4,806 research outputs found
Bisection of Bounded Treewidth Graphs by Convolutions
In the Bisection problem, we are given as input an edge-weighted graph G. The task is to find a partition of V(G) into two parts A and B such that ||A| - |B|| <= 1 and the sum of the weights of the edges with one endpoint in A and the other in B is minimized. We show that the complexity of the Bisection problem on trees, and more generally on graphs of bounded treewidth, is intimately linked to the (min, +)-Convolution problem. Here the input consists of two sequences (a[i])^{n-1}_{i = 0} and (b[i])^{n-1}_{i = 0}, the task is to compute the sequence (c[i])^{n-1}_{i = 0}, where c[k] = min_{i=0,...,k}(a[i] + b[k - i]).
In particular, we prove that if (min, +)-Convolution can be solved in O(tau(n)) time, then Bisection of graphs of treewidth t can be solved in time O(8^t t^{O(1)} log n * tau(n)), assuming a tree decomposition of width t is provided as input. Plugging in the naive O(n^2) time algorithm for (min, +)-Convolution yields a O(8^t t^{O(1)} n^2 log n) time algorithm for Bisection. This improves over the (dependence on n of the) O(2^t n^3) time algorithm of Jansen et al. [SICOMP 2005] at the cost of a worse dependence on t. "Conversely", we show that if Bisection can be solved in time O(beta(n)) on edge weighted trees, then (min, +)-Convolution can be solved in O(beta(n)) time as well. Thus, obtaining a sub-quadratic algorithm for Bisection on trees is extremely challenging, and could even be impossible. On the other hand, for unweighted graphs of treewidth t, by making use of a recent algorithm for Bounded Difference (min, +)-Convolution of Chan and Lewenstein [STOC 2015], we obtain a sub-quadratic algorithm for Bisection with running time O(8^t t^{O(1)} n^{1.864} log n)
Unifying mesh- and tree-based programmable interconnect
We examine the traditional, symmetric, Manhattan mesh design for field-programmable gate-array (FPGA) routing along with tree-of-meshes (ToM) and mesh-of-trees (MoT) based designs. All three networks can provide general routing for limited bisection designs (Rent's rule with p<1) and allow locality exploitation. They differ in their detailed topology and use of hierarchy. We show that all three have the same asymptotic wiring requirements. We bound this tightly by providing constructive mappings between routes in one network and routes in another. For example, we show that a (c,p) MoT design can be mapped to a (2c,p) linear population ToM and introduce a corner turn scheme which will make it possible to perform the reverse mapping from any (c,p) linear population ToM to a (2c,p) MoT augmented with a particular set of corner turn switches. One consequence of this latter mapping is a multilayer layout strategy for N-node, linear population ToM designs that requires only /spl Theta/(N) two-dimensional area for any p when given sufficient wiring layers. We further show upper and lower bounds for global mesh routes based on recursive bisection width and show these are within a constant factor of each other and within a constant factor of MoT and ToM layout area. In the process we identify the parameters and characteristics which make the networks different, making it clear there is a unified design continuum in which these networks are simply particular regions
The Peculiar Phase Structure of Random Graph Bisection
The mincut graph bisection problem involves partitioning the n vertices of a
graph into disjoint subsets, each containing exactly n/2 vertices, while
minimizing the number of "cut" edges with an endpoint in each subset. When
considered over sparse random graphs, the phase structure of the graph
bisection problem displays certain familiar properties, but also some
surprises. It is known that when the mean degree is below the critical value of
2 log 2, the cutsize is zero with high probability. We study how the minimum
cutsize increases with mean degree above this critical threshold, finding a new
analytical upper bound that improves considerably upon previous bounds.
Combined with recent results on expander graphs, our bound suggests the unusual
scenario that random graph bisection is replica symmetric up to and beyond the
critical threshold, with a replica symmetry breaking transition possibly taking
place above the threshold. An intriguing algorithmic consequence is that
although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio
to the optimal value approaches 1 asymptotically) in polynomial time for
typical instances near the phase transition.Comment: substantially revised section 2, changed figures 3, 4 and 6, made
minor stylistic changes and added reference
Stochastic Sensor Scheduling via Distributed Convex Optimization
In this paper, we propose a stochastic scheduling strategy for estimating the
states of N discrete-time linear time invariant (DTLTI) dynamic systems, where
only one system can be observed by the sensor at each time instant due to
practical resource constraints. The idea of our stochastic strategy is that a
system is randomly selected for observation at each time instant according to a
pre-assigned probability distribution. We aim to find the optimal pre-assigned
probability in order to minimize the maximal estimate error covariance among
dynamic systems. We first show that under mild conditions, the stochastic
scheduling problem gives an upper bound on the performance of the optimal
sensor selection problem, notoriously difficult to solve. We next relax the
stochastic scheduling problem into a tractable suboptimal quasi-convex form. We
then show that the new problem can be decomposed into coupled small convex
optimization problems, and it can be solved in a distributed fashion. Finally,
for scheduling implementation, we propose centralized and distributed
deterministic scheduling strategies based on the optimal stochastic solution
and provide simulation examples.Comment: Proof errors and typos are fixed. One section is removed from last
versio
Smoothed Efficient Algorithms and Reductions for Network Coordination Games
Worst-case hardness results for most equilibrium computation problems have
raised the need for beyond-worst-case analysis. To this end, we study the
smoothed complexity of finding pure Nash equilibria in Network Coordination
Games, a PLS-complete problem in the worst case. This is a potential game where
the sequential-better-response algorithm is known to converge to a pure NE,
albeit in exponential time. First, we prove polynomial (resp. quasi-polynomial)
smoothed complexity when the underlying game graph is a complete (resp.
arbitrary) graph, and every player has constantly many strategies. We note that
the complete graph case is reminiscent of perturbing all parameters, a common
assumption in most known smoothed analysis results.
Second, we define a notion of smoothness-preserving reduction among search
problems, and obtain reductions from -strategy network coordination games to
local-max-cut, and from -strategy games (with arbitrary ) to
local-max-cut up to two flips. The former together with the recent result of
[BCC18] gives an alternate -time smoothed algorithm for the
-strategy case. This notion of reduction allows for the extension of
smoothed efficient algorithms from one problem to another.
For the first set of results, we develop techniques to bound the probability
that an (adversarial) better-response sequence makes slow improvements on the
potential. Our approach combines and generalizes the local-max-cut approaches
of [ER14,ABPW17] to handle the multi-strategy case: it requires a careful
definition of the matrix which captures the increase in potential, a tighter
union bound on adversarial sequences, and balancing it with good enough rank
bounds. We believe that the approach and notions developed herein could be of
interest in addressing the smoothed complexity of other potential and/or
congestion games
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