Partial regularity for higher order variational problems under anisotropic growth conditions

We prove a partial regularity result for local minimizers u : \mathbb{R}^{n}\supset\Omega\rightarrow\mathbb{R}^{M} of the variational integral J(u,\Omega)=\int_{\Omega}f(\nabla^{k}u)dx, where k is any integer and f is a strictly convex integrand of anisotropic (p,q)-growth with exponents satisfying the condition q < p(1 + 2/n). This is some extension of the regularity theorem obtained in [BF2] for the case n = 2

Parameterized Algorithms for Graph Partitioning Problems

In parameterized complexity, a problem instance (I, k) consists of an input I and an extra parameter k. The parameter k usually a positive integer indicating the size of the solution or the structure of the input. A computational problem is called fixed-parameter tractable (FPT) if there is an algorithm for the problem with time complexity O(f(k).nc ), where f(k) is a function dependent only on the input parameter k, n is the size of the input and c is a constant. The existence of such an algorithm means that the problem is tractable for fixed values of the parameter. In this thesis, we provide parameterized algorithms for the following NP-hard graph partitioning problems: (i) Matching Cut Problem: In an undirected graph, a matching cut is a partition of vertices into two non-empty sets such that the edges across the sets induce a matching. The matching cut problem is the problem of deciding whether a given graph has a matching cut. The Matching Cut problem is expressible in monadic second-order logic (MSOL). The MSOL formulation, together with Courcelle’s theorem implies linear time solvability on graphs with bounded tree-width. However, this approach leads to a running time of f(||ϕ||, t) · n, where ||ϕ|| is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. The dependency of f(||ϕ||, t) on ||ϕ|| can be as bad as a tower of exponentials. In this thesis we give a single exponential algorithm for the Matching Cut problem with tree-width alone as the parameter. The running time of the algorithm is 2O(t) · n. This answers an open question posed by Kratsch and Le [Theoretical Computer Science, 2016]. We also show the fixed parameter tractability of the Matching Cut problem when parameterized by neighborhood diversity or other structural parameters. (ii) H-Free Coloring Problems: In an undirected graph G for a fixed graph H, the H-Free q-Coloring problem asks to color the vertices of the graph G using at most q colors such that none of the color classes contain H as an induced subgraph. That is every color class is H-free. This is a generalization of the classical q-Coloring problem, which is to color the vertices of the graph using at most q colors such that no pair of adjacent vertices are of the same color. The H-Free Chromatic Number is the minimum number of colors required to H-free color the graph. For a fixed q, the H-Free q-Coloring problem is expressible in monadic secondorder logic (MSOL). The MSOL formulation leads to an algorithm with time complexity f(||ϕ||, t) · n, where ||ϕ|| is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. In this thesis we present the following explicit combinatorial algorithms for H-Free Coloring problems: • An O(q O(t r ) · n) time algorithm for the general H-Free q-Coloring problem, where r = |V (H)|. • An O(2t+r log t · n) time algorithm for Kr-Free 2-Coloring problem, where Kr is a complete graph on r vertices. The above implies an O(t O(t r ) · n log t) time algorithm to compute the H-Free Chromatic Number for graphs with tree-width at most t. Therefore H-Free Chromatic Number is FPT with respect to tree-width. We also address a variant of H-Free q-Coloring problem which we call H-(Subgraph)Free q-Coloring problem, which is to color the vertices of the graph such that none of the color classes contain H as a subgraph (need not be induced). We present the following algorithms for H-(Subgraph)Free q-Coloring problems. • An O(q O(t r ) · n) time algorithm for the general H-(Subgraph)Free q-Coloring problem, which leads to an O(t O(t r ) · n log t) time algorithm to compute the H- (Subgraph)Free Chromatic Number for graphs with tree-width at most t. • An O(2O(t 2 ) · n) time algorithm for C4-(Subgraph)Free 2-Coloring, where C4 is a cycle on 4 vertices. • An O(2O(t r−2 ) · n) time algorithm for {Kr\e}-(Subgraph)Free 2-Coloring, where Kr\e is a graph obtained by removing an edge from Kr. • An O(2O((tr2 ) r−2 ) · n) time algorithm for Cr-(Subgraph)Free 2-Coloring problem, where Cr is a cycle of length r. (iii) Happy Coloring Problems: In a vertex-colored graph, an edge is happy if its endpoints have the same color. Similarly, a vertex is happy if all its incident edges are happy. we consider the algorithmic aspects of the following Maximum Happy Edges (k-MHE) problem: given a partially k-colored graph G, find an extended full k-coloring of G such that the number of happy edges are maximized. When we want to maximize the number of happy vertices, the problem is known as Maximum Happy Vertices (k-MHV). We show that both k-MHE and k-MHV admit polynomial-time algorithms for trees. We show that k-MHE admits a kernel of size k + `, where ` is the natural parameter, the number of happy edges. We show the hardness of k-MHE and k-MHV for some special graphs such as split graphs and bipartite graphs. We show that both k-MHE and k-MHV are tractable for graphs with bounded tree-width and graphs with bounded neighborhood diversity. vii In the last part of the thesis we present an algorithm for the Replacement Paths Problem which is defined as follows: Let G (|V (G)| = n and |E(G)| = m) be an undirected graph with positive edge weights. Let PG(s, t) be a shortest s − t path in G. Let l be the number of edges in PG(s, t). The Edge Replacement Path problem is to compute a shortest s − t path in G\{e}, for every edge e in PG(s, t). The Node Replacement Path problem is to compute a shortest s−t path in G\{v}, for every vertex v in PG(s, t). We present an O(TSP T (G) + m + l 2 ) time and O(m + l 2 ) space algorithm for both the problems, where TSP T (G) is the asymptotic time to compute a single source shortest path tree in G. The proposed algorithm is simple and easy to implement

Quickest Visibility Queries in Polygonal Domains

Let s be a point in a polygonal domain P of h-1 holes and n vertices. We consider the following quickest visibility query problem. Given a query point q in P, the goal is to find a shortest path in P to move from s to see q as quickly as possible. Previously, Arkin et al. (SoCG 2015) built a data structure of size O(n^2 2^alpha(n) log n) that can answer each query in O(K log^2 n) time, where alpha(n) is the inverse Ackermann function and K is the size of the visibility polygon of q in P (and K can be Theta(n) in the worst case). In this paper, we present a new data structure of size O(n log h + h^2) that can answer each query in O(h log h log n) time. Our result improves the previous work when h is relatively small. In particular, if h is a constant, then our result even matches the best result for the simple polygon case (i.e., h = 1), which is optimal. As a by-product, we also have a new algorithm for the following shortest-path-to-segment query problem. Given a query line segment tau in P, the query seeks a shortest path from s to all points of tau. Previously, Arkin et al. gave a data structure of size O(n^2 2^alpha(n) log n) that can answer each query in O(log^2 n) time, and another data structure of size O(n^3 log n) with O(log n) query time. We present a data structure of size O(n) with query time O(h log n/h), which favors small values of h and is optimal when h = O(1)

Transverse-momentum dependent modification of dynamic texture in central Au+Au collisions at sqrt[sNN]=200GeV

