Linear-Time FPT Algorithms via Network Flow

In the area of parameterized complexity, to cope with NP-Hard problems, we introduce a parameter k besides the input size n, and we aim to design algorithms (called FPT algorithms) that run in O(f(k)n^d) time for some function f(k) and constant d. Though FPT algorithms have been successfully designed for many problems, typically they are not sufficiently fast because of huge f(k) and d. In this paper, we give FPT algorithms with small f(k) and d for many important problems including Odd Cycle Transversal and Almost 2-SAT. More specifically, we can choose f(k) as a single exponential (4^k) and d as one, that is, linear in the input size. To the best of our knowledge, our algorithms achieve linear time complexity for the first time for these problems. To obtain our algorithms for these problems, we consider a large class of integer programs, called BIP2. Then we show that, in linear time, we can reduce BIP2 to Vertex Cover Above LP preserving the parameter k, and we can compute an optimal LP solution for Vertex Cover Above LP using network flow. Then, we perform an exhaustive search by fixing half-integral values in the optimal LP solution for Vertex Cover Above LP. A bottleneck here is that we need to recompute an LP optimal solution after branching. To address this issue, we exploit network flow to update the optimal LP solution in linear time.Comment: 20 page

Half-integrality, LP-branching and FPT Algorithms

A recent trend in parameterized algorithms is the application of polytope tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). However, although interesting results have been achieved, the methods require the underlying polytope to have very restrictive properties (half-integrality and persistence), which are known only for few problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a relaxation of the search space from $\{0,1\}^V$ to $\{0,1/2,1\}^V$ such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such relaxations, we provide a much broader class of half-integral polytopes with the required properties, unifying and extending previously known cases. In addition to the insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an $O^*(|\Sigma|^{2k})$-time algorithm for node-deletion Unique Label Cover with label set $\Sigma$ and an $O^*(4^k)$-time algorithm for Group Feedback Vertex Set, including the setting where the group is only given by oracle access. All these significantly improve on previous results. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. (2012). Additionally, we propose a network flow-based approach to solve some cases of the relaxation problem. This gives the first linear-time FPT algorithm to edge-deletion Unique Label Cover.Comment: Added results on linear-time FPT algorithms (not present in SODA paper

Chemical potential jump between hole- and electron-doped sides of ambipolar high-Tc cuprate

In order to study an intrinsic chemical potential jump between the hole- and electron-doped high-Tc superconductors, we have performed core-level X-ray photoemission spectroscopy (XPS) measurements of Y0.38La0.62Ba1.74La0.26Cu3Oy (YLBLCO), into which one can dope both holes and electrons with maintaining the same crystal structure. Unlike the case between the hole-doped system La_2-xSrxCuO4 and the electron-doped system Nd_2-xCexCuO4, we have estimated the true chemical potential jump between the hole- and electron-doped YLBLCO to be ~0.8 eV, which is much smaller than the optical gaps of 1.4-1.7 eV reported for the parent insulating compounds. We attribute the reduced jump to the indirect nature of the charge-excitation gap as well as to the polaronic nature of the doped carriers.Comment: 4 pages, 3 figure

Fast Exact Shortest-Path Distance Queries on Large Networks by Pruned Landmark Labeling

We propose a new exact method for shortest-path distance queries on large-scale networks. Our method precomputes distance labels for vertices by performing a breadth-first search from every vertex. Seemingly too obvious and too inefficient at first glance, the key ingredient introduced here is pruning during breadth-first searches. While we can still answer the correct distance for any pair of vertices from the labels, it surprisingly reduces the search space and sizes of labels. Moreover, we show that we can perform 32 or 64 breadth-first searches simultaneously exploiting bitwise operations. We experimentally demonstrate that the combination of these two techniques is efficient and robust on various kinds of large-scale real-world networks. In particular, our method can handle social networks and web graphs with hundreds of millions of edges, which are two orders of magnitude larger than the limits of previous exact methods, with comparable query time to those of previous methods.Comment: To appear in SIGMOD 201

Substellar Companions to Seven Evolved Intermediate-Mass Stars

We report the detections of substellar companions orbiting around seven evolved intermediate-mass stars from precise Doppler measurements at Okayama Astrophysical Observatory. o UMa (G4 II-III) is a giant with a mass of 3.1 M_sun and hosts a planet with minimum mass of m_2sini=4.1 M_J in an orbit with a period P=1630 d and an eccentricity e=0.13. This is the first planet candidate (< 13 M_J) ever discovered around stars more massive than 3 M_sun. o CrB (K0 III) is a 2.1 M_sun giant and has a planet of m_2sini=1.5 M_J in a 187.8 d orbit with e=0.19. This is one of the least massive planets ever discovered around ~2 M_sun stars. HD 5608 (K0 IV) is an 1.6 M_sun subgiant hosting a planet of m_2sini=1.4 M_J in a 793 d orbit with e=0.19. The star also exhibits a linear velocity trend suggesting the existence of an outer, more massive companion. 75 Cet (G3 III:) is a 2.5 M_sun giant hosting a planet of m_2sini=3.0 M_J in a 692 d orbit with e=0.12. The star also shows possible additional periodicity of about 200 d and 1880 d with velocity amplitude of ~7--10 m/s, although these are not significant at this stage. nu Oph (K0 III) is a 3.0 M_sun giant and has two brown-dwarf companions of m_2sini= 24 M_J and 27 M_J, in orbits with P=530.3 d and 3190 d, and e=0.126 and 0.17, respectively, which were independently announced by Quirrenbach et al. (2011). The ratio of the periods is close to 1:6, suggesting that the companions are in mean motion resonance. We also independently confirmed planets around k CrB (K0 III-IV) and HD 210702 (K1 IV), which had been announced by Johnson et al. (2008) and Johnson et al. (2007a), respectively. All of the orbital parameters we obtained are consistent with the previous results.Comment: 21 pages, 14 figures, accepted for publication in PAS