Fonction sous-modulaire

En optimisation combinatoire, les fonctions sous-modulaires sont des fonctions d'ensemble particulières. Soient E un ensemble et f une fonction qui à tout sous-ensemble X de E associe un réel f(X), on dit que f est sous-modulaire si l'inégalité suivante est vérifiée pour tout sous-ensemble X et Y de E f(X \cap Y) + f(X \cup Y) \le f(X) + f(Y). Les fonctions sous-modulaire peuvent être vues comme l'analogue discret des fonctions convexes. Définition Soient E un ensemble et f une fonction qui à tout sous-ensemble X de E associe un réel f(X), on dit que f est sous-modulaire si l'inégalité suivante est vérifiée pour tout sous-ensemble X et Y de E f(X \cap Y) + f(X \cup Y) \le f(X) + f(Y). Une définition équivalente est que pour tout X, Y \subseteq \Omega avec X \subseteq Y et tout x \in \Omega \backslash Y on a : f(X\cup {x})-f(X)\geq f(Y\cup {x})-f(Y). Cette définition est parfois appelée

Streaming and Matching Problems with Submodular Functions

Paritosh Garg

Submodular functions are a widely studied topic in theoretical computer science. They have found several applications both theoretical and practical in the fields of economics, combinatorial optimization and machine learning. More recently, there have also been numerous works that study combinatorial problems with submodular objective functions. This is motivated by their natural diminishing returns property which is useful in real-world applications. The thesis at hand is concerned with the study of streaming and matching problems with submodular functions.Firstly, motivated by developing robust algorithms, we propose a new adversarial injections model, in which the input is ordered randomly, but an adversary may inject misleading elements at arbitrary positions. We study the maximum matching problem and cardinality constrained monotone submodular maximization. We show that even under this seemingly powerful adversary, it is possible to break the barrier of 1/2 for both these problems in the streaming setting. Our main result is a novel streaming algorithm that computes a 0.55-approximation for cardinality constrained monotone submodular maximization.In the second part of the thesis, we study the problem of matroid intersection in the semi-streaming setting. Our main result is a (2 + e)-approximate semi-streaming algorithm for weighted matroid inter- section improving upon the previous best guarantee of 4 + e. While our algorithm is based on the local ratio technique, its analysis differs from the related problem of weighted maximum matching and uses the concept of matroid kernels. We are also able to generalize our results to work for submodular functions by adapting ideas from a recent result by Levin and Wajc (SODA'21) on submodular maximization subject to matching constraints.Finally, we study the submodular Santa Claus problem in the restricted assignment case. The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application of their structural result. In the mentioned problem n unsplittable resources have to be assigned to m players, each with a monotone submodular utility function fi. The goal is to maximize mini fi(Si) where S1, . . . , Sm is a partition of the resources. The result by Goemans et al. implies a polynomial time O(n1/2+e)-approximation algorithm. In the restricted assignment case, each player is given a set of desired resources Gi and the individual valuation functions are defined as fi(S)= f(SnGi). OurmainresultisaO(loglog(n))-approximation algorithm for the problem. Our proof is inspired by the approach of Bansal and Srividenko (STOC'06) to the Santa Claus problem. Com- pared to the more basic linear setting, the introduction of submodularity requires a much more involved analysis and several new ideas.

EPFL2022

Submodular Stochastic Probing on Matroids

Marek Adamczyk, Justin Dean Ward

In a stochastic probing problem we are given a universe E, and a probability that each element e in E is active. We determine if an element is active by probing it, and whenever a probed element is active, we must permanently include it in our solution. Moreover, throughout the process we need to obey inner constraints on the set of elements taken into the solution, and outer constraints on the set of all probed elements. All previous algorithmic results in this framework have considered only the problem of maximizing a linear function of the active elements. Here, we consider submodular objectives. We provide new, constant-factor approximations for maximizing a monotone submodular function subject to multiple matroid constraints on both the elements that may be taken and the elements that may be probed. We also obtain an improved approximation for linear objective functions, and show how our approach may be generalized to handle k-matchoid constraints.

INFORMS2016

Streaming and Matching Problems with Submodular Functions

Paritosh Garg

EPFL2022

Streaming and Matching Problems with Submodular Functions

Submodular Stochastic Probing on Matroids

Maximizing k-Submodular Functions and Beyond

Streaming and Matching Problems with Submodular Functions

Maximizing k-Submodular Functions and Beyond

Submodular Stochastic Probing on Matroids