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Marco Molinaro

Publications and source records attributed to Marco Molinaro.

At least 73 records · Page 4Linked to original sources
arXiv2016

Online and Random-order Load Balancing Simultaneously

We consider the problem of online load balancing under lp-norms: sequential jobs need to be assigned to one of the machines and the goal is to minimize the lp-norm of the machine loads. This generalizes the classical problem of scheduling for makespan minimization (case l_infty) and has been thoroughly studied. However, despite the recent push for beyond worst-case analyses, no such results are known for this problem. In this paper we provide algorithms with simultaneous guarantees for the worst-case model as well as for the random-order (i.e. secretary) model, where an arbitrary set of jobs comes in random order. First, we show that the greedy algorithm (with restart), known to have optimal O(p) worst-case guarantee, also has a (typically) improved random-order guarantee. However, the behavior of this algorithm in the random-order model degrades with p. We then propose algorithm SIMULTANEOUSLB that has simultaneously optimal guarantees (within constants) in both worst-case and random-order models. In particular, the random-order guarantee of SIMULTANEOUSLB improves as p increases. One of the main components is a new algorithm with improved regret for Online Linear Optimization (OLO) over the non-negative vectors in the lq ball. Interestingly, this OLO algorithm is also used to prove a purely probabilistic inequality that controls the correlations arising in the random-order model, a common source of difficulty for the analysis. Another important component used in both SIMULTANEOUSLB and our OLO algorithm is a smoothing of the lp-norm that may be of independent interest. This smoothness property allows us to see algorithm SIMULTANEOUSLB as essentially a greedy one in the worst-case model and as a primal-dual one in the random-order model, which is instrumental for its simultaneous guarantees.

cs.DS↗
arXiv2016

Improving the Randomization Step in Feasibility Pump

Feasibility pump (FP) is a successful primal heuristic for mixed-integer linear programs (MILP). The algorithm consists of three main components: rounding fractional solution to a mixed-integer one, projection of infeasible solutions to the LP relaxation, and a randomization step used when the algorithm stalls. While many generalizations and improvements to the original Feasibility Pump have been proposed, they mainly focus on the rounding and projection steps. We start a more in-depth study of the randomization step in Feasibility Pump. For that, we propose a new randomization step based on the WalkSAT algorithm for solving SAT instances. First, we provide theoretical analyses that show the potential of this randomization step; to the best of our knowledge, this is the first time any theoretical analysis of running-time of Feasibility Pump or its variants has been conducted. Moreover, we also conduct computational experiments incorporating the proposed modification into a state-of-the-art Feasibility Pump code that reinforce the practical value of the new randomization step.

math.OC↗
arXiv2016

VIALACTEA knowledge base homogenizing access to Milky Way data

The VIALACTEA project has a work package dedicated to Tools and Infrastructure and, inside it, a task for the Database and Virtual Observatory Infrastructure. This task aims at providing an infrastructure to store all the resources needed by the, more purposely, scientific work packages of the project itself. This infrastructure includes a combination of: storage facilities, relational databases and web services on top of them, and has taken, as a whole, the name of VIALACTEA Knowledge Base (VLKB). This contribution illustrates the current status of this VLKB. It details the set of data resources put together; describes the database that allows data discovery through VO inspired metadata maintenance; illustrates the discovery, cutout and access services built on top of the former two for the users to exploit the data content.

astro-ph.IM↗
arXiv2016

Aggregation-based cutting-planes for packing and covering integer programs

In this paper, we study the strength of Chvatal-Gomory (CG) cuts and more generally aggregation cuts for packing and covering integer programs (IPs). Aggregation cuts are obtained as follows: Given an IP formulation, we first generate a single implied inequality using aggregation of the original constraints, then obtain the integer hull of the set defined by this single inequality with variable bounds, and finally use the inequalities describing the integer hull as cutting-planes. Our first main result is to show that for packing and covering IPs, the CG and aggregation closures can be 2-approximated by simply generating the respective closures for each of the original formulation constraints, without using any aggregations. On the other hand, we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general IPs. The proof of the above stated results for the case of covering IPs with bounds require the development of some new structural results, which may be of independent interest. Finally, we examine the strength of cuts based on k different aggregation inequalities simultaneously, the so-called multi-row cuts, and show that every packing or covering IP with a large integrality gap also has a large k-aggregation closure rank. In particular, this rank is always at least of the order of the logarithm of the integrality gap.

