A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing

Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for semi-algebraic $k$-uniform hypergraphs of bounded complexity, showing that for each $\epsilon>0$ the vertex set can be equitably partitioned into a bounded number of parts (in terms of $\epsilon$ and the complexity) so that all but an $\epsilon$-fraction of the $k$-tuples of parts are homogeneous. We prove that the number of parts can be taken to be polynomial in $1/\epsilon$. Our improved regularity lemma can be applied to geometric problems and to the following general question on property testing: is it possible to decide, with query complexity polynomial in the reciprocal of the approximation parameter, whether a hypergraph has a given hereditary property? We give an affirmative answer for testing typical hereditary properties for semi-algebraic hypergraphs of bounded complexity

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones

Uniform hypergraphs containing no grids

A hypergraph is called an r×r grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that Ai∩Aj=Bi∩Bj=φ for 1≤i<j≤r and {pipe}Ai∩Bj{pipe}=1 for 1≤i, j≤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1∩C2{pipe}={pipe}C2∩C3{pipe}={pipe}C3∩C1{pipe}=1, C1∩C2≠C1∩C3. A hypergraph is linear, if {pipe}E∩F{pipe}≤1 holds for every pair of edges E≠F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For r≥. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. © 2013 Elsevier Ltd

Model Counting of Query Expressions: Limitations of Propositional Methods

Query evaluation in tuple-independent probabilistic databases is the problem of computing the probability of an answer to a query given independent probabilities of the individual tuples in a database instance. There are two main approaches to this problem: (1) in `grounded inference' one first obtains the lineage for the query and database instance as a Boolean formula, then performs weighted model counting on the lineage (i.e., computes the probability of the lineage given probabilities of its independent Boolean variables); (2) in methods known as `lifted inference' or `extensional query evaluation', one exploits the high-level structure of the query as a first-order formula. Although it is widely believed that lifted inference is strictly more powerful than grounded inference on the lineage alone, no formal separation has previously been shown for query evaluation. In this paper we show such a formal separation for the first time. We exhibit a class of queries for which model counting can be done in polynomial time using extensional query evaluation, whereas the algorithms used in state-of-the-art exact model counters on their lineages provably require exponential time. Our lower bounds on the running times of these exact model counters follow from new exponential size lower bounds on the kinds of d-DNNF representations of the lineages that these model counters (either explicitly or implicitly) produce. Though some of these queries have been studied before, no non-trivial lower bounds on the sizes of these representations for these queries were previously known.Comment: To appear in International Conference on Database Theory (ICDT) 201

Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays

Understanding the local behaviour of structured multi-dimensional data is a fundamental problem in various areas of computer science. As the amount of data is often huge, it is desirable to obtain sublinear time algorithms, and specifically property testers, to understand local properties of the data. We focus on the natural local problem of testing pattern freeness: given a large $d$-dimensional array $A$ and a fixed $d$-dimensional pattern $P$ over a finite alphabet, we say that $A$ is $P$-free if it does not contain a copy of the forbidden pattern $P$ as a consecutive subarray. The distance of $A$ to $P$-freeness is the fraction of entries of $A$ that need to be modified to make it $P$-free. For any $\epsilon \in [0,1]$ and any large enough pattern $P$ over any alphabet, other than a very small set of exceptional patterns, we design a tolerant tester that distinguishes between the case that the distance is at least $\epsilon$ and the case that it is at most $a_d \epsilon$, with query complexity and running time $c_d \epsilon^{-1}$, where $a_d < 1$ and $c_d$ depend only on $d$. To analyze the testers we establish several combinatorial results, including the following $d$-dimensional modification lemma, which might be of independent interest: for any large enough pattern $P$ over any alphabet (excluding a small set of exceptional patterns for the binary case), and any array $A$ containing a copy of $P$, one can delete this copy by modifying one of its locations without creating new $P$-copies in $A$. Our results address an open question of Fischer and Newman, who asked whether there exist efficient testers for properties related to tight substructures in multi-dimensional structured data. They serve as a first step towards a general understanding of local properties of multi-dimensional arrays, as any such property can be characterized by a fixed family of forbidden patterns

Proton decay into charged leptons

We discuss proton and neutron decays involving three leptons in the final state. Some of these modes could constitute the dominant decay channel because they conserve lepton-flavor symmetries that are broken in all usually considered channels. This includes the particularly interesting and rarely discussed $p\to e^+ e^+ \mu^-$ and $p\to \mu^+ \mu^+ e^-$ modes. As the relevant effective operators arise at dimension 9 or 10, observation of a three-lepton mode would probe energy scales of order 100 TeV. This allows to connect proton decay to other probes such as rare meson decays or collider physics. UV completions of this scenario involving leptoquarks unavoidably violate lepton flavor universality and could provide an explanation to the recent $b\to s\mu\mu$ anomalies observed in $B$ meson decays.Comment: 6 pages, to appear in PR