311 research outputs found
A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover
Given a -uniform hyper-graph, the E-Vertex-Cover problem is to find the
smallest subset of vertices that intersects every hyper-edge. We present a new
multilayered PCP construction that extends the Raz verifier. This enables us to
prove that E-Vertex-Cover is NP-hard to approximate within factor
for any and any . The result is
essentially tight as this problem can be easily approximated within factor .
Our construction makes use of the biased Long-Code and is analyzed using
combinatorial properties of -wise -intersecting families of subsets
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
The next few years will be exciting as prototype universal quantum processors
emerge, enabling implementation of a wider variety of algorithms. Of particular
interest are quantum heuristics, which require experimentation on quantum
hardware for their evaluation, and which have the potential to significantly
expand the breadth of quantum computing applications. A leading candidate is
Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates
between applying a cost-function-based Hamiltonian and a mixing Hamiltonian.
Here, we extend this framework to allow alternation between more general
families of operators. The essence of this extension, the Quantum Alternating
Operator Ansatz, is the consideration of general parametrized families of
unitaries rather than only those corresponding to the time-evolution under a
fixed local Hamiltonian for a time specified by the parameter. This ansatz
supports the representation of a larger, and potentially more useful, set of
states than the original formulation, with potential long-term impact on a
broad array of application areas. For cases that call for mixing only within a
desired subspace, refocusing on unitaries rather than Hamiltonians enables more
efficiently implementable mixers than was possible in the original framework.
Such mixers are particularly useful for optimization problems with hard
constraints that must always be satisfied, defining a feasible subspace, and
soft constraints whose violation we wish to minimize. More efficient
implementation enables earlier experimental exploration of an alternating
operator approach to a wide variety of approximate optimization, exact
optimization, and sampling problems. Here, we introduce the Quantum Alternating
Operator Ansatz, lay out design criteria for mixing operators, detail mappings
for eight problems, and provide brief descriptions of mappings for diverse
problems.Comment: 51 pages, 2 figures. Revised to match journal pape
Parameterized Directed -Chinese Postman Problem and Arc-Disjoint Cycles Problem on Euler Digraphs
In the Directed -Chinese Postman Problem (-DCPP), we are given a
connected weighted digraph and asked to find non-empty closed directed
walks covering all arcs of such that the total weight of the walks is
minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128)
asked for the parameterized complexity of -DCPP when is the parameter.
We prove that the -DCPP is fixed-parameter tractable.
We also consider a related problem of finding arc-disjoint directed
cycles in an Euler digraph, parameterized by . Slivkins (ESA 2003) showed
that this problem is W[1]-hard for general digraphs. Generalizing another
result by Slivkins, we prove that the problem is fixed-parameter tractable for
Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler
digraphs remains W[1]-hard even for Euler digraphs
Entity-Linking via Graph-Distance Minimization
Entity-linking is a natural-language-processing task that consists in
identifying the entities mentioned in a piece of text, linking each to an
appropriate item in some knowledge base; when the knowledge base is Wikipedia,
the problem comes to be known as wikification (in this case, items are
wikipedia articles). One instance of entity-linking can be formalized as an
optimization problem on the underlying concept graph, where the quantity to be
optimized is the average distance between chosen items. Inspired by this
application, we define a new graph problem which is a natural variant of the
Maximum Capacity Representative Set. We prove that our problem is NP-hard for
general graphs; nonetheless, under some restrictive assumptions, it turns out
to be solvable in linear time. For the general case, we propose two heuristics:
one tries to enforce the above assumptions and another one is based on the
notion of hitting distance; we show experimentally how these approaches perform
with respect to some baselines on a real-world dataset.Comment: In Proceedings GRAPHITE 2014, arXiv:1407.7671. The second and third
authors were supported by the EU-FET grant NADINE (GA 288956
ETH-Hardness of Approximating 2-CSPs and Directed Steiner Network
We study the 2-ary constraint satisfaction problems (2-CSPs), which can be
stated as follows: given a constraint graph , an alphabet set
and, for each , a constraint , the goal is to find an assignment
that satisfies as many constraints as possible, where a constraint is
satisfied if .
While the approximability of 2-CSPs is quite well understood when
is constant, many problems are still open when becomes super
constant. One such problem is whether it is hard to approximate 2-CSPs to
within a polynomial factor of . Bellare et al. (1993) suggested
that the answer to this question might be positive. Alas, despite efforts to
resolve this conjecture, it remains open to this day.
In this work, we separate and and ask a related but weaker
question: is it hard to approximate 2-CSPs to within a polynomial factor of
(while may be super-polynomial in )? Assuming the
exponential time hypothesis (ETH), we answer this question positively by
showing that no polynomial time algorithm can approximate 2-CSPs to within a
factor of . Note that our ratio is almost linear, which is
almost optimal as a trivial algorithm gives a -approximation for 2-CSPs.
Thanks to a known reduction, our result implies an ETH-hardness of
approximating Directed Steiner Network with ratio where is
the number of demand pairs. The ratio is roughly the square root of the best
known ratio achieved by polynomial time algorithms (Chekuri et al., 2011;
Feldman et al., 2012).
Additionally, under Gap-ETH, our reduction for 2-CSPs not only rules out
polynomial time algorithms, but also FPT algorithms parameterized by .
Similar statement applies for DSN parameterized by .Comment: 36 pages. A preliminary version appeared in ITCS'1
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
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