Fractional Packing Research Articles

Intersecting and dense restrictions of clutters in polynomial time

A clutter is a family of sets, called members, such that no member contains another. It is called intersecting if every two members intersect, but not all members have a common element. Dense clutters additionally do not have a fractional packing of value 2. We are looking at certain substructures of clutters, namely minors and restrictions. For a family of clutters we introduce a general sufficient condition such that for every clutter we can decide whether the clutter has a restriction in that set in polynomial time. It is known that the sets of intersecting and dense clutters satisfy this condition. For intersecting clutters we generalize the statement to k-wise intersecting clutters using a much simpler proof. We also give a simplified proof that a dense clutter with no proper dense minor is either a delta or the blocker of an extended odd hole. This simplification reduces the running time of the algorithm for finding a delta or the blocker of an extended odd hole minor from previously O(n4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {O}}(n^4)$$\\end{document} to O(n3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {O}}(n^3)$$\\end{document} filter oracle calls.

Clean Clutters and Dyadic Fractional Packings

A vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of $\frac{1}{2^k}$ for some nonnegative integer $k$. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible, for there exist clean clutters with a covering number of three and no dyadic fractional packing of value three. Examples of clean clutters include ideal clutters, binary clutters, and clutters without an intersecting minor. Our proof is constructive and leads naturally to an (albeit exponential) algorithm. We improve the running time to quasi-polynomial in the rank of the input and to polynomial in the binary case.

How Fluvial Deposits Distort Aquifer Recharge Estimated From Groundwater Age: Insights From a Landscape Evolution Model

AbstractSustainable management of groundwater relies on robust estimates of aquifer fluxes. As such, recharge can be estimated from groundwater age using analytical models assuming a homogeneous aquifer. We propose to assess how this assumption affects recharge estimates in fluvial deposits. First, we use CHILD, a landscape evolution model, to generate plausible deposits after tens of thousands of years of river evolution. Then, we use the fractional packing model to compute porosity and permeability of unconsolidated sediments assuming a mixture of a coarse and a fine fraction. Finally, we use PFLOTRAN to simulate groundwater flow and mean age over two thousand years of spatially uniform recharge. This reproduces the boundary conditions underlying Vogel's analytical model to estimate recharge rates. Our results are based on 20,000 realizations from varying inputs: grain sizes, aggradation rate of the river, porosity, recharge, etc. For most models, the mean estimate of the recharge rate remains below an absolute error of 25%. But fluvial heterogeneities can lead to local errors ranging from close to zero to several thousand percent, which means that estimates may vary drastically depending on sampling locations. A sensitivity analysis using the δ‐importance measure and CUSUNORO curves exposes how fluvial processes influence the range and distribution of the error. For instance, a high bank erodibility and small coarse grain size under low aggradation rates favor the reworking of deposits, which increases heterogeneity and leads to high errors. This work paves the way for a more reliable estimation of recharge by considering the sedimentary context.

Clean tangled clutters, simplices, and projective geometries

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ=2 Conjecture.Let C be a clean tangled clutter. It was recently proved that C has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the core of C. The core is a duplication of the cuboid of a set of 0−1 points, called the setcore of C.In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then C has the clutter of the lines of the Fano plane as a minor.Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters.

Duality of graph invariants

We study a new set of duality relations between weighted, combinatoric invariants of a graph G. The dualities arise from a non-linear transform $$\mathcal{B}$$ , acting on the weight function p. We define $$\mathcal{B}$$ on a space of real-valued functions $$\mathcal{O}$$ and investigate its properties. We show that three invariants (the weighted independence number, the weighted Lovász number, and the weighted fractional packing number) are fixed points of $$\mathcal{B}^2$$ , but the weighted Shannon capacity is not. We interpret these invariants in the study of quantum non-locality.

