This paper is an attempt to deal with the recent realization [VY21] that the Hylland-Zeckhauser mechanism, which has remained a classic in economics for one-sided matching markets, is likely to be highly intractable. HZ uses the power of a pricing mechanism, which has endowed it with nice game-theoretic properties.[HV21] define a rich collection of Nash-bargaining-based models for one-sided and twosided matching markets, in both Fisher and Arrow-Debreu settings, together with implementations using available solvers, and very encouraging experimental results. This naturally raises the question of finding efficient combinatorial algorithms for these models.In this paper, we give efficient combinatorial algorithms based on the techniques of multiplicative weights update (MWU) and conditional gradient descent (CGD) for several onesided and two-sided models defined in [HV21]. Additionally, we define for the first time a Nash-bargaining-based model for non-bipartite matching markets and solve it using CGD. Furthermore, in every case, we study not only the Fisher but also the Arrow-Debreu version; the latter is also called the exchange version. We give natural applications for each model studied. These models inherit the game-theoretic and computational properties of Nash bargaining.We also establish a deep connection between HZ and the Nash-bargaining-based models, thereby confirming that the alternative to HZ proposed in [HV21] is a principled one.
We study the classical, randomized Ranking algorithm which is known to be (1 − 1 e )-competitive in expectation for the Online Bipartite Matching Problem. We give a tail inequality bound (Theorem 1), namely that Ranking is (1 − 1 e − α)-competitive with probability at least 1 − e −2α 2 n where n is the size of the maximum matching in the instance. Building on this, we show similar concentration results for the Fully Online Matching Problem and for the Online Vertex-Weighted Bipartite Matching Problem.
We consider the capacitated cycle covering problem: given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the vertices with vertex-disjoint cycles, each serving a demand of at most one. The objective is to minimize a linear combination of the total length and the number of cycles. This problem is closely related to the capacitated vehicle routing problem (CVRP) and other cycle cover problems such as min-max cycle cover and bounded cycle cover. We show that a greedy algorithm followed by a post-processing step yields a $$(2 + \frac{2}{7})$$ ( 2 + 2 7 ) -approximation for this problem by comparing the solution to a polymatroid relaxation. We also show that the analysis of our algorithm is tight and provide a $$2 + \epsilon $$ 2 + ϵ lower bound for the relaxation.
We develop theoretical foundations and practical algorithms for vehicle routing with time-dependent travel times. We also provide new benchmark instances and experimental results.First, we study basic operations on piecewise linear arrival time functions. In particular, we devise a faster algorithm to compute the pointwise minimum of a set of piecewise linear functions and a monotonicity-preserving variant of the Imai-Iri algorithm to approximate an arrival time function with fewer breakpoints.Next, we show how to evaluate insertion and deletion operations in tours efficiently and update the underlying data structure faster than previously known when a tour changes. Evaluating a tour also requires a scheduling step which is non-trivial in the presence of time windows and time-dependent travel times. We show how to perform this in linear time.Based on these results, we develop a local search heuristic to solve real-world vehicle routing problems with various constraints efficiently and report experimental results on classical benchmarks. Since most of these do not have time-dependent travel times, we generate and publish new benchmark instances that are based on real-world data. This data also demonstrates the importance of considering time-dependent travel times in instances with tight time windows.
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