An algorithm for the min concave cost flow problem

Gallo, Giorgio; Sandi, Claudio; Sodini, Claudio

doi:10.1016/0377-2217(80)90109-5

Cited by 54 publications

(18 citation statements)

References 6 publications

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“…(R q ) is a special case of a POP with box constraints (6). Hence, we can use Lemma 2 to ensure Assumption 2 of Theorem 1.…”

Section: Case: Quadratic Concave Functionmentioning

confidence: 99%

“…Network flow problems arising from practical applications in transportation systems, facility location and production planning can be formulated with such mathematical models. For instance, the paper [6] has considered and solved the network flow problems with a square root concave cost function derived from the Italian road network and the U.S. telephone network. We refer the reader to [9,17] for further discussions of the applications.…”

Section: Introductionmentioning

confidence: 99%

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Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables

Mizutani

Yamashita

2012

J Glob Optim

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We present a hierarchy of semidefinite programming (SDP) relaxations for solving the concave cost transportation problem (CCTP), which is known to be NP-hard, with p suppliers and q demanders. In particular, we study cases in which the cost function is quadratic or square-root concave. The key idea of our relaxation methods is in the change of variables to CCTPs, and due to this, we can construct SDP relaxations whose matrix variables are of size O ((min{p, q}) ω ) in the relaxation order ω. The sequence of optimal values of SDP relaxations converges to the global minimum of the CCTP as the relaxation order ω goes to infinity. Furthermore, the size of the matrix variables can be reduced to O ((min{p, q}) ω−1 ), ω ≥ 2 by using Reznick's theorem. Numerical experiments were conducted to assess the performance of the relaxation methods.

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“…(R q ) is a special case of a POP with box constraints (6). Hence, we can use Lemma 2 to ensure Assumption 2 of Theorem 1.…”

Section: Case: Quadratic Concave Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables

Mizutani

Yamashita

2012

J Glob Optim

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“…A number of real-world problems may be formulated as network flow problems involving concave latency functions. Cost functions of this type are useful when dealing with network routing problems in the presence of economy of scale (see Gallo et al [13]). We present a generalized Braess instance that shows that for the concave case, our bound is tight; a similar instance can be used to show that for higher-degree polynomials with nonnegative coefficients, our bounds are almost tight and leave only a small gap for small values of .…”

Section: Techniquesmentioning

confidence: 99%

Stackelberg Routing in Arbitrary Networks

2010

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We investigate the impact of Stackelberg routing to reduce the price of anarchy in network routing games. In this setting, an fraction of the entire demand is first routed centrally according to a predefined Stackelberg strategy and the remaining demand is then routed selfishly by (nonatomic) players. Although several advances have been made recently in proving that Stackelberg routing can, in fact, significantly reduce the price of anarchy for certain network topologies, the central question of whether this holds true in general is still open. We answer this question negatively by constructing a family of singlecommodity instances such that every Stackelberg strategy induces a price of anarchy that grows linearly with the size of the network. Moreover, we prove upper bounds on the price of anarchy of the largest-latency-first (LLF) strategy that only depend on the size of the network. Besides other implications, this rules out the possibility to construct constant-size networks to prove an unbounded price of anarchy. In light of this negative result, we consider bicriteria bounds. We develop an efficiently computable Stackelberg strategy that induces a flow whose cost is at most the cost of an optimal flow with respect to demands scaled by a factor of 1 + √ 1 − . Finally, we analyze the effectiveness of an easy-to-implement Stackelberg strategy, called SCALE. We prove bounds for a general class of latency functions that includes polynomial latency functions as a special case. Our analysis is based on an approach that is simple yet powerful enough to obtain (almost) tight bounds for SCALE in general networks. 1. Introduction. Over the past years, the impact of the behavior of selfish, uncoordinated users in congested networks has been investigated intensively in the theoretical computer science and operations research literature. In this context, network routing games have proved to be an appropriate means of modeling selfish behavior in networks. The basic idea is to model the interaction between the selfish network users as a noncooperative game. We are given a directed graph with latency functions on the arcs and a set of origin-destination pairs, called commodities. Every commodity has a demand associated with it that specifies the amount of flow that needs to be sent from the respective origin to the destination. We assume that every demand represents a large population of players, each controlling an infinitesimal small amount of flow of the entire demand (such players are also called nonatomic). The latency that a player experiences to traverse an arc is given by a (nondecreasing) function of the total flow on that arc. We assume that every player acts selfishly and routes his flow along a minimum-latency path from its origin to the destination; this corresponds to a common solution concept for noncooperative games, that of a Nash equilibrium (here Nash or Wardrop flow; see Wardrop [37]). In a Nash flow, no player can improve his own latency by unilaterally switching to another path.It is well-known that ...

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“…Soland [17] and Kelly et al [10] approximate a concave function by linear functions, whose accuracy is improved during the branching procedures. Also, Gallo [6] proposed a branch and bound procedure which works well when the number of demand nodes is small.…”

Section: Introductionmentioning

confidence: 99%

An algorithm for a concave production cost network flow problem

Yajima

Konno

1999

Japan J. Indust. Appl. Math.

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We will consider a minimum concave cost production-transportation problem on an m X n bipartite network in which the production cost of k out of m supply nodes are concave, while those of m -k supply nodes as well as the transportation costs are linear. We will convert this problem into a concave minimization problem over a polytope in a k + 1-dimensional space and then apply an outer approximation method. This algorithm is finitely convergent and is very efficient for problems with large m and n if k, the number of concave nodes, is less than six.

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An algorithm for the min concave cost flow problem

Cited by 54 publications

References 6 publications

Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables

Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables

Stackelberg Routing in Arbitrary Networks

An algorithm for a concave production cost network flow problem

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