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Seminar on Combinatorics, Games and Optimisation

Below you'll find the programme for the Seminar on Combinatorics, Games and Optimisation (a joining together of the former Seminar on Discrete Mathematics and Game Theory and the former Seminar on Operations Research).

This seminar series covers many of the research areas in the Department: discrete mathematics, algorithms, game theory and operational research.

Unless stated below, this Seminar normally takes place:

Questions, suggestions, etc., about the seminar can be forwarded to the seminar administrator, by sending an e-mail to: seminar@maths.lse.ac.uk.

Upcoming Speakers:

Thursday 7 June - Alex Fink (QMUL)
2.00-3.00 pm, 32L.G.08
Title and abstract tbc

Previous seminars in the series:

Summer Term 2018

Thursday 24 May - Istvan Tomon (EPFL)

On the size of k-cross-free families
Two subsets A,B of a finite ground set X are said to be crossing, if none of the four sets A ∩ B, A \ B, B \ A and X \ (A ∪ B) are empty. A family of subsets of X is k-cross-free, if it does not contain k pairwise crossing elements. The notion of k-cross-free families was first introduced by Karzanov in the context of multicommodity flow problems, but these families also appear in combinatorial geometry, and in evolutionary biology.

It was conjectured by Karzanov and Lomonosov forty years ago that if a family F of subsets of the n-element ground set X does not contain k pairwise crossing elements, then |F| = O(kn). For k = 2 and 3, the conjecture is true, but for larger values of k the best known upper bound, due to Lomonosov, is |F| = O(kn log n). We improve this bound by showing that |F| = O_k(n log^* n) holds, where log^* denotes the iterated logarithm function.

This is joint work with Andrey Kupavskii and János Pach.

Thursday 17 May - Robert Simon (LSE)

Paradoxical decompositions and finitary rules
We colour every point x of a probability space X according to the colours of a finite list x_1 , x_2 , . . . , x_k of points such that each of the x_i, as a function of x, is a measure preserving transformation. We ask two questions about a colouring rule: (1) does there exist a finitely additive extension of the probability measure for which the x_i remain measure preserving and also a colouring obeying the rule almost everywhere that is measurable with respect to this extension?, and (2) does there exist any colouring obeying the rule almost everywhere? If the answer to the first question is no and to the second question yes, we say that the colouring rule is paradoxical. A paradoxical colouring rule not only allows for a paradoxical partition of the space, it requires one.

Thursday 26 April - Georg Loho (EPFL)

Monomial tropical cones for multicriteria optimization
We introduce a special class of tropical cones which we call 'monomial tropical cones'. They arise as a helpful tool in the description of discrete multicriteria optimization problems. After an introduction to tropical convexity with an emphasis on these particular tropical cones, we explain the algorithmic implications. We finish with connections to commutative algebra.

Wednesday 18 April - Fabrizio Grandoni (IDSIA Lugano)

Approximating Geometric Knapsack via L-packings
In the 2-dimensional geometric knapsack problem (2DK) we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is 2 + ε [Jansen and Zhang, SODA 2004]. In this work we break the 2 approximation barrier, achieving a polynomial-time 17/9 + ε < 1.89 approximation.

Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy. We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary of the knapsack, which contains items that are high and narrow or wide and thin.

As a second contribution, we present a PTAS for packing items in this L-shaped region. The previous best approximation factor for this subproblem was 2+ε (obtained via a trivial reduction to 1-dimensional knapsack by considering tall or wide items only).

We achieve better approximation factors for the cardinality case and/or when we are allowed to rotate items by 90 degrees. The previous best approximation factor for all such variants was 2 + ε as well [Jansen and Zhang, SODA 2004].

This is joint work with: Waldo Galvez, Sandy Heydrich, Salvatore Ingala, Arindam Khan, and Andreas Wiese.

Thursday 19 April - Eilon Solan (Tel Aviv University)

Optimal Repeated Inspection
We study a discounted repeated inspection game with two agents and one principal. Both agents may profit by violating certain rules, while the principal can inspect at most one agent in each period, inflicting a punishment on an agent who is caught violating the rules. The goal of the principal is to minimize the discounted number of violations, and she has a Stackelberg leader advantage. We characterize the principal's optimal inspection strategy. It turns out that the optimal history-dependent inspection strategy lowers the discounted loss due to violations to a small fraction of the performance of inspection strategies that were studied in the literature.

Joint work with Chang Zhao.

Wednesday 28 March - Jefferson Huang (Cornell)

Dynamic Scheduling and Maintenance of a Deteriorating Server 
Motivated by a quality control problem in semiconductor manufacturing, we consider a stochastic scheduling problem in the context of a multi-class queue with a single server whose service capacity deteriorates randomly over time. We show that the system may be unstable under a natural extension of the cμ-rule, and provide a sufficient condition for this rule to be optimal. We also consider the problem of jointly deciding whether to perform service or preventive maintenance, for which we provide insights into the structure of optimal policies and heuristics.

