arXiv subjects

László A. Végh

Publications and source records attributed to László A. Végh.

At least 37 records · Page 2Linked to original sources

arXiv2022

Auction Algorithms for Market Equilibrium with Weak Gross Substitute Demands

We consider the Arrow--Debreu exchange market model under the assumption that the agents' demands satisfy the weak gross substitutes (WGS) property. We present a simple auction algorithm that obtains an approximate market equilibrium for WGS demands assuming the availability of a price update oracle. We exhibit specific implementations of such an oracle for WGS demands with bounded price elasticities and for Gale demand systems. As an application of our result, we obtain an efficient algorithm to find an approximate spending-restricted market equilibrium for WGS demands, a model that has been recently introduced as a continuous relaxation of the Nash social welfare (NSW) problem. This leads to a polynomial-time constant factor approximation algorithm for the NSW problem with capped additive separable piecewise linear utility functions; only a pseudopolynomial approximation algorithm was known for this setting previously.

cs.GT↗

arXiv2022

A Strongly Polynomial Algorithm for Linear Exchange Markets

We present a strongly polynomial algorithm for computing an equilibrium in Arrow-Debreu exchange markets with linear utilities. Our algorithm is based on a variant of the weakly-polynomial Duan-Mehlhorn (DM) algorithm. We use the DM algorithm as a subroutine to identify revealed edges, i.e. pairs of agents and goods that must correspond to best bang-per-buck transactions in every equilibrium solution. Every time a new revealed edge is found, we use another subroutine that decides if there is an optimal solution using the current set of revealed edges, or if none exists, finds the solution that approximately minimizes the violation of the demand and supply constraints. This task can be reduced to solving a linear program (LP). Even though we are unable to solve this LP in strongly polynomial time, we show that it can be approximated by a simpler LP with two variables per inequality that is solvable in strongly polynomial time.

cs.DS↗

arXiv2021

Circuit imbalance measures and linear programming

We study properties and applications of various circuit imbalance measures associated with linear spaces. These measures describe possible ratios between nonzero entries of support-minimal nonzero vectors of the space. The fractional circuit imbalance measure turns out to be a crucial parameter in the context of linear programming, and two integer variants can be used to describe integrality properties of associated polyhedra. We give an overview of the properties of these measures, and survey classical and recent applications, in particular, for linear programming algorithms with running time dependence on the constraint matrix only, and for circuit augmentation algorithms. We also present new bounds on the diameter and circuit diameter of polyhedra in terms of the fractional circuit imbalance measure.

math.CO↗

arXiv2021

Approximating Equilibrium under Constrained Piecewise Linear Concave Utilities with Applications to Matching Markets

We study the equilibrium computation problem in the Fisher market model with constrained piecewise linear concave (PLC) utilities. This general class captures many well-studied special cases, including markets with PLC utilities, markets with satiation, and matching markets. For the special case of PLC utilities, although the problem is PPAD-hard, Devanur and Kannan (FOCS 2008) gave a polynomial-time algorithm when the number of items is constant. Our main result is a fixed parameter approximation scheme for computing an approximate equilibrium, where the parameters are the number of agents and the approximation accuracy. This provides an answer to an open question by Devanur and Kannan for PLC utilities, and gives a simpler and faster algorithm for matching markets as the one by Alaei, Jalaly and Tardos (EC 2017). The main technical idea is to work with the stronger concept of thrifty equilibria, and approximating the input utility functions by `robust' utilities that have favorable marginal properties. With some restrictions, the results also extend to the Arrow--Debreu exchange market model.

cs.GT↗

arXiv2021

An Accelerated Newton-Dinkelbach Method and its Application to Two Variables Per Inequality Systems

We present an accelerated, or 'look-ahead' version of the Newton-Dinkelbach method, a well-known technique for solving fractional and parametric optimization problems. This acceleration halves the Bregman divergence between the current iterate and the optimal solution within every two iterations. Using the Bregman divergence as a potential in conjunction with combinatorial arguments, we obtain strongly polynomial algorithms in three applications domains: (i) For linear fractional combinatorial optimization, we show a convergence bound of $O(m \log m)$ iterations; the previous best bound was $O(m^2 \log m)$ by Wang et al. (2006). (ii) We obtain a strongly polynomial label-correcting algorithm for solving linear feasibility systems with two variables per inequality (2VPI). For a 2VPI system with $n$ variables and $m$ constraints, our algorithm runs in $O(mn)$ iterations. Every iteration takes $O(mn)$ time for general 2VPI systems, and $O(m + n \log n)$ time for the special case of deterministic Markov Decision Processes (DMDPs). This extends and strengthens a previous result by Madani (2002) that showed a weakly polynomial bound for a variant of the Newton-Dinkelbach method for solving DMDPs. (iii) We give a simplified variant of the parametric submodular function minimization result by Goemans et al. (2017).

cs.DS↗

arXiv2020

A Constant-Factor Approximation Algorithm for the Asymmetric Traveling Salesman Problem

