An Algorithmic Theory of Integer Programming

We study the general integer programming problem where the number of variables $n$ is a variable part of the input. We consider two natural parameters of the constraint matrix $A$: its numeric measure $a$ and its sparsity measure $d$. We show that integer programming can be solved in time $g(a,d)\textrm{poly}(n,L)$, where $g$ is some computable function of the parameters $a$ and $d$, and $L$ is the binary encoding length of the input. In particular, integer programming is fixed-parameter tractable parameterized by $a$ and $d$, and is solvable in polynomial time for every fixed $a$ and $d$. Our results also extend to nonlinear separable convex objective functions. Moreover, for linear objectives, we derive a strongly-polynomial algorithm, that is, with running time $g(a,d)\textrm{poly}(n)$, independent of the rest of the input data. We obtain these results by developing an algorithmic framework based on the idea of iterative augmentation: starting from an initial feasible solution, we show how to quickly find augmenting steps which rapidly converge to an optimum. A central notion in this framework is the Graver basis of the matrix $A$, which constitutes a set of fundamental augmenting steps. The iterative augmentation idea is then enhanced via the use of other techniques such as new and improved bounds on the Graver basis, rapid solution of integer programs with bounded variables, proximity theorems and a new proximity-scaling algorithm, the notion of a reduced objective function, and others. As a consequence of our work, we advance the state of the art of solving block-structured integer programs. In particular, we develop near-linear time algorithms for $n$-fold, tree-fold, and $2$-stage stochastic integer programs. We also discuss some of the many applications of these classes.Comment: Revision 2: - strengthened dual treedepth lower bound - simplified proximity-scaling algorith

The ZERO Regrets Algorithm: Optimizing over Pure Nash Equilibria via Integer Programming

Designing efficient algorithms to compute Nash equilibria poses considerable challenges in Algorithmic Game Theory (AGT). We shed new light on the intersection between Algorithmic Game Theory and Integer Programming. We introduce ZERO Regrets, a general and efficient cutting plane algorithm to compute, enumerate, and select Pure Nash Equilibria (PNEs) in Integer Programming Games, a class of simultaneous and non-cooperative games. We present a theoretical foundation for our algorithmic reasoning and provide a polyhedral characterization of the convex hull of the Pure Nash Equilibria. We introduce the concept of equilibrium inequality, and devise an equilibrium separation oracle to separate non-equilibrium strategies from PNEs. We evaluate our algorithmic framework on a wide range of problems from the literature and provide a solid benchmark against the existing algorithmic approaches

Estimating the Algorithmic Complexity of Stock Markets

Randomness and regularities in Finance are usually treated in probabilistic terms. In this paper, we develop a completely different approach in using a non-probabilistic framework based on the algorithmic information theory initially developed by Kolmogorov (1965). We present some elements of this theory and show why it is particularly relevant to Finance, and potentially to other sub-fields of Economics as well. We develop a generic method to estimate the Kolmogorov complexity of numeric series. This approach is based on an iterative "regularity erasing procedure" implemented to use lossless compression algorithms on financial data. Examples are provided with both simulated and real-world financial time series. The contributions of this article are twofold. The first one is methodological : we show that some structural regularities, invisible with classical statistical tests, can be detected by this algorithmic method. The second one consists in illustrations on the daily Dow-Jones Index suggesting that beyond several well-known regularities, hidden structure may in this index remain to be identified

Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma

We consider integer programming problems in standard form $\max \{c^Tx : Ax = b, \, x\geq 0, \, x \in Z^n\}$ where $A \in Z^{m \times n}$, $b \in Z^m$ and $c \in Z^n$. We show that such an integer program can be solved in time $(m \Delta)^{O(m)} \cdot \|b\|_\infty^2$, where $\Delta$ is an upper bound on each absolute value of an entry in $A$. This improves upon the longstanding best bound of Papadimitriou (1981) of $(m\cdot \Delta)^{O(m^2)}$, where in addition, the absolute values of the entries of $b$ also need to be bounded by $\Delta$. Our result relies on a lemma of Steinitz that states that a set of vectors in $R^m$ that is contained in the unit ball of a norm and that sum up to zero can be ordered such that all partial sums are of norm bounded by $m$. We also use the Steinitz lemma to show that the $\ell_1$-distance of an optimal integer and fractional solution, also under the presence of upper bounds on the variables, is bounded by $m \cdot (2\,m \cdot \Delta+1)^m$. Here $\Delta$ is again an upper bound on the absolute values of the entries of $A$. The novel strength of our bound is that it is independent of $n$. We provide evidence for the significance of our bound by applying it to general knapsack problems where we obtain structural and algorithmic results that improve upon the recent literature.Comment: We achieve much milder dependence of the running time on the largest entry in $b

Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges

Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multi-agent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problem-specific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context

Mechanism Design via Dantzig-Wolfe Decomposition

In random allocation rules, typically first an optimal fractional point is calculated via solving a linear program. The calculated point represents a fractional assignment of objects or more generally packages of objects to agents. In order to implement an expected assignment, the mechanism designer must decompose the fractional point into integer solutions, each satisfying underlying constraints. The resulting convex combination can then be viewed as a probability distribution over feasible assignments out of which a random assignment can be sampled. This approach has been successfully employed in combinatorial optimization as well as mechanism design with or without money. In this paper, we show that both finding the optimal fractional point as well as its decomposition into integer solutions can be done at once. We propose an appropriate linear program which provides the desired solution. We show that the linear program can be solved via Dantzig-Wolfe decomposition. Dantzig-Wolfe decomposition is a direct implementation of the revised simplex method which is well known to be highly efficient in practice. We also show how to use the Benders decomposition as an alternative method to solve the problem. The proposed method can also find a decomposition into integer solutions when the fractional point is readily present perhaps as an outcome of other algorithms rather than linear programming. The resulting convex decomposition in this case is tight in terms of the number of integer points according to the Carath{\'e}odory's theorem

A constraint-based framework to model harmony for algorithmic composition

Music constraint systems provide a rule-based approach to composition. Existing systems allow users to constrain the harmony, but the constrainable harmonic information is restricted to pitches and intervals between pitches. More abstract analytical information such as chord or scale types, their root, scale degrees, enharmonic note representations, whether a note is the third or fifth of a chord and so forth are not supported. However, such information is important for modelling various music theories. This research proposes a framework for modelling harmony at a high level of abstraction. It explicitly represents various analytical information to allow for complex theories of harmony. It is designed for efficient propagation-based constraint solvers. The framework supports the common 12-tone equal temperament, and arbitrary other equal temperaments. Users develop harmony models by applying user-defined constraints to its music representation. Three examples demonstrate the expressive power of the framework: (1) an automatic melody harmonisation with a simple harmony model; (2) a more complex model implementing large parts of Schoenberg’s tonal theory of harmony; and (3) a composition in extended tonality. Schoenberg’s comprehensive theory of harmony has not been computationally modelled before, neither with constraints programming nor in any other way.