Lent Term 2018

Wednesday 21 March - Jugal Garg (UIUC)
Fisher Markets and Nash Social Welfare 

Fisher market equilibrium is a powerful solution concept that has found several surprising applications even in non-market settings which do not involve any exchange of money but only require the remarkable fairness and efficiency properties of equilibria, e.g., scheduling, auctions, mechanism design, fair division, etc. A very recent new application is approximation algorithms for maximizing the Nash social welfare (NSW) when allocating a set of indivisible items to a set of agents. 

In this talk, I will start with the Fisher market model and its connection with the NSW problem. Then, I will show how to design a constant-factor approximation algorithm for maximizing the NSW when agents have budget-additive valuations. Budget-additive valuations represent an important class of submodular functions. They have attracted a lot of research interest in recent years due to many interesting applications. 

This is based on a joint work with Martin Hoefer and Kurt Mehlhorn. 

Thursday 8 March Robert Simon (LSE)
Games of Incomplete Information and Myopic Equilibria

What happens if the payoffs of an infinitely repeated game are the sums of some payoffs on the first n stages and of some other payoffs from the undiscounted game? If it is a game of incomplete information on one side, there will be a Nash equilibrium. To prove this, a new equilibrium concept is introduced, that of myopic equilibria. With myopic equilibria an agent can become locked into severely suboptimal strategies by the expectations of others.

Joint work with S. Spiez, H. Torunczyk.

Wednesday 7 March - Sergei Chubanov (Siegen)
A polynomial scaling algorithm for linear programming

In this talk we will consider a polynomial algorithm for linear programming based on a scaling technique. As well as the ellipsoid method, this algorithm is applicable to linear problems given by separation oracles. We will also discuss some numerical results concerning machine learning applications and the perspectives of this algorithm in infinite-dimensional linear optimization.

Thursday 1 March - Andrey Kupavskii (Birmingham)
Lower Bounds for Searching Robots, some Faulty

Suppose we are sending out k robots from 0 to search the real line at a constant speed (with turns) to find a target at an unknown location; f of the robots are faulty, meaning that they fail to report the target although visiting its location. The goal is to find the target in time at most lambda |d|, if the target is located at d, |d|>1, for lambda as small as possible. In this work, we find a tight lower bound for lambda.  Joint work with Emo Welzl.

Thursday 22 February - Katherine Staden (Oxford)
The minimum number of triangles in a graph of given order and size

A famous theorem of Mantel from 1907 states that every n-vertex graph with more than n^2/4 edges contains at least one triangle. In the 50s, Erdős asked for a quantitative version of this statement: for every n and e, how many triangles must an n-vertex e-edge graph contain? This question has received a great deal of attention, and a long series of partial results culminated in an asymptotic solution by Razborov, extended to larger cliques by Nikiforov and Reiher. Until recently, an exact solution was only known for a small range of edge densities, due to Lovász and Simonovits. In this talk, I will discuss the history of the problem and some new work which gives an exact solution for almost the entire range of edge densities. This is joint work with Hong Liu and Oleg Pikhurko.

Wednesday 21 February - Peyton Young (LSE and Oxford)
The Speed of Innovation Diffusion in Social Networks

New technologies typically gain a foothold through the actions of a few innovators, and then diffuse more rapidly as more and more people come into contact with prior adopters.

Much of the prior literature focuses on the rate of diffusion as a function of the topology of a given network structure. Here we derive "topology-free" bounds on the expected waiting time until a given fraction of the population has adopted the innovation.

The bounds depend on the payoff gain from using the innovation instead of the status quo, and on the noisiness of the players' response functions, but they do not depend on the network structure per se. In particular, the bounds hold for directed and undirected networks of arbitrary size whose structure may be evolving over time.

Thursday 15 February - Fatemeh Mohammadi (Bristol)
Chip-firing Games: A Combinatorial and Geometric Perspective

The chip firing game is a solitaire game on graphs which has a very beautiful and rich mathematical structure, and it appears in several fields of mathematics and statistics. It starts from a configuration of dollars (chips) on the vertices of a fixed graph: in each step of a chip-firing game we may choose a vertex to lend one dollar to each of its neighbours, or to borrow one dollar from each of its neighbours. The goal of the game is to get all the vertices out of debt (with non-negative integers) by a sequence of legal moves. For any finite graph, there are only finitely many critical configurations whose vertices are not in dept and no subset of vertices can fire without going into debt. I will talk about critical configurations, their corresponding geometric objects and their applications in system reliability theory.

