Ranking on Arbitrary Graphs: Rematch via Continuous Linear Programming

T.-H. Hubert Chan, Fei Chen, Xiaowei Wu, Zhichao Zhao

Abstract: Motivated by online advertisement and exchange settings, greedy randomized algorithms for the maximum matching problem have been studied, in which the algorithm makes (random) decisions that are essentially oblivious to the input graph. Any greedy algorithm can achieve a performance ratio of 0.5, which is the expected number of matched nodes to the number of nodes in a maximum matching. Since Aronson, Dyer, Frieze, and Suen [Random Structures Algorithm, 6 (1991), pp. 29--46] proved that the modified randomized greedy algorithm achieves a performance ratio of $0.5 + \epsilon$ (where $\epsilon = \frac{1}{400000}$) on arbitrary graphs in the midnineties, no further attempts in the literature have been made to improve this theoretical ratio for arbitrary graphs until two papers were published in FOCS 2012 [G. Goel and P. Tripathi, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 718--727; M. Poloczek and M. Szegedy, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 708--717]. In this paper, we revisit the ranking algorithm using the linear programming framework. Special care is given to analyze the structural properties of the ranking algorithm in order to derive the linear programming constraints, of which one known as the boundary constraint requires totally new analysis and is crucial to the success of our linear program (LP). We use continuous linear programming relaxation to analyze the limiting behavior as the finite LP grows. Of particular interest are new duality and complementary slackness characterizations that can handle the monotone and the boundary constraints in continuous linear programming. Improving previous work, this paper achieves a theoretical performance ratio of $\frac{2(5-\sqrt{7})}{9} \approx 0.523$ on arbitrary graphs.