136,708 research outputs found
On Graphs of the Cone Decompositions for the Min-Cut and Max-Cut Problems
We consider maximum and minimum cut problems with nonnegative weights of edges. We define the graphs of the cone decompositions and find a linear clique number for the min-cut problem and a superpolynomial clique number for the max-cut problem. These values characterize the time complexity in a broad class of algorithms based on linear comparisons
Subexponential LPs Approximate Max-Cut
We show that for every , the degree-
Sherali-Adams linear program (with variables
and constraints) approximates the maximum cut problem within a factor of
, for some . Our
result provides a surprising converse to known lower bounds against all linear
programming relaxations of Max-Cut, and hence resolves the extension complexity
of approximate Max-Cut for approximation factors close to (up to
the function ). Previously, only semidefinite
programs and spectral methods were known to yield approximation factors better
than for Max-Cut in time . We also show that
constant-degree Sherali-Adams linear programs (with variables
and constraints) can solve Max-Cut with approximation factor close to on
graphs of small threshold rank: this is the first connection of which we are
aware between threshold rank and linear programming-based algorithms.
Our results separate the power of Sherali-Adams versus Lov\'asz-Schrijver
hierarchies for approximating Max-Cut, since it is known that
approximation of Max Cut requires
rounds in the Lov\'asz-Schrijver hierarchy.
We also provide a subexponential time approximation for Khot's Unique Games
problem: we show that for every the degree- Sherali-Adams linear program distinguishes instances of Unique Games
of value from instances of value , for
some , where is the alphabet size. Such
guarantees are qualitatively similar to those of previous subexponential-time
algorithms for Unique Games but our algorithm does not rely on semidefinite
programming or subspace enumeration techniques
Directed branch-width: A directed analogue of tree-width
We introduce a new digraph width measure called directed branch-width. To do
this, we generalize a characterization of graph classes of bounded tree-width
in terms of their line graphs to digraphs.
Under parameterizations by directed branch-width we obtain linear time
algorithms for many problems, such as directed Hamilton path and Max-Cut, which
are hard when parameterized by other known directed width measures. More
generally, we obtain an algorithmic meta-theorem for the model-checking problem
for a restricted variant of MSO_2-logic on classes of bounded directed
branch-width
Sublinear Algorithm And Lower Bound For Combinatorial Problems
As the scale of the problems we want to solve in real life becomes larger, the input sizes of the problems we want to solve could be much larger than the memory of a single computer. In these cases, the classical algorithms may no longer be feasible options, even when they run in linear time and linear space, as the input size is too large.
In this thesis, we study various combinatorial problems in different computation models that process large input sizes using limited resources. In particular, we consider the query model, streaming model, and massively parallel computation model. In addition, we also study the tradeoffs between the adaptivity and performance of algorithms in these models.We first consider two graph problems, vertex coloring problem and metric traveling salesman problem (TSP). The main results are structure results for these problems, which give frameworks for achieving sublinear algorithms of these problems in different models. We also show that the sublinear algorithms for (∆ + 1)-coloring problem are tight. We then consider the graph sparsification problem, which is an important technique for designing sublinear algorithms. We give proof of the existence of a linear size hypergraph cut sparsifier, along with a polynomial algorithm that calculates one. We also consider sublinear algorithms for this problem in the streaming and query models. Finally, we study the round complexity of submodular function minimization (SFM). In particular, we give a polynomial lower bound on the number of rounds we need to compute s − t max flow - a special case of SFM - in the streaming model. We also prove a polynomial lower bound on the number of rounds we need to solve the general SFM problem in polynomial queries
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