An Improved Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market

We present an improved combinatorial algorithm for the computation of equilibrium prices in the linear Arrow-Debreu model. For a market with n agents and integral utilities bounded by U, the algorithm runs in O(n7 log3(nU)) time. This improves upon the previously best algorithm of Ye by a factor of Ω(n). The algorithm refines the algorithm described by Duan and Mehlhorn and improves it by a factor of Ω(n3). The improvement comes from a better understanding of the iterative price adjustment process, the improved balanced flow computation for nondegenerate instances, and a novel perturbation technique for achieving nondegeneracy.

[1]  D. Gale.  The linear exchange model , 1976 .

[2]  A. Wald.  Über einige Gleichungssysteme der mathematischen Ökonomie , 1936 .

[3]  Raimund Seidel,et al.  The Nature and Meaning of Perturbations in Geometric Computing , 1994, STACS.

[4]  Martin Fürer.  Faster integer multiplication , 2007, STOC '07.

[5]  K. Arrow,et al.  EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY , 1954 .

[6]  Kamal Jain,et al.  A polynomial time algorithm for computing an Arrow-Debreu market equilibrium for linear utilities , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[7]  L. Walras.  Elements of Pure Economics, or The Theory of Social Wealth , 1955 .

[8]  Yinyu Ye,et al.  A path to the Arrow–Debreu competitive market equilibrium , 2007, Math. Program..

[9]  Ruta Mehta,et al.  A complementary pivot algorithm for markets under separable, piecewise-linear concave utilities , 2012, STOC '12.

[10]  Richard Cole,et al.  Approximating the Nash Social Welfare with Indivisible Items , 2018, SIAM J. Comput..

[11]  Kurt Mehlhorn,et al.  A Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market , 2013, ICALP.

[12]  B. Eaves.  A finite algorithm for the linear exchange model , 1976 .

[13]  James B. Orlin,et al.  Improved algorithms for computing fisher's market clearing prices: computing fisher's market clearing prices , 2010, STOC '10.

[14]  Nikhil R. Devanur,et al.  A Rational Convex Program for Linear Arrow-Debreu Markets , 2013, ACM Trans. Economics and Comput..

[15]  Nikhil R. Devanur,et al.  Market equilibrium via a primal--dual algorithm for a convex program , 2008, JACM.