s-t paths using the min-sum algorithm

Ruozzi, Nicholas; Tatikonda, Sekhar

doi:10.1109/allerton.2008.4797655

Cited by 8 publications

(11 citation statements)

References 2 publications

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“…Indeed, the proof methods are different, and results of this paper provide 'implementation' of BP unlike results of [18,19]. We also take note of a work by Ruozzi and Tatikonda [27] that utilizes BP to find source-sink paths in the network.…”

Section: Prior Work On Bpmentioning

confidence: 97%

Belief Propagation for Min-Cost Network Flow: Convergence and Correctness

Gamarnik

Shah

Wei

2012

Operations Research

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Distributed, iterative algorithms operating with minimal data structure while performing little computation per iteration are popularly known as message-passing in the recent literature. Belief Propagation (BP), a prototypical message-passing algorithm, has gained a lot of attention across disciplines including communications, statistics, signal processing and machine learning as an attractive scalable, general purpose heuristic for a wide class of optimization and statistical inference problems. Despite its empirical success, the theoretical understanding of BP is far from complete.With the goal of advancing the state-of-art of our understanding of BP, we study the performance of BP in the context of the capacitated minimum-cost network flow problem -a corner stone in the development of theory of polynomial time algorithms for optimization problems as well as widely used in practice of operations research. As the main result of this paper, we prove that BP converges to the optimal solution in the pseudo-polynomial-time, provided that the optimal solution of the underlying network flow problem instance is unique and the problem parameters are integral. We further provide a simple modification of the BP to obtain a fully polynomial-time randomized approximation scheme (FPRAS) without requiring uniqueness of the optimal solution. This is the first instance where BP is proved to have fully-polynomial running time. Our results thus provide a theoretical justification for the viability of BP as an attractive method to solve an important class of optimization problems.

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Section: Prior Work On Bpmentioning

confidence: 97%

Belief Propagation for Min-Cost Network Flow: Convergence and Correctness

Gamarnik

Shah

Wei

2012

Operations Research

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“…Nonetheless, for graphs with cycles, the BP heuristic often performs very well. In network problems, the min-sum algorithm is applied to find the shortest path between two nodes [22] or minimize path lengths and link congestion [16]. For the min-cost network flow problem with linear or PLC costs on edges, BP was shown in [21] to converge to the correct solution (if the solution is unique).…”

Section: Bp Algorithm For Balanced Routingmentioning

confidence: 99%

Self-organization scheme for balanced routing in large-scale multi-hop networks

Badiu

Saad

Coon

2021

J. Phys. A: Math. Theor.

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We propose a self-organization scheme for cost-effective and load-balanced routing in multi-hop networks. To avoid overloading nodes that provide favourable routing conditions, we assign each node with a cost function that penalizes high loads. Thus, finding routes to sink nodes is formulated as an optimization problem in which the global objective function strikes a balance between route costs and node loads. We apply belief propagation (its min-sum version) to solve the network optimization problem and obtain a distributed algorithm whereby the nodes collectively discover globally optimal routes by performing low-complexity computations and exchanging messages with their neighbours. We prove that the proposed method converges to the global optimum after a finite number of local exchanges of messages. Finally, we demonstrate numerically our framework’s efficacy in balancing the node loads and study the trade-off between load reduction and total cost minimization.

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“…where w e 1 = w e 2 = w e . One can easily observe that solving LP (19) is equivalent to solving LP (18) due to our construction of G and w . Now, construct the following GM for LP (19):…”

Section: Example V: Edge Covermentioning

confidence: 99%

“…In the past years, there have been made extensive research efforts to understand BP performances on loopy GMs under connections to combinatorial optimization [3,21,12,20,2,24,19,10,6,1,22]. In particular, it has been studied about the BP convergence to the correct answer under a few classes of loopy GM formulations of combinatorial optimization problems: matching [3,21,12,20], perfect matching [2], matching with odd cycles [24], shortest path [19] and network flow [10]. The important common feature of these instances is that BP converges to a correct MAP assignment if linear programming (LP) relaxation of the MAP inference problem is tight, i.e., it has no integrality gap.…”

Section: Introductionmentioning

confidence: 99%

Convergence and Correctness of Max-Product Belief Propagation for Linear Programming

Park¹,

Shin²

2017

SIAM J. Discrete Math.

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The max-product belief propagation (BP) is a popular message-passing heuristic for approximating a maximum-a-posteriori (MAP) assignment in a joint distribution represented by a graphical model (GM). In the past years, it has been shown that BP can solve a few classes of linear programming (LP) formulations to combinatorial optimization problems including maximum weight matching, shortest path and network flow, i.e., BP can be used as a message-passing solver for certain combinatorial optimizations. However, those LPs and corresponding BP analysis are very sensitive to underlying problem setups, and it has been not clear what extent these results can be generalized to. In this paper, we obtain a generic criteria that BP converges to the optimal solution of given LP, and show that it is satisfied in LP formulations associated to many classical combinatorial optimization problems including maximum weight perfect matching, shortest path, traveling salesman, cycle packing, vertex/edge cover and network flow. methods [14] for large-scale inputs. However, these theoretical results on BP are very sensitive to underlying structural properties depending on specific problems and it is not clear what extent they can be generalized to, e.g., the BP analysis for matching problems [3,21,12,20] does not extend to even for perfect matching ones [2]. In this paper, we overcome such technical difficulties for enhancing the power of BP as a LP solver. ContributionWe establish a generic criteria for GM formulations of given LP so that BP converges to the optimal LP solution given arbitrary initialization. Consequently, it also provides a sufficient condition for guaranteeing that a BP fixed point is unique. As one can naturally expect given prior results, one of our conditions requires the LP tightness. Our main contribution is finding other sufficient generic conditions so that BP converges to the correct MAP assignment of GM. First of all, our generic criteria can rediscover all prior BP results on this line, including matching [3, 21, 12], perfect matching [2], matching with odd cycles [24] and shortest path [19], i.e., we provide a unified framework on establishing the convergence and correctness of BPs in relation to associated LPs. Furthermore, we provide new instances under our framework: we show that BP can solve LP formulations associated to other popular combinatorial optimizations including perfect matching with odd cycles, traveling salesman, cycle packing, network flow and vertex/edge cover, which are not known in the literature. Here, we remark that the same network flow problem was already studied using BP by Gamarnik et al. [10]. However, our BP is different from theirs and much simpler to implement/analyze: the authors study BP on continuous GMs, and we do BP on discrete GMs. While most prior known BP results on this line focused on the case when the associated LP has an integral solution, the proposed criteria naturally guides the BP design to compute fractional LP solutions as well (see Section 4.2 and Section 4.4...

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s-t paths using the min-sum algorithm

Cited by 8 publications

References 2 publications

Belief Propagation for Min-Cost Network Flow: Convergence and Correctness

Belief Propagation for Min-Cost Network Flow: Convergence and Correctness

Self-organization scheme for balanced routing in large-scale multi-hop networks

Convergence and Correctness of Max-Product Belief Propagation for Linear Programming

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