Correlations in the hadron distributions produced in relativistic Au+Au collisions are studied in the discrete wavelet expansion method. The analysis is performed in the space of pseudorapidity (| eta | <= 1) and azimuth(full 2 pi ) in bins of transverse momentum (pt) from 0.14 <= pt <= 2.1GeV/c. In peripheral Au+Au collisions a correlation structure ascribed to minijet fragmentation is observed. It evolves with collision centrality and pt in a way not seen before, which suggests strong dissipation of minijet fragmentation in the longitudinally expanding medium.Alle Autoren: J. Adams, M. M. Aggarwal, Z. Ahammed, J. Amonett, B. D. Anderson, D. Arkhipkin, G. S. Averichev, S. K. Badyal, Y. Bai, J. Balewski, O. Barannikova, L. S. Barnby, J. Baudot, S. Bekele, V. V. Belaga, R. Bellwied, J. Berger, B. I. Bezverkhny, S. Bharadwaj, A. Bhasin, A. K. Bhati, V. S. Bhatia, H. Bichsel, A. Billmeier, L. C. Bland, C. O. Blyth, B. E. Bonner, M. Botje, A. Boucham, A. Brandin, A. Bravar, M. Bystersky, R. V. Cadman, X. Z. Cai, H. Caines, M. Calderón de la Barca Sánchez, J. Castillo, D. Cebra, Z. Chajecki, P. Chaloupka, S. Chattopdhyay, H. F. Chen, Y. Chen, J. Cheng, M. Cherney, A. Chikanian, W. Christie, J. P. Coffin, T. M. Cormier, J. G. Cramer, H. J. Crawford, D. Das, S. Das, M. M. de Moura, A. A. Derevschikov, L. Didenko, T. Dietel, S. M. Dogra, W. J. Dong, X. Dong, J. E. Draper, F. Du, A. K. Dubey, V. B. Dunin, J. C. Dunlop, M. R. Dutta Mazumdar, V. Eckardt, W. R. Edwards, L. G. Efimov, V. Emelianov, J. Engelage, G. Eppley, B. Erazmus, M. Estienne, P. Fachini, J. Faivre, R. Fatemi, J. Fedorisin, K. Filimonov, P. Filip, E. Finch, V. Fine, Y. Fisyak, K. Fomenko, J. Fu, C. A. Gagliardi, J. Gans, M. S. Ganti, L. Gaudichet, F. Geurts, V. Ghazikhanian, P. Ghosh, J. E. Gonzalez, O. Grachov, O. Grebenyuk, D. Grosnick, S. M. Guertin, Y. Guo, A. Gupta, T. D. Gutierrez, T. J. Hallman, A. Hamed, D. Hardtke, J. W. Harris, M. Heinz, T. W. Henry, S. Hepplemann, B. Hippolyte, A. Hirsch, E. Hjort, G. W. Hoffmann, H. Z. Huang, S. L. Huang, E. W. Hughes, T. J. Humanic, G. Igo, A. Ishihara, P. Jacobs, W. W. Jacobs, M. Janik, H. Jiang, P. G. Jones, E. G. Judd, S. Kabana, K. Kang, M. Kaplan, D. Keane, V. Yu. Khodyrev, J. Kiryluk, A. Kisiel, E. M. Kislov, J. Klay, S. R. Klein, A. Klyachko, D. D. Koetke, T. Kollegger, M. Kopytine, L. Kotchenda, M. Kramer, P. Kravtsov, V. I. Kravtsov, K. Krueger, C. Kuhn, A. I. Kulikov, A. Kumar, R. Kh. Kutuev, A. A. Kuznetsov, M. A. C. Lamont, J. M. Landgraf, S. Lange, F. Laue, J. Lauret, A. Lebedev, R. Lednicky, S. Lehocka, M. J. LeVine, C. Li, Q. Li, Y. Li, G. Lin, S. J. Lindenbaum, M. A. Lisa, F. Liu, L. Liu, Q. J. Liu, Z. Liu, T. Ljubicic, W. J. Llope, H. Long, R. S. Longacre, M. Lopez-Noriega, W. A. Love, Y. Lu, T. Ludlam, D. Lynn, G. L. Ma, J. G. Ma, Y. G. Ma, D. Magestro, S. Mahajan, D. P. Mahapatra, R. Majka, L. K. Mangotra, R. Manweiler, S. Margetis, C. Markert, L. Martin, J. N. Marx, H. S. Matis, Yu. A. Matulenko, C. J. McClain, T. S. McShane, F. Meissner, Yu. Melnick, A. Meschanin, M. L. Miller, N. G. Minaev, C. Mironov, A. Mischke, D. K. Mishra, J. Mitchell, B. Mohanty, L. Molnar, C. F. Moore, D. A. Morozov, M. G. Munhoz, B. K. Nandi, S. K. Nayak, T. K. Nayak, J. M. Nelson, P. K. Netrakanti, V. A. Nikitin, L. V. Nogach, S. B. Nurushev, G. Odyniec, A. Ogawa, V. Okorokov, M. Oldenburg, D. Olson, S. K. Pal, Y. Panebratsev, S. Y. Panitkin, A. I. Pavlinov, T. Pawlak, T. Peitzmann, V. Perevoztchikov, C. Perkins, W. Peryt, V. A. Petrov, S. C. Phatak, R. Picha, M. Planinic, J. Pluta, N. Porile, J. Porter, A. M. Poskanzer, M. Potekhin, E. Potrebenikova, B. V. K. S. Potukuchi, D. Prindle, C. Pruneau, J. Putschke, G. Rakness, R. Raniwala, S. Raniwala, O. Ravel, R. L. Ray, S. V. Razin, D. Reichhold, J. G. Reid, G. Renault, F. Retiere, A. Ridiger, H. G. Ritter, J. B. Roberts, O. V. Rogachevskiy, J. L. Romero, A. Rose, C. Roy, L. Ruan, R. Sahoo, I. Sakrejda, S. Salur, J. Sandweiss, I. Savin, P. S. Sazhin, J. Schambach, R. P. Scharenberg, N. Schmitz, K. Schweda, J. Seger, P. Seyboth, E. Shahaliev, M. Shao, W. Shao, M. Sharma, W. Q. Shen, K. E. Shestermanov, S. S. Shimanskiy, E. Sichtermann, F. Simon, R. N. Singaraju, G. Skoro, N. Smirnov, R. Snellings, G. Sood, P. Sorensen, J. Sowinski, J. Speltz, H. M. Spinka, B. Srivastava, A. Stadnik, T. D. S. Stanislaus, R. Stock, A. Stolpovsky, M. Strikhanov, B. Stringfellow, A. A. P. Suaide, E. Sugarbaker, C. Suire, M. Sumbera, B. Surrow, T. J. M. Symons, A. Szanto de Toledo, P. Szarwas, A. Tai, J. Takahashi, A. H. Tang, T. Tarnowsky, D. Thein, J. H. Thomas, S. Timoshenko, M. Tokarev, T. A. Trainor, S. Trentalange, R. E. Tribble, O. D. Tsai, J. Ulery, T. Ullrich, D. G. Underwood, A. Urkinbaev, G. Van Buren, M. van Leeuwen, A. M. Vander Molen, R. Varma, I. M. Vasilevski, A. N. Vasiliev, R. Vernet, S. E. Vigdor, Y. P. Viyogi, S. Vokal, S. A. Voloshin, M. Vznuzdaev, W. T. Waggoner, F. Wang, G. Wang, G. Wang, X. L. Wang, Y. Wang, Y. Wang, Z. M. Wang, H. Ward, J. W. Watson, J. C. Webb, R. Wells, G. D. Westfall, A. Wetzler, C. Whitten Jr., H. Wieman, S. W. Wissink, R. Witt, J. Wood, J. Wu, N. Xu, Z. Xu, Z. Z. Xu, E. Yamamoto, P. Yepes, V. I. Yurevich, Y. V. Zanevsky, H. Zhang, W. M. Zhang, Z. P. Zhang, P. A. Zolnierczuk, R. Zoulkarneev, Y. Zoulkarneeva, and A. N. Zubare