math.OC↗
arXiv2016

Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions

We investigate how well the graph of a bilinear function $b:[0,1]^n\to\mathbb{R}$ can be approximated by its McCormick relaxation. In particular, we are interested in the smallest number $c$ such that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is at most $c$ times the difference between the concave and convex envelopes. Answering a question of Luedtke, Namazifar and Linderoth, we show that this factor $c$ cannot be bounded by a constant independent of $n$. More precisely, we show that for a random bilinear function $b$ we have asymptotically almost surely $c\geqslant\sqrt n/4$. On the other hand, we prove that $c\leqslant 600\sqrt{n}$, which improves the linear upper bound proved by Luedtke, Namazifar and Linderoth. In addition, we present an alternative proof for a result of Misener, Smadbeck and Floudas characterizing functions $b$ for which the McCormick relaxation is equal to the convex hull.

math.OC↗
arXiv2016

Analysis of Sparse Cutting-planes for Sparse MILPs with Applications to Stochastic MILPs

In this paper, we present an analysis of the strength of sparse cutting-planes for mixed integer linear programs (MILP) with sparse formulations. We examine three kinds of problems: packing problems, covering problems, and more general MILPs with the only assumption that the objective function is non-negative. Given a MILP instance of one of these three types, assume that we decide on the support of cutting-planes to be used and the strongest inequalities on these supports are added to the linear programming relaxation. Call the optimal objective function value of the linear programming relaxation together with these cuts as $z^{cut}$. We present bounds on the ratio of $z^{cut}$ and the optimal objective function value of the MILP that depends only on the sparsity structure of the constraint matrix and the support of sparse cuts selected, that is, these bounds are completely data independent. These results also shed light on the strength of scenario-specific cuts for two stage stochastic MILPs.

math.OC↗
arXiv2015

How the Experts Algorithm Can Help Solve LPs Online

We consider the problem of solving packing/covering LPs online, when the columns of the constraint matrix are presented in random order. This problem has received much attention and the main focus is to figure out how large the right-hand sides of the LPs have to be (compared to the entries on the left-hand side of the constraints) to allow $(1+ε)$-approximations online. It is known that the right-hand sides have to be $Ω(ε^{-2} \log m)$ times the left-hand sides, where $m$ is the number of constraints. In this paper we give a primal-dual algorithm that achieve this bound for mixed packing/covering LPs. Our algorithms construct dual solutions using a regret-minimizing online learning algorithm in a black-box fashion, and use them to construct primal solutions. The adversarial guarantee that holds for the constructed duals helps us to take care of most of the correlations that arise in the algorithm; the remaining correlations are handled via martingale concentration and maximal inequalities. These ideas lead to conceptually simple and modular algorithms, which we hope will be useful in other contexts.

cs.DS↗
arXiv2015

IVOA Recommendation: Registry Relational Schema Version 1.0

Registries provide a mechanism with which VO applications can discover and select resources - first and foremost data and services - that are relevant for a particular scientific problem. This specification defines an interface for searching this resource metadata based on the IVOA's TAP protocol. It specifies a set of tables that comprise a useful subset of the information contained in the registry records, as well as the table's data content in terms of the XML VOResource data model. The general design of the system is geared towards allowing easy authoring of queries.

astro-ph.IM↗
arXiv2014

Some lower bounds on sparse outer approximations of polytopes

Motivated by the need to better understand the properties of sparse cutting-planes used in mixed integer programming solvers, the paper [2] studied the idealized problem of how well a polytope is approximated by the use of sparse valid inequalities. As an extension to this work, we study the following less idealized questions in this paper: (1) Are there integer programs, such that sparse inequalities do not approximate the integer hull well even when added to a linear programming relaxation? (2) Are there polytopes, where the quality of approximation by sparse inequalities cannot be significantly improved by adding a budgeted number of arbitrary (possibly dense) valid inequalities? (3) Are there polytopes that are difficult to approximate under every rotation? (4) Are there polytopes that are difficult to approximate in all directions using sparse inequalities? We answer each of the above questions in the positive.

math.OC↗
arXiv2014

Mixed-integer Quadratic Programming is in NP

Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP and integer linear programming is in NP.

cs.DM↗
arXiv2014

How Good Are Sparse Cutting-Planes?

