László A. Végh

Sharat Ibrahimpur, László A. Végh

In the Flexible Graph Connectivity (FGC) problem, we are given an undirected multigraph on $n$ vertices with nonnegative edge costs, where each edge is classified as either safe or unsafe. Given integer parameters $p$ and $q$, the goal in $(p,q)$-FGC is to purchase a minimum-cost set of edges such that the resulting spanning subgraph remains $p$-edge-connected after the removal of any set of up to $q$ unsafe edges. Our main contribution is an $O(\log n)$-approximation algorithm based on independent rounding, improving the previous best approximation ratio of $O(q \log n)$. Central to our approach is a new linear programming formulation of feasible solutions that encodes knapsack cover inequalities as cut-capacity constraints. Unlike prior work, the capacity of an edge in a cut may depend on the partially purchased solution for this cut. We show that the resulting linear program admits a polynomial-time separation oracle. Scaling the fractional solution by $Θ(\log n)$ and applying independent rounding yields a feasible integral solution with constant probability; here, we leverage the knapsack cover inequalities to obtain strong concentration bounds for the rounded solution relative to any given partial solution. A key ingredient in both separation and rounding is the use of Karger's bound on the number of near-minimum cuts. We also extend the $(p,q)$-FGC problem to model more than two safety tiers and show that our results and techniques extend naturally to this setting, albeit with increased approximation ratios and running times that scale with the number of tiers.

Marianna E. -Nagy, László A. Végh

Linear Complementarity Problems (LCPs) with sufficient matrices form an important subclass of LCPs, and it remains a significant open question whether problems in this class can be solved in polynomial time. Kojima, Megiddo, Noma, and Yoshise gave an Interior Point Algorithm (IPA) in 1991, that can solve LCPs with sufficient matrices in time bounded polynomially in the input size and the so-called handicap number $\hatκ(M)$ of the coefficient matrix $M$. However, this value can be exponentially large in the bit encoding length. In fact, no upper bounds were previously known on $\hatκ(M)$. Settling an open question raised in de Klerk and E.-Nagy (Math Programming, 2011), we give an exponential upper bound on $\hatκ(M)$ in the bit-complexity of $M$. This is based on a new characterization of sufficient matrices. The new characterization also leads to a simple new proof of Väliaho's theorem on the equivalence of sufficient and $\mathcal{P}^*$-matrices (Linear Algebra and its Applications, 1996). Noting that one can obtain an equivalent LCP by rescaling the rows and columns by a positive diagonal matrix, we define $\hatκ^\star(M)$ as the best possible handicap number achievable under such rescalings. Our second main result is an algorithm for LCPs with sufficient matrices, where the running time is polynomially bounded in the input size and in the optimized value $\hatκ^\star(M)$. This algorithm is based on the observation that the set of near-optimal row-rescalings forms a convex set. Our algorithm combines the Ellipsoid Method over the set of row rescalings, and an IPA with running time dependent on the handicap number of the matrix. If the IPA fails to solve the LCP in the desired running time, it provides a separation oracle to the Ellipsoid Method to find a better rescaling.

Christoph Hertrich, Yixin Tao, László A. Végh

Optimal auction design is a fundamental problem in algorithmic game theory. This problem is notoriously difficult already in very simple settings. Recent work in differentiable economics showed that neural networks can efficiently learn known optimal auction mechanisms and discover interesting new ones. In an attempt to theoretically justify their empirical success, we focus on one of the first such networks, RochetNet, and a generalized version for affine maximizer auctions. We prove that they satisfy mode connectivity, i.e., locally optimal solutions are connected by a simple, piecewise linear path such that every solution on the path is almost as good as one of the two local optima. Mode connectivity has been recently investigated as an intriguing empirical and theoretically justifiable property of neural networks used for prediction problems. Our results give the first such analysis in the context of differentiable economics, where neural networks are used directly for solving non-convex optimization problems.

Alina Ene, Huy L. Nguyen, László A. Végh

This paper investigates connections between discrete and continuous approaches for decomposable submodular function minimization. We provide improved running time estimates for the state-of-the-art continuous algorithms for the problem using combinatorial arguments. We also provide a systematic experimental comparison of the two types of methods, based on a clear distinction between level-0 and level-1 algorithms.