Co-density and fractional edge cover packing

Given a multigraph $$G=(V,E)$$, the edge cover packing problem (ECPP) on G is to find a coloring of edges of G using the maximum number of colors such that at each vertex all colors occur. ECPP can be formulated as an integer program and is NP-hard in general. In this paper, we consider the fractional edge cover packing problem, the LP relaxation of ECPP. We focus on the more general weighted setting, the weighted fractional edge cover packing problem (WFECPP), which can be formulated as the following linear program $$\begin{aligned} \begin{array}{ll} \hbox {Maximize} \ \ \ &{} {{\varvec{1}}}^T {{\varvec{x}}} \\ \hbox {subject to} &{} A{{\varvec{x}}} \le {{\varvec{w}}} \\ &{} \quad {{\varvec{x}}} \ge {{\varvec{0}}}, \end{array} \end{aligned}$$where A is the edge–edge cover incidence matrix of G, $${\varvec{w}}=(w(e): e\in E)$$, and w(e) is a positive rational weight on each edge e of G. The weighted co-density problem, closely related to WFECPP, is to find a subset $$S\subseteq V$$ with $$|S|\ge 3$$ and odd, such that $$\frac{2w(E^{+}(S))}{|S|+1}$$ is minimized, where $$E^{+}(S)$$ is the set of all edges of G with at least one end in S and $$w(E^{+}(S))$$ is the total weight of all edges in $$E^{+}(S)$$. We present polynomial combinatorial algorithms for solving these two problems exactly.

A Competitive Analysis of Online Knapsack Problems with Unit Density

We study an online knapsack problem where the items arrive sequentially and must be either immediately packed into the knapsack or irrevocably discarded. Each item has a different size and the objective is to maximize the total size of items packed. We focus on the class of randomized algorithms which initially draw a threshold from some distribution, and then pack every fitting item whose size is at least that threshold. Threshold policies satisfy many desiderata including simplicity, fairness, and incentive-alignment. We derive two optimal threshold distributions, the first of which implies a competitive ratio of 0.432 relative to the optimal offline packing, and the second of which implies a competitive ratio of 0.428 relative to the optimal fractional packing. We also consider the generalization to multiple knapsacks, where an arriving item has a different size in each knapsack and must be placed in at most one. This is equivalent to the AdWords problem where item truncation is not allowed. We derive a randomized threshold algorithm for this problem which is 0.214-competitive. We also show that any randomized algorithm for this problem cannot be more than 0.461-competitive, providing the first upper bound which is strictly less than 0.5. This online knapsack problem finds applications in many areas, like supply chain ordering, online advertising, and healthcare scheduling, refugee integration, and crowdsourcing. We show how our optimal threshold distributions can be naturally implemented in the warehouses for a Latin American chain department store. We run simulations on their large-scale order data, which demonstrate the robustness of our proposed algorithms.

A new method for mechanical damage characterization in proppant packs using nuclear magnetic resonance measurements

Proppant performance can significantly affect the success of hydraulic fracturing treatments and production in unconventional plays. Performance of proppants can decrease rapidly by mechanical damage. Quantifying the impact of mechanical damage on proppant performance can be challenging either in the laboratory or in-situ condition. This paper introduces a new method based on Nuclear Magnetic Resonance (NMR) Transverse relaxation (T2) measurements to characterize pore-size distribution in proppant packs. Quantifying the pore-size distribution in proppant packs enables enhanced evaluation of porosity and conductivity of proppant packs and possible mechanical damage to the packs. The objectives of this paper are (a) to evaluate the performance of this new NMR-based method for characterizing pore structure of the proppant packs, and (b) to isolate and quantify the mechanical damage in proppant packs due to generating of fines in the laboratory using this new NMR-based method.First, we prepared proppant pack test samples in three categories including (a) different types of proppants with different surface relaxivity, (b) mixture of proppants with different sizes, and (c) proppants undergone different levels of mechanical damage. We measured NMR response in proppant packs and quantified the impacts of proppant type, size, and mechanical damage on NMR T2 distribution. Next, we used the fractional packing model to quantify the efficiency of packing of proppant mixture and to quantify the changes in porosity of proppant packs caused by variation in size and fraction of fines. Finally, we used Kozeny-Carman (KC) equation and modified Schlumberger Doll Research (SDR) equation to correlate the variation in T2 distribution to the decrease in proppant pack permeability.Results showed that NMR T2 distribution is sensitive to pore-size distribution in the proppant packs. The variable pore-size distribution was either a consequence of having a mixture of proppants with different sizes or different levels of mechanical damage. The impacts of fines on the pore-size distribution of the proppant packs was measureable, which is promising for the successful application of the introduced method for time-lapse quantification of generated fines. We also observed a measurable sensitivity of T2 distribution to mechanical damage in the proppant packs, which enables quantifying the level of damage in proppants through quantifying pore structure in the proppant packs. The NMR T2 measurements were, however, sensitive to the presence of paramagnetic materials such as iron in proppants. This sensitivity was more significant in cases where paramagnetic materials were distributed on the proppant surface. The proposed method is considered as an indirect measurement of pore structure and could potentially provide a quantitative assessment of the proppant pack conductivity both in the lab-scale and in-situ conditions. Improved assessment of proppant conductivity can help in enhancing the design and execution of fracture treatments for successful development of unconventional plays.

