This paper presents a new method and a constraint-based objective function to solve two problemsrelated to the design of optical telecommunication networks, namely the Synchronous Optical Net-work Ring Assignment Problem (SRAP) and the Intra-ring Synchronous Optical Network DesignProblem (IDP). These network topology problems can be represented as a graph partitioning withcapacity constraints as shown in previous works. We present here a new objective function and anew local search algorithm to solve these problems. Experiments conducted in C
These two problems have been well studied. It has been proven that they are both @math -hard ( @cite_4 , @cite_1 ).
@cite_1 @cite_4
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We present a detailed experimental investigation of the easy-hard-easy phase transition for randomly generated instances of satisfiability problems. Problems in the hard part of the phase transition have been extensively used for benchmarking satisfiability algorithms. This study demonstrates that problem classes and regions of the phase transition previously thought to be easy can sometimes be orders of magnitude more difficult than the worst problems in problem classes and regions of the phase transition considered hard. These difficult problems are either hard unsatisfiable problems or are satisfiable problems which give a hard unsatisfiable subproblem following a wrong split. Whilst these hard unsatisfiable problems may have short proofs, these appear to be difficult to find, and other proofs are long and hard. The following paper is reproduced from a Russian journal of the character of our own Proceedings of the National Academy of Sciences, Comptes Rendus (Doklady) de I'Academie des Sciences de I'URSS, 1942, Volume XXXVII, No. 7-8. The author is one of the most distinguished of Russian mathematicians. He has made very important contributions in pure mathematics in the theory of functional analysis, and has made equally important contributions to applied mathematics in numerical analysis and the theory and practice of computation. Although his exposition in this paper is quite terse and couched in mathematical language which may be difficult for some readers of Management Science to follow, it is thought that this presentation will: (1) make available to American readers generally an important work in the field of linear programming, (2) provide an indication of the type of analytic work which has been done and is being done in connection with rational planning in Russia, (3) through the specific examples mentioned indicate the types of interpretation which the Russians have made of the abstract mathematics (for example, the potential and field interpretations adduced in this country recently by W. Prager were anticipated in this paper). It is to be noted, however, that the problem of determining an effective method of actually acquiring the solution to a specific problem is not solved in this paper. In the category of development of such methods we seem to be, currently, ahead of the Russians.--A. Charnes, Northwestern Technological Institute and The Transportation Center. Let @math be a @math -ary predicate over a finite alphabet. Consider a random CSP @math instance @math over @math variables with @math constraints. When @math the instance @math will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When @math is the @math -ary OR predicate, this is the well studied problem of refuting random @math -SAT formulas, and an efficient algorithm is known only when @math . Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a @math -ary predicate, having @math constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if @math fails to support a @math -wise uniform distribution, then there is an efficient algorithm that refutes random CSP @math instances @math whp when @math . Indeed, our algorithm will "somewhat strongly" refute @math , certifying @math , if @math then we get the strongest possible refutation, certifying @math . This last result is new even in the context of random @math -SAT. Regarding the optimality of our @math requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP @math may be impossible when @math . Thus there is an indication our algorithm's dependence on @math is optimal for every @math , at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the @math -round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz. The class of @math -complete problems is a class of computationally eqivalent counting problems (defined by the author in a previous paper) that are at least as difficult as the @math -complete problems. Here we show, for a large number of natural counting problems for which there was no previous indication of intractability, that they belong to this class. The technique used is that of polynomial time reduction with oracles via translations that are of algebraic or arithmetic nature. This paper presents a novel approach to learning to solve simple arithmetic word problems. Our system, ARIS, analyzes each of the sentences in the problem statement to identify the relevant variables and their values. ARIS then maps this information into an equation that represents the problem, and enables its (trivial) solution as shown in Figure 1. The paper analyzes the arithmetic-word problems “genre”, identifying seven categories of verbs used in such problems. ARIS learns to categorize verbs with 81.2 accuracy, and is able to solve 77.7 of the problems in a corpus of standard primary school test questions. We report the first learning results on this task without reliance on predefined templates and make our data publicly available. 