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Michel X. Goemans

Publications and source records attributed to Michel X. Goemans.

12 recordsLinked to original sources
arXiv2026

Bicriteria Approximation Algorithms for Demand Matching

The demand matching problem generalizes both the knapsack problem and the $b$-matching problem. In this problem, each edge of a graph has a demand and a weight, each vertex has a capacity, and the goal is to find a maximum weight subset of edges whose total incident demand at every vertex does not exceed its capacity. We study $(α, β)$-bicriteria approximation algorithms, which return a solution of weight at least $1/α$ times the optimum while allowing an additive capacity violation of at most $β$ times the maximum edge demand. We give an iterative relaxation algorithm for the demand matching problem that exploits a structural characterization of strictly fractional extreme points of the natural LP relaxation, which reduces the residual rounding problem to odd-cycle instances. Combined with a better-of-two rounding strategy, this yields $(7/6, 1)$- and $(1, 1)$-bicriteria approximation algorithms for general and bipartite graphs, respectively. We further generalize this approach to obtain a parametric family of algorithms, including a $(1, 4/3)$-bicriteria approximation. Separately, for the more general $k$-hypergraph demand matching problem, we give a greedy, combinatorial $(k, 1)$-bicriteria approximation algorithm. We complement these algorithmic results with matching lower bounds relative to the natural LP relaxation for $β= 0$ and all $β\geq 1$, completely characterizing the trade-off between weight approximation and additive capacity violation in this range.

cs.DS↗
arXiv2022

Shrunk subspaces via operator Sinkhorn iteration

A recent breakthrough in Edmonds' problem showed that the noncommutative rank can be computed in deterministic polynomial time, and various algorithms for it were devised. However, only quite complicated algorithms are known for finding a so-called shrunk subspace, which acts as a dual certificate for the value of the noncommutative rank. In particular, the operator Sinkhorn algorithm, perhaps the simplest algorithm to compute the noncommutative rank with operator scaling, does not find a shrunk subspace. Finding a shrunk subspace plays a key role in applications, such as separation in the Brascamp-Lieb polytope, one-parameter subgroups in the null-cone membership problem, and primal-dual algorithms for matroid intersection and fractional matroid matching. In this paper, we provide a simple Sinkhorn-style algorithm to find the smallest shrunk subspace over the complex field in deterministic polynomial time. To this end, we introduce a generalization of the operator scaling problem, where the spectra of the marginals must be majorized by specified vectors. Then we design an efficient Sinkhorn-style algorithm for the generalized operator scaling problem. Applying this to the shrunk subspace problem, we show that a sufficiently long run of the algorithm also finds an approximate shrunk subspace close to the minimum exact shrunk subspace. Finally, we show that the approximate shrunk subspace can be rounded if it is sufficiently close. Along the way, we also provide a simple randomized algorithm to find the smallest shrunk subspace. As applications, we design a faster algorithm for fractional linear matroid matching and efficient weak membership and optimization algorithms for the rank-2 Brascamp-Lieb polytope.

cs.DS↗
arXiv2020

Polynomiality for Bin Packing with a Constant Number of Item Types

We consider the bin packing problem with d different item sizes s_i and item multiplicities a_i, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem. In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d >= 3. In fact, for constant d our algorithm solves the following problem in polynomial time: given two d-dimensional polytopes P and Q, find the smallest number of integer points in P whose sum lies in Q. Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.

cs.DS↗
arXiv2018

Stochastic Block Model for Hypergraphs: Statistical limits and a semidefinite programming approach

We study the problem of community detection in a random hypergraph model which we call the stochastic block model for $k$-uniform hypergraphs ($k$-SBM). We investigate the exact recovery problem in $k$-SBM and show that a sharp phase transition occurs around a threshold: below the threshold it is impossible to recover the communities with non-vanishing probability, yet above the threshold there is an estimator which recovers the communities almost asymptotically surely. We also consider a simple, efficient algorithm for the exact recovery problem which is based on a semidefinite relaxation technique.

math.PR↗
arXiv2018

Community Detection in Hypergraphs, Spiked Tensor Models, and Sum-of-Squares

We study the problem of community detection in hypergraphs under a stochastic block model. Similarly to how the stochastic block model in graphs suggests studying spiked random matrices, our model motivates investigating statistical and computational limits of exact recovery in a certain spiked tensor model. In contrast with the matrix case, the spiked model naturally arising from community detection in hypergraphs is different from the one arising in the so-called tensor Principal Component Analysis model. We investigate the effectiveness of algorithms in the Sum-of-Squares hierarchy on these models. Interestingly, our results suggest that these two apparently similar models exhibit significantly different computational to statistical gaps.

