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The mixture of data in real-life exhibits structure or connection property in nature. Typical data include biological data, communication network data, image data, etc. Graphs provide a natural way to represent and analyze these types of data and their relationships. Unfortunately, the related algorithms usually suffer from high computational complexity, since some of these problems are NP-hard. Therefore, in recent years, many graph models and optimization algorithms have been proposed to achieve a better balance between efficacy and efficiency. This book contains some papers reporting recent achievements regarding graph models, algorithms, and applications to problems in the real world, with some focus on optimization and computational complexity.

planar graphs --- k-planarity --- NP-hardness --- polynomial time reduction --- cliques --- paths --- computational social choice --- election control --- multi-winner election --- social influence --- influence maximization --- congestion games --- pure Nash equilibrium --- potential games --- price of anarchy --- price of stability --- phylogenetic tree --- evolutionary tree --- ancestral mixture model --- mixture tree --- mixture distance --- tree comparison --- clique independent set --- clique transversal number --- signed clique transversal function --- minus clique transversal function --- k-fold clique transversal set --- distance-hereditary graphs --- stretch number --- recognition problem --- forbidden subgraphs --- hole detection --- analysis and design or graph algorithms --- distributed graph and network algorithms --- graph theory with algorithmic applications --- computational complexity of graph problems --- experimental evaluation of graph algorithms

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How the concept of proof has enabled the creation of mathematical knowledgeThe Story of Proof investigates the evolution of the concept of proof—one of the most significant and defining features of mathematical thought—through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge.Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as “infinitesimal algebra,” and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved.Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field’s power and progress.

Proof theory. --- Mathematicians. --- Scientists --- Logic, Symbolic and mathematical --- Accuracy and precision. --- Addition. --- Aleph number. --- Algorithm. --- Analogy. --- Analysis. --- Archimedean property. --- Associative property. --- Axiom of choice. --- Axiom schema. --- Axiom. --- Bijection. --- Calculation. --- Certainty. --- Coefficient. --- Commutative property. --- Computability theory. --- Computability. --- Computable function. --- Computation. --- Constructible number. --- Constructive analysis. --- Continuous function (set theory). --- Corollary. --- Countable set. --- Credential. --- Dedekind cut. --- Desargues's theorem. --- Determinant. --- Direct proof. --- Equation. --- Equinumerosity. --- Estimation. --- Estimator. --- Extreme value theorem. --- Fundamental theorem. --- Gentzen's consistency proof. --- Geometry. --- Hypotenuse. --- Hypothesis. --- Identifiability. --- Inference. --- Infimum and supremum. --- Infinitesimal. --- Intermediate value theorem. --- Intuitionism. --- Logic. --- Logical connective. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Maximal element. --- Natural number. --- Number theory. --- Obstacle. --- Ordinal number. --- Peano axioms. --- Permutation group. --- Permutation. --- Planarity. --- Playfair's axiom. --- Polygon. --- Polynomial. --- Power set. --- Predicate logic. --- Prediction. --- Prime factor. --- Prime number. --- Proof by infinite descent. --- Pythagorean theorem. --- Quantifier (logic). --- Quantity. --- Quaternion. --- Quintic function. --- Rational number. --- Real number. --- Reason. --- Recursively enumerable set. --- Rule of inference. --- Satisfiability. --- Self-reference. --- Sequence. --- Set theory. --- Special case. --- Staffing. --- Subsequence. --- Subset. --- Summation. --- Symbolic computation. --- Symmetry group. --- Theorem. --- Theory. --- Total order. --- Truth value. --- Turing machine. --- Unit square. --- Vector space. --- Well-order. --- Zorn's lemma.