FOCS 2011
TechTalks from event: FOCS 2011
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On Range Searching in the Group Model and Combinatorial DiscrepancyIn this paper we establish an intimate connection between dynamic range searching in the group model and combinatorial discrepancy. Our result states that, for a broad class of range searching data structures (including all known upper bounds), it must hold that $t_u t_q = \Omega(\disc^2/\lg n)$ where $t_u$ is the worst case update time, $t_q$ the worst case query time and $\disc$ is the combinatorial discrepancy of the range searching problem in question. This relation immediately implies a whole range of exceptionally high and neartight lower bounds for all of the basic range searching problems. We list a few of them in the following: \begin{itemize} \item For halfspace range searching in $d$dimensional space, we get a lower bound of $t_u t_q = \Omega(n^{11/d}/\lg n)$. This comes within a $\lg n \lg \lg n$ factor of the best known upper bound. \item For orthogonal range searching in $d$dimensional space, we get a lower bound of $t_u t_q = \Omega(\lg^{d2+\mu(d)}n)$, where $\mu(d)>0$ is some small but strictly positive function of $d$. \item For ball range searching in $d$dimensional space, we get a lower bound of $t_u t_q = \Omega(n^{11/d}/\lg n)$. \end{itemize} We note that the previous highest lower bound for any explicit problem, due to Patrascu [STOC'07], states that $t_q =\Omega((\lg n/\lg(\lg n+t_u))^2)$, which does however hold for a less restrictive class of data structures. Our result also has implications for the field of combinatorial discrepancy. Using textbook range searching solutions, we improve on the best known discrepancy upper bound for axisaligned rectangles in dimensions $d \geq 3$.

A Randomized Rounding Approach to the Traveling Salesman ProblemFor some positive constant $\epsilon_0$, we give a $(\frac{3}{2}\epsilon_0$)approximation algorithm for the following problem: given a graph $G_0=(V,E_0)$, find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in $G_0$. The result improves on the $\frac{3}{2}$approximation algorithm due to Christofides for this special case.Similar to Christofides, our algorithm finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or Tjoin) of that tree. The main difference is in the selection of the spanning tree. Except in certain cases where the solution of LP is nearly integral, we select the spanning tree randomly by sampling from a maximum entropy distribution defined by the linear programming relaxation. Despite the simplicity of the algorithm, the analysis builds on a variety of ideas such as properties of strongly Rayleigh measures from probability theory, graph theoretical results on the structure of near minimum cuts, and the integrality of the Tjoin polytope from polyhedral theory. Also, as a byproduct of our result, we show new properties of the near minimum cuts of any graph, which may be of independent interest.

Approximating Graphic TSP by MatchingsWe present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graphTSP), the approach yields a 1.461approximation algorithm with respect to the HeldKarp lower bound. For graphTSP restricted to a class of graphs that contains degree three bounded and clawfree graphs, we show that the integrality gap of the HeldKarp relaxation matches the conjectured ratio 4/3. The framework allows for generalizations in a natural way and also leads to a 1.586approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.