10,272 research outputs found
Approximating Dense Max 2-CSPs
In this paper, we present a polynomial-time algorithm that approximates
sufficiently high-value Max 2-CSPs on sufficiently dense graphs to within
approximation ratio for any constant .
Using this algorithm, we also achieve similar results for free games,
projection games on sufficiently dense random graphs, and the Densest
-Subgraph problem with sufficiently dense optimal solution. Note, however,
that algorithms with similar guarantees to the last algorithm were in fact
discovered prior to our work by Feige et al. and Suzuki and Tokuyama.
In addition, our idea for the above algorithms yields the following
by-product: a quasi-polynomial time approximation scheme (QPTAS) for
satisfiable dense Max 2-CSPs with better running time than the known
algorithms
Exact Algorithms for Solving Stochastic Games
Shapley's discounted stochastic games, Everett's recursive games and
Gillette's undiscounted stochastic games are classical models of game theory
describing two-player zero-sum games of potentially infinite duration. We
describe algorithms for exactly solving these games
Compositional Algorithms for Succinct Safety Games
We study the synthesis of circuits for succinct safety specifications given
in the AIG format. We show how AIG safety specifications can be decomposed
automatically into sub specifications. Then we propose symbolic compositional
algorithms to solve the synthesis problem compositionally starting for the
sub-specifications. We have evaluated the compositional algorithms on a set of
benchmarks including those proposed for the first synthesis competition
organised in 2014 by the Synthesis Workshop affiliated to the CAV conference.
We show that a large number of benchmarks can be decomposed automatically and
solved more efficiently with the compositional algorithms that we propose in
this paper.Comment: In Proceedings SYNT 2015, arXiv:1602.0078
Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results
We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in
time , where is the treewidth of the graph. This improves on the
previous -approximation in time \poly(n) 2^{O(k)} due to
Chlamt\'a\v{c} et al.
To complement this algorithm, we show the following hardness results: If the
Non-Uniform Sparsest Cut problem has a -approximation for series-parallel
graphs (where ), then the Max Cut problem has an algorithm with
approximation factor arbitrarily close to . Hence, even for such
restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard
to approximate better than for ; assuming the
Unique Games Conjecture the hardness becomes . For
graphs with large (but constant) treewidth, we show a hardness result of assuming the Unique Games Conjecture.
Our algorithm rounds a linear program based on (a subset of) the
Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for
treewidth-2 graphs, the LP has an integrality gap close to 2 even after
polynomially many rounds of Sherali-Adams. Hence our approach cannot be
improved even on such restricted graphs without using a stronger relaxation
Towards a better approximation for sparsest cut?
We give a new -approximation for sparsest cut problem on graphs
where small sets expand significantly more than the sparsest cut (sets of size
expand by a factor bigger, for some small ; this
condition holds for many natural graph families). We give two different
algorithms. One involves Guruswami-Sinop rounding on the level- Lasserre
relaxation. The other is combinatorial and involves a new notion called {\em
Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which
we show exists in the input graph. Both algorithms run in time . We also show similar approximation algorithms in graphs with
genus with an analogous local expansion condition. This is the first
algorithm we know of that achieves -approximation on such general
family of graphs
On the NP-Hardness of Approximating Ordering Constraint Satisfaction Problems
We show improved NP-hardness of approximating Ordering Constraint
Satisfaction Problems (OCSPs). For the two most well-studied OCSPs, Maximum
Acyclic Subgraph and Maximum Betweenness, we prove inapproximability of
and .
An OCSP is said to be approximation resistant if it is hard to approximate
better than taking a uniformly random ordering. We prove that the Maximum
Non-Betweenness Problem is approximation resistant and that there are width-
approximation-resistant OCSPs accepting only a fraction of
assignments. These results provide the first examples of
approximation-resistant OCSPs subject only to P \NP
Improved Hardness of Approximating Chromatic Number
We prove that for sufficiently large K, it is NP-hard to color K-colorable
graphs with less than 2^{K^{1/3}} colors. This improves the previous result of
K versus K^{O(log K)} in Khot [14]
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