73 research outputs found
On Coloring Resilient Graphs
We introduce a new notion of resilience for constraint satisfaction problems,
with the goal of more precisely determining the boundary between NP-hardness
and the existence of efficient algorithms for resilient instances. In
particular, we study -resiliently -colorable graphs, which are those
-colorable graphs that remain -colorable even after the addition of any
new edges. We prove lower bounds on the NP-hardness of coloring resiliently
colorable graphs, and provide an algorithm that colors sufficiently resilient
graphs. We also analyze the corresponding notion of resilience for -SAT.
This notion of resilience suggests an array of open questions for graph
coloring and other combinatorial problems.Comment: Appearing in MFCS 201
Hardness of Finding Independent Sets in 2-Colorable Hypergraphs and of Satisfiable CSPs
This work revisits the PCP Verifiers used in the works of Hastad [Has01],
Guruswami et al.[GHS02], Holmerin[Hol02] and Guruswami[Gur00] for satisfiable
Max-E3-SAT and Max-Ek-Set-Splitting, and independent set in 2-colorable
4-uniform hypergraphs. We provide simpler and more efficient PCP Verifiers to
prove the following improved hardness results: Assuming that NP\not\subseteq
DTIME(N^{O(loglog N)}),
There is no polynomial time algorithm that, given an n-vertex 2-colorable
4-uniform hypergraph, finds an independent set of n/(log n)^c vertices, for
some constant c > 0.
There is no polynomial time algorithm that satisfies 7/8 + 1/(log n)^c
fraction of the clauses of a satisfiable Max-E3-SAT instance of size n, for
some constant c > 0.
For any fixed k >= 4, there is no polynomial time algorithm that finds a
partition splitting (1 - 2^{-k+1}) + 1/(log n)^c fraction of the k-sets of a
satisfiable Max-Ek-Set-Splitting instance of size n, for some constant c > 0.
Our hardness factor for independent set in 2-colorable 4-uniform hypergraphs
is an exponential improvement over the previous results of Guruswami et
al.[GHS02] and Holmerin[Hol02]. Similarly, our inapproximability of (log
n)^{-c} beyond the random assignment threshold for Max-E3-SAT and
Max-Ek-Set-Splitting is an exponential improvement over the previous bounds
proved in [Has01], [Hol02] and [Gur00]. The PCP Verifiers used in our results
avoid the use of a variable bias parameter used in previous works, which leads
to the improved hardness thresholds in addition to simplifying the analysis
substantially. Apart from standard techniques from Fourier Analysis, for the
first mentioned result we use a mixing estimate of Markov Chains based on
uniform reverse hypercontractivity over general product spaces from the work of
Mossel et al.[MOS13].Comment: 23 Page
Derandomized Graph Product Results using the Low Degree Long Code
In this paper, we address the question of whether the recent derandomization
results obtained by the use of the low-degree long code can be extended to
other product settings. We consider two settings: (1) the graph product results
of Alon, Dinur, Friedgut and Sudakov [GAFA, 2004] and (2) the "majority is
stablest" type of result obtained by Dinur, Mossel and Regev [SICOMP, 2009] and
Dinur and Shinkar [In Proc. APPROX, 2010] while studying the hardness of
approximate graph coloring.
In our first result, we show that there exists a considerably smaller
subgraph of which exhibits the following property (shown for
by Alon et al.): independent sets close in size to the
maximum independent set are well approximated by dictators.
The "majority is stablest" type of result of Dinur et al. and Dinur and
Shinkar shows that if there exist two sets of vertices and in
with very few edges with one endpoint in and another in
, then it must be the case that the two sets and share a single
influential coordinate. In our second result, we show that a similar "majority
is stablest" statement holds good for a considerably smaller subgraph of
. Furthermore using this result, we give a more efficient
reduction from Unique Games to the graph coloring problem, leading to improved
hardness of approximation results for coloring
Approximating the Orthogonality Dimension of Graphs and Hypergraphs
A t-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in R^t to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of a hypergraph H, denoted by overline{xi}(H), is the smallest integer t for which there exists a t-dimensional orthogonal representation of H. In this paper we study computational aspects of the orthogonality dimension of graphs and hypergraphs. We prove that for every k >= 4, it is NP-hard (resp. quasi-NP-hard) to distinguish n-vertex k-uniform hypergraphs H with overline{xi}(H) = Omega(log^delta n) for some constant delta>0 (resp. overline{xi}(H) >= Omega(log^{1-o(1)} n)). For graphs, we relate the NP-hardness of approximating the orthogonality dimension to a variant of a long-standing conjecture of Stahl. We also consider the algorithmic problem in which given a graph G with overline{xi}(G) <= 3 the goal is to find an orthogonal representation of G of as low dimension as possible, and provide a polynomial time approximation algorithm based on semidefinite programming
Linear Index Coding via Semidefinite Programming
In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the
goal is to broadcast an n bit word to n receivers (one bit per receiver), where
the receivers have side information represented by a graph G. The objective is
to minimize the length of a codeword sent to all receivers which allows each
receiver to learn its bit. For linear index coding, the minimum possible length
is known to be equal to a graph parameter called minrank (Bar-Yossef et al.,
FOCS, 2006).
We show a polynomial time algorithm that, given an n vertex graph G with
minrank k, finds a linear index code for G of length ,
where f(k) depends only on k. For example, for k=3 we obtain f(3) ~ 0.2574. Our
algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani
and Sudan (J. ACM, 1998) for graph coloring and its refined analysis due to
Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a
relaxation of the minimization problem we consider, a crucial component of our
analysis is an upper bound on the objective value of the SDP in terms of the
minrank.
At the heart of our analysis lies a combinatorial result which may be of
independent interest. Namely, we show an exact expression for the maximum
possible value of the Lovasz theta-function of a graph with minrank k. This
yields a tight gap between two classical upper bounds on the Shannon capacity
of a graph.Comment: 24 page
Robust Factorizations and Colorings of Tensor Graphs
Since the seminal result of Karger, Motwani, and Sudan, algorithms for
approximate 3-coloring have primarily centered around SDP-based rounding.
However, it is likely that important combinatorial or algebraic insights are
needed in order to break the threshold. One way to develop new
understanding in graph coloring is to study special subclasses of graphs. For
instance, Blum studied the 3-coloring of random graphs, and Arora and Ge
studied the 3-coloring of graphs with low threshold-rank.
In this work, we study graphs which arise from a tensor product, which appear
to be novel instances of the 3-coloring problem. We consider graphs of the form
with and ,
where is any edge set such that no vertex has
more than an fraction of its edges in . We show that one can
construct with that is close to . For arbitrary , satisfies . Additionally when is a
mild expander, we provide a 3-coloring for in polynomial time. These
results partially generalize an exact tensor factorization algorithm of Imrich.
On the other hand, without any assumptions on , we show that it is NP-hard
to 3-color .Comment: 34 pages, 3 figure
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