Logic Integer Programming Models for Signaling Networks

We propose a static and a dynamic approach to model biological signaling networks, and show how each can be used to answer relevant biological questions. For this we use the two different mathematical tools of Propositional Logic and Integer Programming. The power of discrete mathematics for handling qualitative as well as quantitative data has so far not been exploited in Molecular Biology, which is mostly driven by experimental research, relying on first-order or statistical models. The arising logic statements and integer programs are analyzed and can be solved with standard software. For a restricted class of problems the logic models reduce to a polynomial-time solvable satisfiability algorithm. Additionally, a more dynamic model enables enumeration of possible time resolutions in poly-logarithmic time. Computational experiments are included

Stochastic Budget Optimization in Internet Advertising

Internet advertising is a sophisticated game in which the many advertisers "play" to optimize their return on investment. There are many "targets" for the advertisements, and each "target" has a collection of games with a potentially different set of players involved. In this paper, we study the problem of how advertisers allocate their budget across these "targets". In particular, we focus on formulating their best response strategy as an optimization problem. Advertisers have a set of keywords ("targets") and some stochastic information about the future, namely a probability distribution over scenarios of cost vs click combinations. This summarizes the potential states of the world assuming that the strategies of other players are fixed. Then, the best response can be abstracted as stochastic budget optimization problems to figure out how to spread a given budget across these keywords to maximize the expected number of clicks. We present the first known non-trivial poly-logarithmic approximation for these problems as well as the first known hardness results of getting better than logarithmic approximation ratios in the various parameters involved. We also identify several special cases of these problems of practical interest, such as with fixed number of scenarios or with polynomial-sized parameters related to cost, which are solvable either in polynomial time or with improved approximation ratios. Stochastic budget optimization with scenarios has sophisticated technical structure. Our approximation and hardness results come from relating these problems to a special type of (0/1, bipartite) quadratic programs inherent in them. Our research answers some open problems raised by the authors in (Stochastic Models for Budget Optimization in Search-Based Advertising, Algorithmica, 58 (4), 1022-1044, 2010).Comment: FINAL versio

Mixed integer predictive control and shortest path reformulation

Mixed integer predictive control deals with optimizing integer and real control variables over a receding horizon. The mixed integer nature of controls might be a cause of intractability for instances of larger dimensions. To tackle this little issue, we propose a decomposition method which turns the original $n$-dimensional problem into $n$ indipendent scalar problems of lot sizing form. Each scalar problem is then reformulated as a shortest path one and solved through linear programming over a receding horizon. This last reformulation step mirrors a standard procedure in mixed integer programming. The approximation introduced by the decomposition can be lowered if we operate in accordance with the predictive control technique: i) optimize controls over the horizon ii) apply the first control iii) provide measurement updates of other states and repeat the procedure

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte