Preprints
The online matching problem was introduced by Karp, Vazirani and Vazirani (STOC 1990) on bipartite graphs with vertex arrivals. It is well-known that the optimal competitive ratio is 1−1/e for both integral and fractional versions of the problem. Since then, there has been considerable effort to find optimal competitive ratios for other related settings. In this work, we go beyond the graph case and study the online matching problem on k-uniform hypergraphs. For k = 3, we provide an optimal primal-dual fractional algorithm, which achieves a competitive ratio of (e−1)/(e+1) ≈ 0.4621. As our main technical contribution, we present a carefully constructed adversarial instance, which shows that this ratio is in fact optimal. It combines ideas from known hard instances for bipartite graphs under the edge-arrival and vertex-arrival models. For k≥3, we give a simple integral algorithm which performs better than greedy when the online nodes have bounded degree. As a corollary, it achieves the optimal competitive ratio of 1/2 on 3-uniform hypergraphs when every online node has degree at most 2. This is because the special case where every online node has degree 1 is equivalent to the edge-arrival model on graphs, for which an upper bound of 1/2 is known.
Publications
We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Primal and dual feasibility were shown by Megiddo (SICOMP ’83) and Végh (MOR ’17) respectively. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale’s 9th problem.
Our approach is based on the recent primal-dual interior point method (IPM) due to Allamigeon, Dadush, Loho, Natura and Végh (FOCS ’22). The number of iterations needed by the IPM is bounded, up to a polynomial factor in the number of inequalities, by the straight line complexity of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path.
As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By applying a reduction of Hochbaum (ORL ’04), the same bound applies to any linear program with at most two non-zeros per column or per row.
To be able to run the IPM, one requires a suitable initial point. For this purpose, we develop a novel multistage approach, where each stage can be solved in strongly polynomial time given the result of the previous stage. Beyond this, substantial work is needed to ensure that the bit complexity of each iterate remains bounded during the execution of the algorithm. For this purpose, we show that one can maintain a representation of the iterates as a low complexity convex combination of vertices. Our approach is black-box and can be applied to any log-barrier path following method.
A set function can be extended to the unit cube in various ways; the correlation gap measures the ratio between two natural extensions. This quantity has been identified as the performance guarantee in a range of approximation algorithms and mechanism design settings. It is known that the correlation gap of a monotone submodular function is 1−1/e, and this is tight even for simple matroid rank functions.
We initiate a fine-grained study of correlation gaps of matroid rank functions. In particular, we present improved lower bounds on the correlation gap as parametrized by the rank and the girth of the matroid. We also show that the worst correlation gap of a weighted matroid rank function is achieved under uniform weights. Such improved lower bounds have direct applications for submodular maximization under matroid constraints, mechanism design, and contention resolution schemes. Previous work relied on implicit correlation gap bounds for problems such as list decoding and approval voting.
Parity games have witnessed several new quasi-polynomial algorithms since the breakthrough result of Calude et al. (2017). The combinatorial object underlying these approaches is a universal tree, as identified by Czerwiński et al. (2019). By providing a quasi-polynomial lower bound on the size of a universal tree, they have highlighted a barrier that must be overcome by all existing approaches to attain polynomial running time. This is due to the existence of worst case instances which force these algorithms to explore a large portion of the tree.
As an attempt to overcome this barrier, we propose a strategy iteration framework which can be applied on any universal tree. It is at least as fast as its value iteration counterparts, while allowing one to take bigger leaps in the universal tree. Our main technical contribution is an efficient method for computing the least fixed point of 1-player games. This is achieved via a careful adaptation of shortest path algorithms to the setting of ordered trees. By plugging in the universal tree of Jurdziński and Lazić (2017), or the Strahler universal tree of Daviaud et al. (2020), we obtain instantiations of the general framework that take time O(mn2log n log d) and O(mn2log3n log d) respectively per iteration.
We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold and Hemmecke (SIDMA 2015) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system {x∈ℝn: Ax = b, 0 ≤ x ≤ u} for A∈ℝm×n is bounded as O(m2log(m+κA) + n log n), where κA is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound e.g. if all entries of A have polynomially bounded encoding length in n. Further, we present circuit augmentation algorithms for LPs using the minimum-ratio circuit cancelling rule. Even though the standard minimum-ratio circuit cancelling algorithm is not finite in general, our variant can solve an optimization LP in O(n3log(n+κA)) augmentation steps.
An Accelerated Newton–Dinkelbach Method and its Application to Two Variables Per Inequality Systems
with Daniel Dadush, Bento Natura and László A. Végh
Mathematics of Operations Research 48(4): 1934–1958, 2023.
Conference version: ESA 2021.
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(m2log 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).
Cooperative games form an important class of problems in game theory, where the goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial time characterization of submodular instances, for prominent cooperative games that are in general non-convex. In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game. An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial-time characterization of submodular spanning tree games.
An edge-weighted graph G is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings. In particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with the minimum number of odd cycles in its support. This generalizes a fundamental and classical result on unweighted matchings given by Balas more than 30 years ago, which we expect to prove useful beyond this particular application. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P = NP. In this setting, we develop an O(Δ)-approximation algorithm for the problem, where Δ is the maximum degree of a node in G.
I am a postdoctoral researcher with Daniel Dadush at CWI Amsterdam, working on the ERC project Towards a Quantitative Theory of Integer Programming.
Previously, I completed my PhD in mathematics at the London School of Economics, under the supervision of László Végh.
I received my BMath and MMath degrees from the University of Waterloo, where I was supervised by Laura Sanità.