1 1. Summary In Part I, four ostensibly different theoretical models of induction are presented, in which the problem dealt with is the extrapolation of a very long sequence of symbols—presumably containing all of the information to be used in the induction. Almost all, if not all problems in induction can be put in this form. Some strong heuristic arguments have been obtained for the equivalence of the last three models. One of these models is equivalent to a Bayes formulation, in which a priori probabilities are assigned to sequences of symbols on the basis of the lengths of inputs to a universal Turing machine that are required to produce the sequence of interest as output. Though it seems likely, it is not certain whether the first of the four models is equivalent to the other three. Few rigorous results are presented. Informal investigations are made of the properties of these models. There are discussions of their consistency and meaningfulness, of their degree of independence of the exact nature of the Turing machine used, and of the accuracy of their predictions in comparison to those of other induction methods. In Part II these models are applied to the solution of three problems—prediction of the Bernoulli sequence, extrapolation of a certain kind of Markov chain, and the use of phrase structure grammars for induction. Though some approximations are used, the first of these problems is treated most rigorously. The result is Laplace's rule of succession. The solution to the second problem uses less certain approximations, but the properties of the solution that are discussed, are fairly independent of these approximations. The third application, using phrase structure grammars, is least exact of the three. First a formal solution is presented. Though it appears to have certain deficiencies, it is hoped that presentation of this admittedly inadequate model will suggest acceptable improvements in it. This formal solution is then applied in an approximate way to the determination of the “optimum” phrase structure grammar for a given set of strings. The results that are obtained are plausible, but subject to the uncertainties of the approximation used. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . Despite @l-calculus is now three quarters of a century old, no formula counting @l-terms has been proposed yet, and the combinatorics of @l-calculus is considered a hard problem. The difficulty lies in the fact that the recursive expression of the numbers of terms of size n with at most m free variables contains the number of terms of size n-1 with at most m+1 variables. This leads to complex recurrences that cannot be handled by classical analytic methods. Here based on de Bruijn indices (another presentation of @l-calculus) we propose several results on counting untyped lambda terms, i.e., on telling how many terms belong to such or such class, according to the size of the terms and or to the number of free variables. We extend the results to normal forms. Papadimitriou introduced several classes of NP search problems based on combinatorial principles which guarantee the existence of solutions to the problems. Many interesting search problems not known to be solvable in polynomial time are contained in these classes, and a number of them are complete problems. We consider the question of the relative complexity of these search problem classes. We prove several separations which show that in a generic relativized world the search classes are distinct and there is a standard search problem in each of them that is not computationally equivalent to any decision problem. (Naturally, absolute separations would imply thatP?NP.) Our separation proofs have interesting combinatorial content and go to the heart of the combinatorial principles on which the classes are based. We derive one result via new lower bounds on the degrees of poly- nomials asserted to exist by Hilbert's nullstellensatz over finite fields. We study inherent structural properties of a strongly NP-hard problem of scheduling @math jobs with release times and due dates on a single machine to minimize the number of late jobs. Our study leads to two polynomial-time algorithms. The first algorithm with the time complexity @math solves the problem if during its execution no job with some special property occurs. The second algorithm solves the version of the problem when all jobs have the same length. The time complexity of the latter algorithm is @math , which is an improvement over the earlier known algorithm with the time complexity @math . Editors' preface Acknowledgments Author's introduction 1. A problem and a conjecture 2. A proof 3. Criticism of the proof by counterexamples which are local but not global 4. Criticism of the conjecture by global counterexamples 5. Criticism of the proof-analysis by counterexamples which are global but not local: the problem of rigour 6. Return to criticism of the proof by counterexamples which are local but not global: the problem of content 7. The problem of content revisited 8. Concept-formation 9. How criticism may turn mathematical truth into logical truth Appendices Bibliography Index of names Index of subjects. Identifying mathematical relations expressed in text is essential to understanding a broad range of natural language text from election reports, to financial news, to sport commentaries to mathematical word problems. This paper focuses on identifying and understanding mathematical relations described within a single sentence. We introduce the problem of Equation Parsing -- given a sentence, identify noun phrases which represent variables, and generate the mathematical equation expressing the relation described in the sentence. We introduce the notion of projective equation parsing and provide an efficient algorithm to parse text to projective equations. Our system makes use of a high precision lexicon of mathematical expressions and a pipeline of structured predictors, and generates correct equations in @math of the cases. In @math of the time, it also identifies the correct noun phrase @math variables mapping, significantly outperforming baselines. We also release a new annotated dataset for task evaluation. Given two documents A and B we define two mathematical notions: their resemblance r(A, B) and their containment c(A, B) that seem to capture well the informal notions of "roughly the same" and "roughly contained." The basic idea is to reduce these issues to set intersection problems that can be easily evaluated by a process of random sampling that can be done independently for each document. Furthermore, the resemblance can be evaluated using a fixed size sample for each document. This paper discusses the mathematical properties of these measures and the efficient implementation of the sampling process using Rabin (1981) fingerprints. This paper presents, to our knowledge, the first study on analyzing mathematical expressions to detect academic plagiarism. We make the following contributions. First, we investigate confirmed cases of plagiarism to categorize the similarities of mathematical content commonly found in plagiarized publications. From this investigation, we derive possible feature selection and feature comparison strategies for developing math-based detection approaches and a ground truth for our experiments. Second, we create a test collection by embedding confirmed cases of plagiarism into the NTCIR-11 MathIR Task dataset, which contains approx. 