The influence of grain physicochemistry and biomass on hydraulic conductivity in sand-filled treatment wetlands

The flow of effluent through treatment wetlands is influenced by the infrastructure set-up, the effluent character, the type of hydraulic flow, the mode of operation, the type of substrate, and the type and quantity of biomass. Current flow models have not been well validated, and/or do not accurately account for biomass clogging. In this study, treatment wetlands containing Dune or River sand with similar particle size distributions exhibited significant disparities in achievable flow rates. To gain insight into this phenomenon, further investigations were conducted to compare: (i) sand particle characteristics (size, elemental and mineral composition, grain morphology), (ii) the relationships between mineral composition and shape of the sand particles, (iii) the hydraulic conductivity of the different sand types before and after inducement of biomass growth, and (iv) the measured hydraulic conductivities with those predicted using the fractional packing Kozeny-Carman model.Using automated scanning electron microscopy (QEMSCAN™) it was determined that the shape of the quartz particles of the River sand (98% quartz) and calcite particles of the Dune sand (81% quartz, 18% calcite) were less round and more angular than the quartz particles of the Dune sand, and that the River sand particles were conglomerate in nature and/or fractured. The hydraulic conductivities of the Dune and River sands were significantly different (0.284 and 0.015 mm s−1, respectively), and the hydraulic conductivity of the Dune sand decreased by 51% due to biomass accumulation. The fractional packing model overestimated the measured values.

Separation Between Quantum Lovász Number and Entanglement-Assisted Zero-Error Classical Capacity

Quantum Lov\'asz number is a quantum generalization of the Lov\'asz number in graph theory. It is the best known efficiently computable upper bound of the entanglement-assisted zero-error classical capacity of a quantum channel. However, it remains an intriguing open problem whether quantum entanglement can always enhance the zero-error capacity to achieve the quantum Lov\'asz number. In this paper, by constructing a particular class of qutrit-to-qutrit channels, we show that there exists a strict gap between the entanglement-assisted zero-error capacity and the quantum Lov\'asz number. Interestingly, for this class of quantum channels, the quantum generalization of fractional packing number is strictly larger than the zero-error capacity assisted with feedback or no-signalling correlations, which differs from the case of classical channels.

A generalized approximation framework for fractional network flow and packing problems

We generalize the fractional packing framework of Garg and Koenemann (SIAM J Comput 37(2):630–652, 2007) to the case of linear fractional packing problems over polyhedral cones. More precisely, we provide approximation algorithms for problems of the form $$\max \{c^T x : Ax \le b, x \in C \}$$ , where the matrix A contains no negative entries and C is a cone that is generated by a finite set S of non-negative vectors. While the cone is allowed to require an exponential-sized representation, we assume that we can access it via one of three types of oracles. For each of these oracles, we present positive results for the approximability of the packing problem. In contrast to other frameworks, the presented one allows the use of arbitrary linear objective functions and can be applied to a large class of packing problems without much effort. In particular, our framework instantly allows to derive fast and simple fully polynomial-time approximation algorithms (FPTASs) for a large set of network flow problems, such as budget-constrained versions of traditional network flows, multicommodity flows, or generalized flows. Some of these FPTASs represent the first ones of their kind, while others match existing results but offer a much simpler proof.

MISO Networks With Imperfect CSIT: A Topological Rate-Splitting Approach

Recently, the Degrees-of-Freedom (DoF) region of multiple-input-single-output (MISO) networks with imperfect channel state information at the transmitter (CSIT) has attracted significant attentions. An achievable scheme is known as rate-splitting (RS) that integrates common-message-multicasting and private-message-unicasting. In this paper, focusing on the general $K$-cell MISO IC where the CSIT of each interference link has an arbitrary quality of imperfectness, we firstly identify the DoF region achieved by RS. Secondly, we introduce a novel scheme, so called Topological RS (TRS), whose novelties compared to RS lie in a multi-layer structure and transmitting multiple common messages to be decoded by groups of users rather than all users. The design of TRS is motivated by a novel interpretation of the $K$-cell IC with imperfect CSIT as a weighted-sum of a series of partially connected networks. We show that the DoF region achieved by TRS covers that achieved by RS. Also, we find the maximal sum DoF achieved by TRS via hypergraph fractional packing, which yields the best sum DoF so far. Lastly, for a realistic scenario where each user is connected to three dominant transmitters, we identify the sufficient condition where TRS strictly outperforms conventional schemes.

