The Complexity of Non-Monotone Markets

We introduce the notion of non-monotone utilities, which covers a wide variety of utility functions in economic theory. We then prove that it is PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets with linear and non-monotone utilities. Building on this result, we settle the long-standing open problem regarding the computation of an approximate Arrow-Debreu market equilibrium in markets with CES utility functions, by proving that it is PPAD-complete when the Constant Elasticity of Substitution parameter \rho is any constant less than -1

Searching, Sorting, and Cake Cutting in Rounds

We study sorting and searching in rounds motivated by a cake cutting problem. The search problem we consider is: we are given an array $x = (x_1, \ldots, x_n)$ and an element $z$ promised to be in the array. We have access to an oracle that answers comparison queries: "How is $x_i$ compared to $x_j$?", where the answer can be "$$". The goal is to find the location of $z$ with success probability at least $p \in [0,1]$ in at most $k$ rounds of interaction with the oracle. The problem is called ordered or unordered search, depending on whether the array $x$ is sorted or unsorted, respectively. For ordered search, we show the expected query complexity of randomized algorithms is $\Theta\bigl(k\cdot p \cdot n^{1/k}\bigr)$ in the worst case. In contrast, the expected query complexity of deterministic algorithms searching for a uniformly random element is $\Theta\bigl(k\cdot p^{1/k} \cdot n^{1/k}\bigr)$. The uniform distribution is the worst case for deterministic algorithms. For unordered search, the expected query complexity of randomized algorithms is $np\bigl(\frac{k+1}{2k}\bigr) \pm 1$ in the worst case, while the expected query complexity of deterministic algorithms searching for a uniformly random element is $np \bigl(1 - \frac{k-1}{2k}p \bigr) \pm 1$. We also discuss the connections of these search problems to the rank query model, where the array $x$ can be accessed via queries of the form "Is rank$(x_i) \leq k$?". Unordered search is equivalent to Select with rank queries (given $q$, find $x_i$ with rank $q$) and ordered search to Locate with rank queries (given $x_i$, find its rank). We show an equivalence between sorting with rank queries and proportional cake cutting with contiguous pieces for any number of rounds, as well as an improved lower bound for deterministic sorting in rounds with rank queries.Comment: 33 pages, 4 figure

The Complexity of Optimal Multidimensional Pricing

We resolve the complexity of revenue-optimal deterministic auctions in the unit-demand single-buyer Bayesian setting, i.e., the optimal item pricing problem, when the buyer's values for the items are independent. We show that the problem of computing a revenue-optimal pricing can be solved in polynomial time for distributions of support size 2, and its decision version is NP-complete for distributions of support size 3. We also show that the problem remains NP-complete for the case of identical distributions

Clustering on k-Edge-Colored Graphs

International audienceWe study the Max k-colored clustering problem, where, given an edge-colored graph with k colors, we seek to color the vertices of the graph so as to find a clustering of the vertices maximizing the number (or the weight) of matched edges, i.e. the edges having the same color as their extremities. We show that the cardinality problem is NP-hard even for edge-colored bipartite graphs with a chromatic degree equal to two and k ≥ 3. Our main result is a constant approximation algorithm for the weighted version of the Max k-colored clustering problem which is based on a rounding of a natural linear programming relaxation. For graphs with chromatic degree equal to two, we improve this ratio by exploiting the relation of our problem with the Max 2-and problem. We also present a reduction to the maximum-weight independent set (IS) problem in bipartite graphs which leads to a polynomial time algorithm for the case of two colors

Clustering on kk-edge-colored graphs

International audienceWe study the Max kk-colored clustering problem, where given an edge-colored graph with kk colors, we seek to color the vertices of the graph so as to find a clustering of the vertices maximizing the number (or the weight) of matched edges, i.e. the edges having the same color as their extremities. We show that the cardinality problem is NP-hard even for edge-colored bipartite graphs with a chromatic degree equal to two and k≥3k≥3. Our main result is a constant approximation algorithm for the weighted version of the Max kk-colored clustering problem which is based on a rounding of a natural linear programming relaxation. For graphs with chromatic degree equal to two we improve this ratio by exploiting the relation of our problem with the Max 2-and problem. We also present a reduction to the maximum-weight independent set (IS) problem in bipartite graphs which leads to a polynomial time algorithm for the case of two colors