Thursday 8 February - Trine Tornoe Platz (Copenhagen Business School)
On totally balanced, submodular and PMAS-admissible weighted minimum colouring games

We introduce the weighted minimum colouring (WMC) games, which is a class of cooperative combinatorial optimization games. A graph G = (N,E) and a positive integer weight vector w that assigns a weight to each vertex in N induce a WMC game. Our aim is to characterize classes of graphs that induce WMC games with specific properties for either any choice of weight vector or for at least one weight vector. A graph G is said to be globally (respectively, locally) WMC totally balanced, submodular, or PMAS-admissible, if for all positive integer weight vectors w (respectively, for at least one positive integer weight vector w), the corresponding WMC game is totally balanced, submodular, or admits a PMAS (population monotonic allocation scheme). We show that a graph G is globally WMC totally balanced if and only if it is perfect, and that any graph G is locally WMC totally balanced. Furthermore, we show that G is globally (respectively, locally) WMC submodular if and only if it is complete r-partite (respectively, (2K2, P4)-free). Finally, we show that G is globally PMAS-admissible if and only if it is (2K2, P4)-free.

Joint work with Herbert Hamers, Nayat Horozoglu, and Henk Norde.

Wednesday 7 February - Tamas Kiraly (Eotvos University)
Tight √2-approximation for Linear 3-Cut

In the linear 3-cut problem, the input is a node-weighted directed graph and three specified terminal nodes s,r,t, and the goal is to find a minimum weight subset of non-terminal nodes whose removal ensures that s cannot reach r and t, and r cannot reach t. This problem is approximation-equivalent to the following arborescence blocking problem: given a node-weighted directed graph with a specified root node r, remove a minimum weight subset of non-root nodes such that the remaining digraph has no in-arborescence and no out-arborescence rooted at r.

Linear 3-cut contains undirected 3-way node cut as a special case, and can be reduced to directed 2-way cut. Under the Unique Games Conjecture, the best approximation ratios of these problems are 4/3 and 2, respectively. We show that the linear 3-cut problem has a √2-approximation algorithm and this is tight under UGC. The proof involves showing that, somewhat surprisingly, the integrality gap of the natural LP-relaxation is √2. Joint work with Kristóf Bérczi, Karthekeyan Chandrasekaran, and Vivek Madan.

Thursday 1 February - Igal Milchtaich (Bar-Ilan University)
Polyequilibrium

Polyequilibrium is a generalization of Nash equilibrium that is applicable to any strategic game, whether finite or otherwise, and to dynamic games, with perfect or imperfect information. It differs from equilibrium in specifying strategies that players do not choose and by requiring an after-the-fact justification for the exclusion of these strategies rather than the retainment of the non-excluded ones.

Specifically, for each excluded strategy of each player there must be a non-excluded one that responds to every profile of non-excluded strategies of the other players at least as well as the first strategy does. A polyequilibrium's description of the outcome of the game may be more or less specific, depending on the number and the identities of the retained, non-excluded strategy profiles. A particular result (e.g., Pareto efficiency of the payoffs) is said to hold in a polyequilibrium if it holds for all non-excluded profiles. Such a result does not necessarily hold in any Nash equilibrium in the game. In this sense, the generalization proposed in this work extends the set of justifiable predictions concerning a game's results.

Wednesday 17 January - David C. Parkes (Paulson School of Engineering and Applied Sciences, Harvard University)
Optimal Economic Design through Deep Learning
***This seminar was part of the Social and Economic Data Science Seminars Series***

Designing an auction that maximizes expected revenue is an intricate task. Despite major efforts, only the single-item case is fully understood. We explore the use of tools from deep learning on this topic. The design objective that we adopt is revenue optimal, dominant-strategy incentive compatible auctions. For a baseline, we show that multi-layer neural networks can learn almost-optimal auctions for a variety of settings for which there are analytical solutions, and even without leveraging characterization results. We also show that deep learning can be used to derive auctions for poorly understood problems, including settings with multiple items and budget constraints. Our research also demonstrates that the deep learning framework is quite general, being applicable to other problems of optimal economic design.

Joint work with Paul Duetting (LSE), Zhe Feng (Harvard University), and Harikrishna Narasimhan (Harvard University). Working paper: https://arxiv.org/abs/1706.03459

Thursday 11 January - Johannes Carmesin (Cambridge)
Embedding simply connected 2-complexes in 3-space

We characterise the embeddability of simply connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski’s characterisation of graph planarity, by excluded minors. This answers questions of Lovász, Pardon and Wagner.

2017, 2016

The Seminar on Combinatorics, Games and Optimisation started in 2016.  It is a combination of two previous seminar series:

Seminar on Discrete Mathematics and Game Theory: 

2016, 2015, 2014, 2013, 2012, 2011, 2010, 2009, 2008, 2007, 2006, 2005, 2004, 2003

Seminar on Operations Research: 

2016, 2015, 2014, 2013, 2012, 2011, 2010