DEM simulation of binary sphere packing densification under vertical vibration

ABSTRACTThe densification of random binary sphere packings subjected to vertical vibration was modeled by using the discrete element method (DEM). The influences of operating parameters such as the vibration conditions, sphere size ratio (diameter ratio of larger versus small spheres), and composition (volume fraction of large spheres) of the binary mixture on the fractional packing density φ (defined as the volume of spheres divided by the volume of container) were studied. Two packing states, i.e., random loose packing (RLP) and random close packing (RCP), were reproduced and their micro properties such as the coordination number (CN), radial distribution function (RDF), and force structure were characterized and compared. The results indicate that properly controlling vibration conditions can realize the transition of binary packing structure from the RLP to RCP state when the sphere size ratio and composition are fixed, and the fractional packing density for RCP after vibration can reach φRCP ≈ 0.86. Different packing characteristics from RLP and RCP identify that RCP shows much denser and more uniform structure than RLP. The current modeling results are in good agreement with those obtained from both the physical experiments and the proposed empirical models in the literature.

A new property of the Lovász number and duality relations between graph parameters

We show that for any graph G, by considering “activation” through the strong product with another graph H, the relation α(G)≤ϑ(G) between the independence number and the Lovász number of G can be made arbitrarily tight: Precisely, the inequality α(G⊠H)⩽ϑ(G⊠H)=ϑ(G)ϑ(H) becomes asymptotically an equality for a suitable sequence of ancillary graphs H.This motivates us to look for other products of graph parameters of G and H on the right hand side of the above relation. For instance, a result of Rosenfeld and Hales states that α(G⊠H)⩽α∗(G)α(H), with the fractional packing number α∗(G), and for every G there exists H that makes the above an equality; conversely, for every graph H there is a G that attains equality.These findings constitute some sort of duality of graph parameters, mediated through the independence number, under which α and α∗ are dual to each other, and the Lovász number ϑ is self-dual. We also show duality of Schrijver’s and Szegedy’s variants ϑ− and ϑ+ of the Lovász number, and explore analogous notions for the chromatic number under strong and disjunctive graph products.

No-Signalling-Assisted Zero-Error Capacity of Quantum Channels and an Information Theoretic Interpretation of the Lovász Number

We study the one-shot zero-error classical capacity of a quantum channel assisted by quantum no-signalling correlations, and the reverse problem of exact simulation of a prescribed channel by a noiseless classical one. Quantum no-signalling correlations are viewed as two-input and two-output completely positive and trace preserving maps with linear constraints enforcing that the device cannot signal. Both problems lead to simple semidefinite programmes (SDPs) that depend only on the Choi-Kraus (operator) space of the channel. In particular, we show that the zero-error classical simulation cost is precisely the conditional min-entropy of the Choi-Jamiołkowski matrix of the given channel. The zero-error classical capacity is given by a similar-looking but different SDP; the asymptotic zero-error classical capacity is the regularization of this SDP, and in general, we do not know of any simple form. Interestingly, however, for the class of classical-quantum channels, we show that the asymptotic capacity is given by a much simpler SDP, which coincides with a semidefinite generalization of the fractional packing number suggested earlier by Aram Harrow. This finally results in an operational interpretation of the celebrated Lovász ϑ function of a graph as the zero-error classical capacity of the graph assisted by quantum no-signalling correlations, the first information theoretic interpretation of the Lovász number.

A Fast Parallel Implementation of a PTAS for Fractional Packing and Covering Linear Programs

We present a parallel implementation of the randomized $$(1+\varepsilon )$$(1+?) approximation algorithm for packing and covering linear programs presented by Koufogiannakis and Young (2007). Their approach builds on ideas of the sublinear time algorithm of Grigoriadis and Khachiyan's (Oper Res Lett 18(2):53---58, 1995) and Garg and Konemann's (SIAM J Comput 37(2):630---652, 2007) non-uniform-increment amortization scheme. With high probability it computes a feasible primal and dual solution whose costs are within a factor of $$1+\varepsilon $$1+? of the optimal cost. In order to make their algorithm more parallelizable we also implemented a deterministic version of the algorithm, i.e. instead of updating a single random entry at each iteration we updated deterministically many entries at once. This slowed down a single iteration of the algorithm but allowed for larger step-sizes which lead to fewer iterations. We use NVIDIA's parallel computing architecture CUDA for the parallel environment. We report a speedup between one and two orders of magnitude over the times reported by Koufogiannakis and Young (2007).