60 million mathematical expressions in 105,120 documents from arXiv.org. Third, we develop a first math-based detection approach by implementing and evaluating different feature comparison approaches using an open source parallel data processing pipeline built using the Apache Flink framework. The best performing approach identifies all but two of our real-world test cases at the top rank and achieves a mean reciprocal rank of 0.86. The results show that mathematical expressions are promising text-independent features to identify academic plagiarism in large collections. To facilitate future research on math-based plagiarism detection, we make our source code and data available. 1. Summary Bohnenblust, Shapley, and Sherman [2] have introduced a method of comparing two sampling procedures or experiments; essentially their concept is that one experiment a is more informative than a second experiment ,, a v ,S, if, for every possible risk function, any risk attainable with , is also attainable with a. If a is a sufficient statistic for a procedure equivalent to ,S, a >,, it is shown that a v j3. In the case of dichotomies, the converse is proved. Whether > and v are equivalent in general is not known. Various properties of > and n are obtained, such as the following: if a > , and y is independent of both, then the combination (a, -y) > (#, y). An application to a problem in 2 X 2 tables is discussed. In the (d,n) group testing problem n items have to be identified as either good or defective and the number of defective items is known to be d. A test on an arbitrary group (subset) of items reveals either that all items in the group are good or that at least one of the items is defective, but not how many or which items are defective. We present a new algorithm which in the worst case needs less than @math tests more than the information lower bound @math for n d≥2. For n d≥38, the difference decreases to less than @math tests. For d≥10, this is a considerable improvement over the d−1 additional tests given for the best previously known algorithm by Hwang. We conjecture that the behaviour for large n and d of the difference is optimal for @math . This implies that the @math tests per defective given in the bound above are the best possible. It is shown that two sorts of problems belong to the NP-complete class. First, it is proven that for a given @k-colorable graph and a given @k-coloring of that graph, determining whether the graph is or is not uniquely @k-colorable is NP-complete. Second, a result by Garey, Johnson, and Stockmeyer is extended with a proof that the coloring of four-regular planar graphs is NP-complete. Let Q be any algebraic structure and the set of all total programs over Q using the instruction set z ,,-1, z , , x + y, z ,,-x y , z x * y , z -x y . (A program is total if no division by zero occurs during any computation ) Let the equivalence problem for be the problem of deciding for two given programs in whether or not they compute the same funcuon The following results are proved: (1) If Q is an inftmte field (e.g, the rauonal numbers or the complex numbers), then the equwalence problem for is probabilistlcally decidable in polynomml time. The result also holds for programs with no dwlslon instructions and Q an infimte integral domain (e.g., the integers). (2) If Q is a finite field, or if Q is a fimte set of integers of cardmahty _>2, then the equivalence problem is NP-hard. The case when the field Q is finite but its cardinality is a funcuon of the size of the instance to the eqmvalence problem is also considered An example is shown for which a sharp boundary between the classes NP-hard and probabihsticaUy decidable exists (provided they are not identical classes). The 3SUM problem is to decide, given a set of @math real numbers, whether any three sum to zero. It is widely conjectured that a trivial @math -time algorithm is optimal and over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies @math lower bounds on numerous problems in computational geometry and a variant of the conjecture implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures. In this paper we refute the 3SUM conjecture. We prove that the decision tree complexity of 3SUM is @math and give two subquadratic 3SUM algorithms, a deterministic one running in @math time and a randomized one running in @math time with high probability. Our results lead directly to improved bounds for @math -variate linear degeneracy testing for all odd @math . The problem is to decide, given a linear function @math and a set @math , whether @math . We show the decision tree complexity of this problem is @math . Finally, we give a subcubic algorithm for a generalization of the @math -product over real-valued matrices and apply it to the problem of finding zero-weight triangles in weighted graphs. We give a depth- @math decision tree for this problem, as well as an algorithm running in time @math . This article tackles a new challenging task in computational argumentation. Given a pair of two arguments to a certain controversial topic, we aim to directly assess qualitative properties of the arguments in order to explain why one argument is more convincing than the other one. We approach this task in a fully empirical manner by annotating 26k explanations written in natural language. These explanations describe convincingness of arguments in the given argument pair, such as their strengths or flaws. We create a new crowd-sourced corpus containing 9,111 argument pairs, multi-labeled with 17 classes, which was cleaned and curated by employing several strict quality measures. We propose two tasks on this data set, namely (1) predicting the full label distribution and (2) classifying types of flaws in less convincing arguments. Our experiments with feature-rich SVM learners and Bidirectional LSTM neural networks with convolution and attention mechanism reveal that such a novel fine-grained analysis of Web argument convincingness is a very challenging task. We release the new UKPConvArg2 corpus and software under permissive licenses to the research community.