Sparse cutting-planes are often the ones used in mixed-integer programing (MIP) solvers, since they help in solving the linear programs encountered during branch-&-bound more efficiently. However, how well can we approximate the integer hull by just using sparse cutting-planes? In order to understand this question better, given a polyope $P$ (e.g. the integer hull of a MIP), let $P^k$ be its best approximation using cuts with at most $k$ non-zero coefficients. We consider $d(P, P^k) = \max_{x \in P^k} \left(min_{y \in P} \| x - y\|\right)$ as a measure of the quality of sparse cuts. In our first result, we present general upper bounds on $d(P, P^k)$ which depend on the number of vertices in the polytope and exhibits three phases as $k$ increases. Our bounds imply that if $P$ has polynomially many vertices, using half sparsity already approximates it very well. Second, we present a lower bound on $d(P, P^k)$ for random polytopes that show that the upper bounds are quite tight. Third, we show that for a class of hard packing IPs, sparse cutting-planes do not approximate the integer hull well, that is $d(P, P^k)$ is large for such instances unless $k$ is very close to $n$. Finally, we show that using sparse cutting-planes in extended formulations is at least as good as using them in the original polyhedron, and give an example where the former is actually much better.

math.OC↗
arXiv2013

A statistical analysis of circumstellar material in Type Ia supernovae

A key tracer of the elusive progenitor systems of Type Ia supernovae (SNe Ia) is the detection of narrow blueshifted time-varying Na I D absorption lines, interpreted as evidence of circumstellar material (CSM) surrounding the progenitor system. The origin of this material is controversial, but the simplest explanation is that it results from previous mass loss in a system containing a white dwarf and a non-degenerate companion star. We present new single-epoch intermediate-resolution spectra of 17 low-redshift SNe Ia taken with XShooter on the ESO Very Large Telescope. Combining this sample with events from the literature, we confirm an excess (~20 per cent) of SNe Ia displaying blueshifted narrow Na I D absorption features compared to non-blueshifted Na I D features. The host galaxies of SNe Ia displaying blueshifted absorption profiles are skewed towards later-type galaxies, compared to SNe Ia that show no Na I D absorption, and SNe Ia displaying blueshifted narrow Na I D absorption features have broader light curves. The strength of the Na I D absorption is stronger in SNe Ia displaying blueshifted Na I D absorption features than those without blueshifted features, and the strength of the blueshifted Na I D is correlated with the B-V colour of the SN at maximum light. This strongly suggests the absorbing material is local to the SN. In the context of the progenitor systems of SNe Ia, we discuss the significance of these findings and other recent observational evidence on the nature of SN Ia progenitors. We present a summary that suggests there are at least two distinct populations of normal, cosmologically useful SNe Ia.

astro-ph.SR↗
arXiv2013

SN 2009ip á la PESSTO: No evidence for core-collapse yet

We present observations of the interacting transient SN 2009ip, from the start of the outburst in October 2012 until the end of the 2012 observing season. The transient reached a peak of $M_V$=-17.7 mag before fading rapidly, with a total integrated luminosity of 1.9$\times10^{49}$ erg over the period of August-December 2012. The optical and near infrared spectra are dominated by narrow emission lines, signaling a dense circumstellar environment, together with multiple components of broad emission and absorption in H and He at velocities between 0.5-1.2$\times10^4$ km s$^{-1}$\. We see no evidence for nucleosynthesized material in SN 2009ip, even in late-time pseudo-nebular spectra. We set a limit of $<$0.02 M$_{\odot}$\ on the mass of any synthesized $^{56}$Ni from the late time lightcurve. A simple model for the narrow Balmer lines is presented, and used to derive number densities for the circumstellar medium of between $\sim 10^{9}-10^{10}$ cm$^{-3}$. Our near-infrared data does not show any excess at longer wavelengths. Our last data, taken in December 2012, shows that SN 2009ip has spectroscopically evolved to something quite similar to its appearance in late 2009, albeit with higher velocities. It is possible that neither of the eruptive and high luminosity events of SN 2009ip were induced by a core-collapse. We show that the peak and total integrated luminosity can be due to the efficient conversion of kinetic energy from colliding ejecta, and that around 0.05-0.1 M$_{\odot}$\ of material moving at 0.5-1$\times10^4$ km s$^{-1}$\ could comfortably produce the observed luminosity. The ejection of multiple shells, lack of evidence for nucleosynthesied elements and broad nebular lines, are all consistent with the pulsational-pair instability scenario. In this case the progenitor star may still exist, and will be observed after the current outburst fades.