Graph-Theoretic Approach to Quantum Correlations

Correlations in Bell and noncontextuality inequalities can be expressed as a positive linear combination of probabilities of events. Exclusive events can be represented as adjacent vertices of a graph, so correlations can be associated to a subgraph. We show that the maximum value of the correlations for classical, quantum, and more general theories is the independence number, the Lovász number, and the fractional packing number of this subgraph, respectively. We also show that, for any graph, there is always a correlation experiment such that the set of quantum probabilities is exactly the Grötschel-Lovász-Schrijver theta body. This identifies these combinatorial notions as fundamental physical objects and provides a method for singling out experiments with quantum correlations on demand.

Exclusivity structures and graph representatives of local complementation orbits

We describe a construction that maps any connected graph G on three or more vertices into a larger graph, H(G), whose independence number is strictly smaller than its Lovász number which is equal to its fractional packing number. The vertices of H(G) represent all possible events consistent with the stabilizer group of the graph state associated with G, and exclusive events are adjacent. Mathematically, the graph H(G) corresponds to the orbit of G under local complementation. Physically, the construction translates into graph-theoretic terms the connection between a graph state and a Bell inequality maximally violated by quantum mechanics. In the context of zero-error information theory, the construction suggests a protocol achieving the maximum rate of entanglement-assisted capacity, a quantum mechanical analogue of the Shannon capacity, for each H(G). The violation of the Bell inequality is expressed by the one-shot version of this capacity being strictly larger than the independence number. Finally, given the correspondence between graphs and exclusivity structures, we are able to compute the independence number for certain infinite families of graphs with the use of quantum non-locality, therefore highlighting an application of quantum theory in the proof of a purely combinatorial statement.

Exact and heuristic solutions for combinatorial optimization problems

This is a summary of the Ph.D. thesis defended by the author on July 2012 at the University of Modena and Reggio Emilia. The thesis was supervised by Mauro Dell’Amico and co-supervised by Manuel Iori. The manuscript is written in English and is available from the author upon request at jose.diaz@unimore.it. The thesis addresses relevant optimization problems: the Bin Packing Problem, the Quadratic Assignment Problem, the large-scale Energy Management Problem, and the Node, Edge and Arc Routing Problem. To solve these problems exact, heuristic and local search techniques are proposed. The first problem studied is the Bin Packing Problem with Precedence Constraints (BPP-P): given a set of identical capacitated bins, a set of weighted items and a set of precedences among those items, we are interested in determining the minimum number of bins that can accommodate all items and can be ordered in such a way that all precedences are satisfied. The problem has a very intriguing combinatorial structure and models many assembly and scheduling issues. According to our knowledge the BPP-P has received little attention in the literature, and in the thesis we address it for the first time with exact solution methods. In particular, we develop reduction criteria, a large set of lower bounds, a variable neighborhood search upper bounding technique and a branch-and-bound algorithm (see Dell’Amico M., Diaz Diaz J.C. and Iori M. (2012): The Bin Packing Problem with Precedence Constraints, Operations Research, doi:10.1287/opre.1120.1109). We show the effectiveness of the proposed algorithms by means of extensive computational tests on benchmark instances and comparison with standard integer linear programming techniques. The second problem studied is to find the Friendly Bin Packing Instances without Integer Round-up Property. It is well known that the gap between the optimal values of bin packing and fractional bin packing, if the latter is rounded up to the closest integer, is almost always null. Known counterexamples to this for integer input values involve fairly large numbers. Specifically, the first one was derived in 1986 and involved a bin capacity of the order of a billion. Later in 1998 a counterexample with a bin capacity of the order of a million was found. In the thesis we show a large number of counterexamples with bin capacity of the order of a hundred, showing that the gap may be positive even for numbers which arise in customary applications. The associated instances are constructed starting from the Petersen graph and taking advantage of the fact that it is fractionally, but not integrally, 3-edge colorable. The third problem studied is the Single-finger Keyboard Layout Problem. The problem of designing new keyboards layouts able to improve the typing speed of an

A Nearly Linear-Time PTAS for Explicit Fractional Packing and Covering Linear Programs

We give an approximation algorithm for packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm computes feasible primal and dual solutions whose costs are within a factor of 1+eps of the optimal cost in time O((r+c)log(n)/eps^2 + n).