László A. Végh

Augmenting Undirected Node-Connectivity by One

We present a min-max formula for the problem of augmenting the node-connectivity of a graph by one and give a polynomial time algorithm for finding an optimal solution. We also solve the minimum-cost version for node-induced cost functions.

Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives

A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective ∑ij∈E Cij(fij) over feasible flows f, where on every arc ij of the network, Cij is a convex function. We give a strongly polynomial algorithm for finding an exact optimal solution for a broad class of such problems. The key characteristic of this class is that an optimal solution can be computed exactly provided its support. This includes separable convex quadratic objectives and also certain market equilibria problems: Fishers market with linear and with spending constraint utilities. We thereby give the first strongly polynomial algorithms for separable quadratic minimum-cost flows and for Fishers market with spending constraint utilities, settling open questions posed e.g. in [15] and in [35], respectively. The running time is O(m4 log m) for quadratic costs, O(n4+n2(m+n log n) log n) for Fishers markets with linear utilities and O(mn3 +m2(m+n log n) log m) for spending constraint utilities.

Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs

We present a 6-approximation algorithm for the minimum-cost k-node connected spanning sub graph problem, assuming that the number of nodes is at least k3(k-1)+k. We apply a combinatorial preprocessing, based on the Frank-Tardos algorithm for k-out connectivity, to transform any input into an instance such that the iterative rounding method gives a 2-approximation guarantee. This is the first constant-factor approximation algorithm even in the asymptotic setting of the problem, that is, the restriction to instances where the number of nodes is lower bounded by a function of k.

Augmenting undirected node-connectivity by one

We present a min-max formula for the problem of augmenting the node-connectivity of a graph by one and give a polynomial time algorithm for finding an optimal solution. We also solve the minimum cost version for node-induced cost functions.

A Rational Convex Program for Linear Arrow-Debreu Markets

We present a new flow-type convex program describing equilibrium solutions to linear Arrow-Debreu markets. Whereas convex formulations were previously known ([Nenakov and Primak 1983; Jain 2007; Cornet 1989]), our program exhibits several new features. It provides a simple necessary and sufficient condition and a concise proof of the existence and rationality of equilibria, settling an open question raised by Vazirani [2012]. As a consequence, we also obtain a simple new proof of the result in Mertens [2003] that the equilibrium prices form a convex polyhedral set.

Restricted b -matchings in degree-bounded graphs

We present a min-max formula and a polynomial time algorithm for a slight generalization of the following problem: in a simple undirected graph in which the degree of each node is at most t+1, find a maximum t-matching containing no member of a list

Oriented Euler complexes and signed perfect matchings

\mathcal{K}

Concave Generalized Flows with Applications to Market Equilibria

of forbidden Kt,t and Kt+1 subgraphs. An analogous problem for bipartite graphs without degree bounds was solved by Makai [15], while the special case of finding a maximum square-free 2-matching in a subcubic graph was solved in [1].

An algorithm to increase the node-connectivity of a digraph by one

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.

Fixed-Parameter Algorithms for Minimum-Cost Edge-Connectivity Augmentation

We consider a nonlinear extension of the generalized network How model, with the How leaving an arc being an increasing concave function of the How entering it, as proposed by Truemper [1] and Shigeno [2]. We give a polynomial time combinatorial algorithm for solving corresponding How maximization problems, finding an ε-approximate solution in O(m(m + log n) log(MUm/ε)) arithmetic operations and value oracle queries, where M and U are upper bounds on simple parameters. This also gives a new algorithm for linear generalized Hows, an efficient, purely scaling variant of the Fat-Path algorithm by Goldberg, Plotkin and Tardos [3], not using any cycle cancellations. We show that this general convex programming model serves as a common framework for several market equilibrium problems, including the linear Fisher market model and its various extensions. Our result immediately provides combinatorial algorithms for various extensions of these market models. This includes nonsymmetric Arrow-Debreu Nash bargaining, settling an open question by Vazirani [4].