astro-ph.SR↗
arXiv2012

Geometry of Online Packing Linear Programs

We consider packing LP's with $m$ rows where all constraint coefficients are normalized to be in the unit interval. The n columns arrive in random order and the goal is to set the corresponding decision variables irrevocably when they arrive so as to obtain a feasible solution maximizing the expected reward. Previous (1 - ε)-competitive algorithms require the right-hand side of the LP to be Omega((m/ε^2) log (n/ε)), a bound that worsens with the number of columns and rows. However, the dependence on the number of columns is not required in the single-row case and known lower bounds for the general case are also independent of n. Our goal is to understand whether the dependence on n is required in the multi-row case, making it fundamentally harder than the single-row version. We refute this by exhibiting an algorithm which is (1 - ε)-competitive as long as the right-hand sides are Omega((m^2/ε^2) log (m/ε)). Our techniques refine previous PAC-learning based approaches which interpret the online decisions as linear classifications of the columns based on sampled dual prices. The key ingredient of our improvement comes from a non-standard covering argument together with the realization that only when the columns of the LP belong to few 1-d subspaces we can obtain small such covers; bounding the size of the cover constructed also relies on the geometry of linear classifiers. General packing LP's are handled by perturbing the input columns, which can be seen as making the learning problem more robust.

cs.DS↗
arXiv2011

The Query-commit Problem

In the query-commit problem we are given a graph where edges have distinct probabilities of existing. It is possible to query the edges of the graph, and if the queried edge exists then its endpoints are irrevocably matched. The goal is to find a querying strategy which maximizes the expected size of the matching obtained. This stochastic matching setup is motivated by applications in kidney exchanges and online dating. In this paper we address the query-commit problem from both theoretical and experimental perspectives. First, we show that a simple class of edges can be queried without compromising the optimality of the strategy. This property is then used to obtain in polynomial time an optimal querying strategy when the input graph is sparse. Next we turn our attentions to the kidney exchange application, focusing on instances modeled over real data from existing exchange programs. We prove that, as the number of nodes grows, almost every instance admits a strategy which matches almost all nodes. This result supports the intuition that more exchanges are possible on a larger pool of patient/donors and gives theoretical justification for unifying the existing exchange programs. Finally, we evaluate experimentally different querying strategies over kidney exchange instances. We show that even very simple heuristics perform fairly well, being within 1.5% of an optimal clairvoyant strategy, that knows in advance the edges in the graph. In such a time-sensitive application, this result motivates the use of committing strategies.

cs.DS↗
arXiv2011

A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation

We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k+1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k=1, and Cornuejols and Molinaro for k=2.

math.OC↗
arXiv2011

Approximation Algorithms for Correlated Knapsacks and Non-Martingale Bandits

In the stochastic knapsack problem, we are given a knapsack of size B, and a set of jobs whose sizes and rewards are drawn from a known probability distribution. However, we know the actual size and reward only when the job completes. How should we schedule jobs to maximize the expected total reward? We know O(1)-approximations when we assume that (i) rewards and sizes are independent random variables, and (ii) we cannot prematurely cancel jobs. What can we say when either or both of these assumptions are changed? The stochastic knapsack problem is of interest in its own right, but techniques developed for it are applicable to other stochastic packing problems. Indeed, ideas for this problem have been useful for budgeted learning problems, where one is given several arms which evolve in a specified stochastic fashion with each pull, and the goal is to pull the arms a total of B times to maximize the reward obtained. Much recent work on this problem focus on the case when the evolution of the arms follows a martingale, i.e., when the expected reward from the future is the same as the reward at the current state. What can we say when the rewards do not form a martingale? In this paper, we give constant-factor approximation algorithms for the stochastic knapsack problem with correlations and/or cancellations, and also for budgeted learning problems where the martingale condition is not satisfied. Indeed, we can show that previously proposed LP relaxations have large integrality gaps. We propose new time-indexed LP relaxations, and convert the fractional solutions into distributions over strategies, and then use the LP values and the time ordering information from these strategies to devise a randomized adaptive scheduling algorithm. We hope our LP formulation and decomposition methods may provide a new way to address other correlated bandit problems with more general contexts.

cs.DS↗