Generalized Perron--Frobenius Theorem for Nonsquare Matrices

The celebrated Perron--Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems theory, economics, statistics and optimization. However, many real-life scenarios give rise to nonsquare matrices. A natural question is whether the PF Theorem (along with its applications) can be generalized to a nonsquare setting. Our paper provides a generalization of the PF Theorem to nonsquare matrices. The extension can be interpreted as representing client-server systems with additional degrees of freedom, where each client may choose between multiple servers that can cooperate in serving it (while potentially interfering with other clients). This formulation is motivated by applications to power control in wireless networks, economics and others, all of which extend known examples for the use of the original PF Theorem. We show that the option of cooperation between servers does not improve the situation, in the sense that in the optimal solution no cooperation is needed, and only one server needs to serve each client. Hence, the additional power of having several potential servers per client translates into \emph{choosing} the best single server and not into \emph{sharing} the load between the servers in some way, as one might have expected. The two main contributions of the paper are (i) a generalized PF Theorem that characterizes the optimal solution for a non-convex nonsquare problem, and (ii) an algorithm for finding the optimal solution in polynomial time

A Characterization Theorem and An Algorithm for A Convex Hull Problem

Given $S= \{v_1, \dots, v_n\} \subset \mathbb{R} ^m$ and $p \in \mathbb{R} ^m$, testing if $p \in conv(S)$, the convex hull of $S$, is a fundamental problem in computational geometry and linear programming. First, we prove a Euclidean {\it distance duality}, distinct from classical separation theorems such as Farkas Lemma: $p$ lies in $conv(S)$ if and only if for each $p' \in conv(S)$ there exists a {\it pivot}, $v_j \in S$ satisfying $d(p',v_j) \geq d(p,v_j)$. Equivalently, $p \not \in conv(S)$ if and only if there exists a {\it witness}, $p' \in conv(S)$ whose Voronoi cell relative to $p$ contains $S$. A witness separates $p$ from $conv(S)$ and approximate $d(p, conv(S))$ to within a factor of two. Next, we describe the {\it Triangle Algorithm}: given $\epsilon \in (0,1)$, an {\it iterate}, $p' \in conv(S)$, and $v \in S$, if $d(p, p') < \epsilon d(p,v)$, it stops. Otherwise, if there exists a pivot $v_j$, it replace $v$ with $v_j$ and $p'$ with the projection of $p$ onto the line $p'v_j$. Repeating this process, the algorithm terminates in $O(mn \min \{\epsilon^{-2}, c^{-1}\ln \epsilon^{-1} \})$ arithmetic operations, where $c$ is the {\it visibility factor}, a constant satisfying $c \geq \epsilon^2$ and $\sin (\angle pp'v_j) \leq 1/\sqrt{1+c}$, over all iterates $p'$. Additionally, (i) we prove a {\it strict distance duality} and a related minimax theorem, resulting in more effective pivots; (ii) describe $O(mn \ln \epsilon^{-1})$-time algorithms that may compute a witness or a good approximate solution; (iii) prove {\it generalized distance duality} and describe a corresponding generalized Triangle Algorithm; (iv) prove a {\it sensitivity theorem} to analyze the complexity of solving LP feasibility via the Triangle Algorithm. The Triangle Algorithm is practical and competitive with the simplex method, sparse greedy approximation and first-order methods.Comment: 42 pages, 17 figures, 2 tables. This revision only corrects minor typo

On Computability of Equilibria in Markets with Production

Although production is an integral part of the Arrow-Debreu market model, most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. This paper takes a significant step towards understanding computational aspects of markets with production. We first define the notion of separable, piecewise-linear concave (SPLC) production by analogy with SPLC utility functions. We then obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production, and we give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 of extending his complementary pivot algorithm to markets with production. Since this is a path-following algorithm, we obtain a proof of membership of this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of existence of equilibrium (i.e., without using a fixed point theorem), rationality, and oddness of the number of equilibria. We further give a proof of PPAD-hardness for this problem and also for its restriction to markets with linear utilities and SPLC production. Experiments show that our algorithm runs fast on randomly chosen examples, and unlike previous approaches, it does not suffer from issues of numerical instability. Additionally, it is strongly polynomial when the number of goods or the number of agents and firms is constant. This extends the result of Devanur and Kannan (2008) to markets with production. Finally, we show that an LCP-based approach cannot be extended to PLC (non-separable) production, by constructing an example which has only irrational equilibria.Comment: An extended abstract will appear in SODA 201