cs.DS↗
arXiv2017

Matroids are Immune to Braess Paradox

The famous Braess paradox describes the following phenomenon: It might happen that the improvement of resources, like building a new street within a congested network, may in fact lead to larger costs for the players in an equilibrium. In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur. In a nutshell, bases of matroids are exactly this maximal structure. We prove our characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra which may be of independent interest.

cs.GT↗
arXiv2011

Matroids and Integrality Gaps for Hypergraphic Steiner Tree Relaxations

Until recently, LP relaxations have played a limited role in the design of approximation algorithms for the Steiner tree problem. In 2010, Byrka et al. presented a ln(4)+epsilon approximation based on a hypergraphic LP relaxation, but surprisingly, their analysis does not provide a matching bound on the integrality gap. We take a fresh look at hypergraphic LP relaxations for the Steiner tree problem - one that heavily exploits methods and results from the theory of matroids and submodular functions - which leads to stronger integrality gaps, faster algorithms, and a variety of structural insights of independent interest. More precisely, we present a deterministic ln(4)+epsilon approximation that compares against the LP value and therefore proves a matching ln(4) upper bound on the integrality gap. Similarly to Byrka et al., we iteratively fix one component and update the LP solution. However, whereas they solve an LP at every iteration after contracting a component, we show how feasibility can be maintained by a greedy procedure on a well-chosen matroid. Apart from avoiding the expensive step of solving a hypergraphic LP at each iteration, our algorithm can be analyzed using a simple potential function. This gives an easy means to determine stronger approximation guarantees and integrality gaps when considering restricted graph topologies. In particular, this readily leads to a 73/60 bound on the integrality gap for quasi-bipartite graphs. For the case of quasi-bipartite graphs, we present a simple algorithm to transform an optimal solution to the bidirected cut relaxation to an optimal solution of the hypergraphic relaxation, leading to a fast 73/60 approximation for quasi-bipartite graphs. Furthermore, we show how the separation problem of the hypergraphic relaxation can be solved by computing maximum flows, providing a fast independence oracle for our matroids.

cs.DM↗
arXiv2010

Symmetric Submodular Function Minimization Under Hereditary Family Constraints

We present an efficient algorithm to find non-empty minimizers of a symmetric submodular function over any family of sets closed under inclusion. This for example includes families defined by a cardinality constraint, a knapsack constraint, a matroid independence constraint, or any combination of such constraints. Our algorithm make $O(n^3)$ oracle calls to the submodular function where $n$ is the cardinality of the ground set. In contrast, the problem of minimizing a general submodular function under a cardinality constraint is known to be inapproximable within $o(\sqrt{n/\log n})$ (Svitkina and Fleischer [2008]). The algorithm is similar to an algorithm of Nagamochi and Ibaraki [1998] to find all nontrivial inclusionwise minimal minimizers of a symmetric submodular function over a set of cardinality $n$ using $O(n^3)$ oracle calls. Their procedure in turn is based on Queyranne's algorithm [1998] to minimize a symmetric submodular

cs.DS↗
arXiv2002

Wide partitions, Latin tableaux, and Rota's basis conjecture

Say that mu is a ``subpartition'' of an integer partition lambda if the multiset of parts of mu is a submultiset of the parts of lambda, and define an integer partition lambda to be ``wide'' if for every subpartition mu of lambda, mu >= mu' in dominance order (where mu' denotes the conjugate or transpose of mu). Then Brian Taylor and the first author have conjectured that an integer partition lambda is wide if and only if there exists a tableau of shape lambda such that (1) for all i, the entries in the ith row of the tableau are precisely the integers from 1 to lambda_i inclusive, and (2) for all j, the entries in the jth column of the tableau are pairwise distinct. This conjecture was originally motivated by Rota's basis conjecture and, if true, yields a new class of integer multiflow problems that satisfy max-flow min-cut and integrality. Wide partitions also yield a class of graphs that satisfy ``delta-conjugacy'' (in the sense of Greene and Kleitman), and the above conjecture implies that these graphs furthermore have a completely saturated stable set partition. We present several partial results, but the conjecture remains very much open.

math.CO↗
arXiv2000

When Does the Positive Semidefiniteness Constraint Help In Lifting Procedures

We study the lift-and-project procedures of Lovász and Schrijver for 0-1 integer programming problems. We prove that the procedure using the positive semidefiniteness constraint is not better than the one without it, in the worst case. Various examples are considered. We also provide geometric conditions characterizing when the positive semidefiniteness constraint does not help.

math.CO↗