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem (ATSP). Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. The main idea of our approach is a reduction to Subtour Partition Cover, an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. We first show that any algorithm for Subtour Partition Cover can be turned into an algorithm for ATSP while only losing a small constant factor in the performance guarantee. Next, we present a reduction from general ATSP instances to structured instances, on which we then solve Subtour Partition Cover, yielding our constant-factor approximation algorithm for ATSP.

cs.DS↗

arXiv2020

Revisiting Tardos's Framework for Linear Programming: Faster Exact Solutions using Approximate Solvers

In breakthrough work, Tardos (Oper. Res. '86) gave a proximity based framework for solving linear programming (LP) in time depending only on the constraint matrix in the bit complexity model. In Tardos's framework, one reduces solving the LP $\min \langle c,{x}\rangle$, $Ax=b$, $x \geq 0$, $A \in \mathbb{Z}^{m \times n}$, to solving $O(nm)$ LPs in $A$ having small integer coefficient objectives and right-hand sides using any exact LP algorithm. This gives rise to an LP algorithm in time poly$(n,m\logΔ_A)$, where $Δ_A$ is the largest subdeterminant of $A$. A significant extension to the real model of computation was given by Vavasis and Ye (Math. Prog. '96), giving a specialized interior point method that runs in time poly$(n,m,\log\barχ_A)$, depending on Stewart's $\barχ_A$, a well-studied condition number. In this work, we extend Tardos's original framework to obtain such a running time dependence. In particular, we replace the exact LP solves with approximate ones, enabling us to directly leverage the tremendous recent algorithmic progress for approximate linear programming. More precisely, we show that the fundamental "accuracy" needed to exactly solve any LP in $A$ is inverse polynomial in $n$ and $\log\barχ_A$. Plugging in the recent algorithm of van den Brand (SODA '20), our method computes an optimal primal and dual solution using ${O}(m n^{ω+1} \log (n)\log(\barχ_A+n))$ arithmetic operations, outperforming the specialized interior point method of Vavasis and Ye and its recent improvement by Dadush et al (STOC '20). At a technical level, our framework combines together approximate LP solutions to compute exact ones, making use of constructive proximity theorems -- which bound the distance between solutions of "nearby" LPs -- to keep the required accuracy low.

math.OC↗

arXiv2020

A Simpler and Faster Strongly Polynomial Algorithm for Generalized Flow Maximization

We present a new strongly polynomial algorithm for generalized flow maximization that is significantly simpler and faster than the previous strongly polynomial algorithm [Végh16]. For the uncapacitated problem formulation, the complexity bound $O(mn(m+n\log n)\log (n^2/m))$ improves on the previous estimate by almost a factor $O(n^2)$. Even for small numerical parameter values, our running time bound is comparable to the best weakly polynomial algorithms. The key new technical idea is relaxing the primal feasibility conditions. This allows us to work almost exclusively with integral flows, in contrast to all previous algorithms for the problem.

cs.DS↗

arXiv2020

Geometric Rescaling Algorithms for Submodular Function Minimization

We present a new class of polynomial-time algorithms for submodular function minimization (SFM), as well as a unified framework to obtain strongly polynomial SFM algorithms. Our algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige-Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. Firstly, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound $O(({n}^4\cdot \textrm{EO} + {n}^5)\log ({n} L))$. Secondly, we exhibit a general combinatorial black-box approach to turn $\varepsilon L$-approximate SFM oracles into strongly polynomial exact SFM algorithms. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige-Wolfe algorithm. Combined with the geometric rescaling technique, the black-box approach provides an $O(({n}^5\cdot \textrm{EO} +{n}^6)\log^2{n})$ algorithm. Finally, we show that one of the techniques we develop in the paper can also be combined with the cutting-plane method of Lee, Sidford, and Wong \cite{LSW}, yielding a simplified variant of their $O(n^3 \log^2 n \cdot \textrm{EO} + n^4\log^{O(1)} n)$ algorithm.

math.OC↗

arXiv2019

Signed tropical convexity

We establish a new notion of tropical convexity for signed tropical numbers. We provide several equivalent descriptions involving balance relations and intersections of open halfspaces as well as the image of a union of polytopes over Puiseux series and hyperoperations. Along the way, we deduce a new Farkas lemma and Fourier-Motzkin elimination without the non-negativity restriction on the variables. This leads to a Minkowski-Weyl theorem for polytopes over the signed tropical numbers.

math.OC↗

arXiv2019

Rescaling Algorithms for Linear Conic Feasibility

We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix $A\in \mathbb{R}^{m\times n}$, the kernel problem requires a positive vector in the kernel of $A$, and the image problem requires a positive vector in the image of $A^\top$. Both algorithms iterate between simple first order steps and rescaling steps. These rescalings improve natural geometric potentials. If Goffin's condition measure $ρ_A$ is negative, then the kernel problem is feasible and the worst-case complexity of the kernel algorithm is $O\left((m^3n+mn^2)\log{|ρ_A|^{-1}}\right)$; if $ρ_A>0$, then the image problem is feasible and the image algorithm runs in time $O\left(m^2n^2\log{ρ_A^{-1}}\right)$. We also extend the image algorithm to the oracle setting. We address the degenerate case $ρ_A=0$ by extending our algorithms to find maximum support nonnegative vectors in the kernel of $A$ and in the image of $A^\top$. In this case the running time bounds are expressed in the bit-size model of computation: for an input matrix $A$ with integer entries and total encoding length $L$, the maximum support kernel algorithm runs in time $O\left((m^3n+mn^2)L\right)$, while the maximum support image algorithm runs in time $O\left(m^2n^2L\right)$. The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for Linear Programming.

