3,454 research outputs found
Approximately Counting Triangles in Sublinear Time
We consider the problem of estimating the number of triangles in a graph.
This problem has been extensively studied in both theory and practice, but all
existing algorithms read the entire graph. In this work we design a {\em
sublinear-time\/} algorithm for approximating the number of triangles in a
graph, where the algorithm is given query access to the graph. The allowed
queries are degree queries, vertex-pair queries and neighbor queries.
We show that for any given approximation parameter , the
algorithm provides an estimate such that with high constant
probability, , where
is the number of triangles in the graph . The expected query complexity of
the algorithm is , where
is the number of vertices in the graph and is the number of edges, and
the expected running time is . We also prove
that queries are necessary, thus establishing that
the query complexity of this algorithm is optimal up to polylogarithmic factors
in (and the dependence on ).Comment: To appear in the 56th Annual IEEE Symposium on Foundations of
Computer Science (FOCS 2015
Streaming Verification of Graph Properties
Streaming interactive proofs (SIPs) are a framework for outsourced
computation. A computationally limited streaming client (the verifier) hands
over a large data set to an untrusted server (the prover) in the cloud and the
two parties run a protocol to confirm the correctness of result with high
probability. SIPs are particularly interesting for problems that are hard to
solve (or even approximate) well in a streaming setting. The most notable of
these problems is finding maximum matchings, which has received intense
interest in recent years but has strong lower bounds even for constant factor
approximations.
In this paper, we present efficient streaming interactive proofs that can
verify maximum matchings exactly. Our results cover all flavors of matchings
(bipartite/non-bipartite and weighted). In addition, we also present streaming
verifiers for approximate metric TSP. In particular, these are the first
efficient results for weighted matchings and for metric TSP in any streaming
verification model.Comment: 26 pages, 2 figure, 1 tabl
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More
We consider questions that arise from the intersection between the areas of
polynomial-time approximation algorithms, subexponential-time algorithms, and
fixed-parameter tractable algorithms. The questions, which have been asked
several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a
non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and
Minimum Dominating Set (DomSet) problems parameterized by the size of the
optimal solution. In particular, letting be the optimum and be
the size of the input, is there an algorithm that runs in
time and outputs a solution of size
, for any functions and that are independent of (for
Clique, we want )?
In this paper, we show that both Clique and DomSet admit no non-trivial
FPT-approximation algorithm, i.e., there is no
-FPT-approximation algorithm for Clique and no
-FPT-approximation algorithm for DomSet, for any function
(e.g., this holds even if is the Ackermann function). In fact, our results
imply something even stronger: The best way to solve Clique and DomSet, even
approximately, is to essentially enumerate all possibilities. Our results hold
under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which
states that no -time algorithm can distinguish between a satisfiable
3SAT formula and one which is not even -satisfiable for some
constant .
Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for
Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and
Maximum Induced Matching in bipartite graphs. Additionally, we rule out
-FPT-approximation algorithm for Densest -Subgraph although this
ratio does not yet match the trivial -approximation algorithm.Comment: 43 pages. To appear in FOCS'1
Approximability of Connected Factors
Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by
Tutte's reduction to the matching problem. By the same reduction, it is easy to
find a minimal or maximal d-factor of a graph. However, if we require that the
d-factor is connected, these problems become NP-hard - finding a minimal
connected 2-factor is just the traveling salesman problem (TSP).
Given a complete graph with edge weights that satisfy the triangle
inequality, we consider the problem of finding a minimal connected -factor.
We give a 3-approximation for all and improve this to an
(r+1)-approximation for even d, where r is the approximation ratio of the TSP.
This yields a 2.5-approximation for even d. The same algorithm yields an
(r+1)-approximation for the directed version of the problem, where r is the
approximation ratio of the asymmetric TSP. We also show that none of these
minimization problems can be approximated better than the corresponding TSP.
Finally, for the decision problem of deciding whether a given graph contains
a connected d-factor, we extend known hardness results.Comment: To appear in the proceedings of WAOA 201
Minimum Weight Perfect Matching via Blossom Belief Propagation
Max-product Belief Propagation (BP) is a popular message-passing algorithm
for computing a Maximum-A-Posteriori (MAP) assignment over a distribution
represented by a Graphical Model (GM). It has been shown that BP can solve a
number of combinatorial optimization problems including minimum weight
matching, shortest path, network flow and vertex cover under the following
common assumption: the respective Linear Programming (LP) relaxation is tight,
i.e., no integrality gap is present. However, when LP shows an integrality gap,
no model has been known which can be solved systematically via sequential
applications of BP. In this paper, we develop the first such algorithm, coined
Blossom-BP, for solving the minimum weight matching problem over arbitrary
graphs. Each step of the sequential algorithm requires applying BP over a
modified graph constructed by contractions and expansions of blossoms, i.e.,
odd sets of vertices. Our scheme guarantees termination in O(n^2) of BP runs,
where n is the number of vertices in the original graph. In essence, the
Blossom-BP offers a distributed version of the celebrated Edmonds' Blossom
algorithm by jumping at once over many sub-steps with a single BP. Moreover,
our result provides an interpretation of the Edmonds' algorithm as a sequence
of LPs
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