potential argument yields the stated bound on the number of InteriorPoint solver calls.
C.3 Finding the Central Path
FindCentralPath initializes path following by reversing the direction of progress from a feasible interior point, following an auxiliary central path until it reaches the original path’s neighborhood.
- Initialization: FindCentralPath uses the auxiliary vector b̄ = A(S0)^−1 1m derived from the initial feasible point (y0, s0).The initialization starts from a feasible interior point and defines an auxiliary path.
- Auxiliary path: The procedure runs path following in reverse, stepping away from the optimum associated with b̄ rather than toward the original optimum associated with b.This constructs a point near the central path required by the main algorithm.
- Auxiliary path: Starting at z̄0, FindCentralPath decreases z̄ along the b̄ central path until the auxiliary analytic center is sufficiently close to the original central path.The resulting point is then used to initialize InteriorPoint.
- Complexity analysis: The analysis tracks progress with a potential function B̄(z̄) and proves progress for the Unshift procedure near the auxiliary central path.The proof parallels the potential analysis used for the forward path-following algorithm.
- Complexity analysis: FindCentralPath makes O iterations before its auxiliary gap slack reaches the threshold needed for initialization.The supplied complexity statements omit the full asymptotic expression.
C.4 Calls to the Solver
The section bounds the interior-point algorithm’s solver calls and analyzes the matrix and slack conditions underlying those bounds.
- The analysis bounds the slack vector from below to control the condition number of the solver matrices.The slacks remain within a constant factor of the current central-path slacks, and the initial point remains interior during FindCentralPath.
- The algorithm makes O calls to the approximate solver and includes one final call of the latter form.
- The solver’s error parameter and matrix condition number are controlled using bounds on the slack vector and Lemma C.18.
- All solves use matrices of the form AS−2AT + vvT.
- The respective solve bounds follow from sgap = Ω(ǫ) and s̄gap = Ω(m).