math.OC↗

arXiv2018

On Submodular Search and Machine Scheduling

Suppose some objects are hidden in a finite set $S$ of hiding places which must be examined one-by-one. The cost of searching subsets of $S$ is given by a submodular function and the probability that all objects are contained in a subset is given by a supermodular function. We seek an ordering of $S$ that finds all the objects in minimal expected cost. This problem is NP-hard and we give an efficient combinatorial $2$-approximation algorithm, generalizing analogous results in scheduling theory. We also give a new scheduling application $1|prec|\sum w_A h(C_A)$, where a set of jobs must be ordered subject to precedence constraints to minimize the weighted sum of some concave function $h$ of the completion times of {\em subsets} of jobs. We go on to give better approximations for submodular functions with low {\em total curvature} and we give a full solution when the problem is what we call {\em series-parallel decomposable}. Next, we consider a zero-sum game between a cost-maximizing Hider and a cost-minimizing Searcher. We prove that the equilibrium mixed strategies for the Hider are in the base polyhedron of the cost function, suitably scaled, and we solve the game in the series-parallel decomposable case, giving approximately optimal strategies in other cases.

math.OC↗

arXiv2017

Approximating Minimum Cost Connectivity Orientation and Augmentation

We investigate problems addressing combined connectivity augmentation and orientations settings. We give a polynomial-time 6-approximation algorithm for finding a minimum cost subgraph of an undirected graph $G$ that admits an orientation covering a nonnegative crossing $G$-supermodular demand function, as defined by Frank. An important example is $(k,\ell)$-edge-connectivity, a common generalization of global and rooted edge-connectivity. Our algorithm is based on a non-standard application of the iterative rounding method. We observe that the standard linear program with cut constraints is not amenable and use an alternative linear program with partition and co-partition constraints instead. The proof requires a new type of uncrossing technique on partitions and co-partitions. We also consider the problem setting when the cost of an edge can be different for the two possible orientations. The problem becomes substantially more difficult already for the simpler requirement of $k$-edge-connectivity. Khanna, Naor, and Shepherd showed that the integrality gap of the natural linear program is at most $4$ when $k=1$ and conjectured that it is constant for all fixed $k$. We disprove this conjecture by showing an $Ω(|V|)$ integrality gap even when $k=2$.

cs.DS↗

arXiv2017

Constant Factor Approximation for ATSP with Two Edge Weights

We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently by Svensson. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held-Karp relaxation, which may be of independent interest.

cs.DS↗

arXiv2017

Decomposable Submodular Function Minimization: Discrete and Continuous

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.

cs.LG↗

arXiv2016

A strongly polynomial algorithm for generalized flow maximization

A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every dual optimal solution, and thus can be contracted. As a consequence of the result, we also obtain a strongly polynomial algorithm for the linear feasibility problem with at most two nonzero entries per column in the constraint matrix.

cs.DS↗

arXiv2015

A 7/3-Approximation for Feedback Vertex Sets in Tournaments

We consider the minimum-weight feedback vertex set problem in tournaments: given a tournament with non-negative vertex weights, remove a minimum-weight set of vertices that intersects all cycles. This problem is $\mathsf{NP}$-hard to solve exactly, and Unique Games-hard to approximate by a factor better than 2. We present the first $7/3$ approximation algorithm for this problem, improving on the previously best known ratio $5/2$ given by Cai et al. [FOCS 1998, SICOMP 2001].

cs.DS↗

arXiv2014

Oriented Euler Complexes and Signed Perfect Matchings

This paper presents "oriented pivoting systems" as an abstract framework for complementary pivoting. It gives a unified simple proof that the endpoints of complementary pivoting paths have opposite sign. A special case are the Nash equilibria of a bimatrix game at the ends of Lemke-Howson paths, which have opposite index. For Euler complexes or "oiks", an orientation is defined which extends the known concept of oriented abstract simplicial manifolds. Ordered "room partitions" for a family of oriented oiks come in pairs of opposite sign. For an oriented oik of even dimension, this sign property holds also for unordered room partitions. In the case of a two-dimensional oik, these are perfect matchings of an Euler graph, with the sign as defined for Pfaffian orientations of graphs. A near-linear time algorithm is given for the following problem: given a graph with an Eulerian orientation with a perfect matching, find another perfect matching of opposite sign. In contrast, the complementary pivoting algorithm for this problem may be exponential.

cs.DM↗