416 research outputs found
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
FPTAS for Counting Monotone CNF
A monotone CNF formula is a Boolean formula in conjunctive normal form where
each variable appears positively. We design a deterministic fully
polynomial-time approximation scheme (FPTAS) for counting the number of
satisfying assignments for a given monotone CNF formula when each variable
appears in at most clauses. Equivalently, this is also an FPTAS for
counting set covers where each set contains at most elements. If we allow
variables to appear in a maximum of clauses (or sets to contain
elements), it is NP-hard to approximate it. Thus, this gives a complete
understanding of the approximability of counting for monotone CNF formulas. It
is also an important step towards a complete characterization of the
approximability for all bounded degree Boolean #CSP problems. In addition, we
study the hypergraph matching problem, which arises naturally towards a
complete classification of bounded degree Boolean #CSP problems, and show an
FPTAS for counting 3D matchings of hypergraphs with maximum degree .
Our main technique is correlation decay, a powerful tool to design
deterministic FPTAS for counting problems defined by local constraints among a
number of variables. All previous uses of this design technique fall into two
categories: each constraint involves at most two variables, such as independent
set, coloring, and spin systems in general; or each variable appears in at most
two constraints, such as matching, edge cover, and holant problem in general.
The CNF problems studied here have more complicated structures than these
problems and require new design and proof techniques. As it turns out, the
technique we developed for the CNF problem also works for the hypergraph
matching problem. We believe that it may also find applications in other CSP or
more general counting problems.Comment: 24 pages, 2 figures. version 1=>2: minor edits, highlighted the
picture of set cover/packing, and an implication of our previous result in 3D
matchin
New Inapproximability Bounds for TSP
In this paper, we study the approximability of the metric Traveling Salesman
Problem (TSP) and prove new explicit inapproximability bounds for that problem.
The best up to now known hardness of approximation bounds were 185/184 for the
symmetric case (due to Lampis) and 117/116 for the asymmetric case (due to
Papadimitriou and Vempala). We construct here two new bounded occurrence CSP
reductions which improve these bounds to 123/122 and 75/74, respectively. The
latter bound is the first improvement in more than a decade for the case of the
asymmetric TSP. One of our main tools, which may be of independent interest, is
a new construction of a bounded degree wheel amplifier used in the proof of our
results
Non-approximability and Polylogarithmic Approximations of the Single-Sink Unsplittable and Confluent Dynamic Flow Problems
Dynamic Flows were introduced by Ford and Fulkerson in 1958 to model flows over time. They define edge capacities to be the total amount of flow that can enter an edge in one time unit. Each edge also has a length, representing the time needed to traverse it. Dynamic Flows have been used to model many problems including traffic congestion, hop-routing of packets and evacuation protocols in buildings. While the basic problem of moving the maximal amount of supplies from sources to sinks is polynomial time solvable, natural minor modifications can make it NP-hard.
One such modification is that flows be confluent, i.e., all flows leaving a vertex must leave along the same edge. This corresponds to natural conditions in, e.g., evacuation planning and hop routing.
We investigate the single-sink Confluent Quickest Flow problem. The input is a graph with edge capacities and lengths, sources with supplies and a sink. The problem is to find a confluent flow minimizing the time required to send supplies to the sink. Our main results include:
a) Logarithmic Non-Approximability: Directed Confluent Quickest Flows cannot be approximated in polynomial time with an O(log n) approximation factor, unless P=NP.
b) Polylogarithmic Bicriteria Approximations: Polynomial time (O(log^8 n), O(log^2 kappa)) bicritera approximation algorithms for the Confluent Quickest Flow problem where kappa is the number of sinks, in both directed and undirected graphs.
Corresponding results are also developed for the Confluent Maximum Flow over time problem. The techniques developed also improve recent approximation algorithms for static confluent flows
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
On the characterization of models of H*: The semantical aspect
We give a characterization, with respect to a large class of models of
untyped lambda-calculus, of those models that are fully abstract for
head-normalization, i.e., whose equational theory is H* (observations for head
normalization). An extensional K-model is fully abstract if and only if it
is hyperimmune, {\em i.e.}, not well founded chains of elements of D cannot be
captured by any recursive function.
This article, together with its companion paper, form the long version of
[Bre14]. It is a standalone paper that presents a purely semantical proof of
the result as opposed to its companion paper that presents an independent and
purely syntactical proof of the same result
Some results on more flexible versions of Graph Motif
The problems studied in this paper originate from Graph Motif, a problem
introduced in 2006 in the context of biological networks. Informally speaking,
it consists in deciding if a multiset of colors occurs in a connected subgraph
of a vertex-colored graph. Due to the high rate of noise in the biological
data, more flexible definitions of the problem have been outlined. We present
in this paper two inapproximability results for two different optimization
variants of Graph Motif: one where the size of the solution is maximized, the
other when the number of substitutions of colors to obtain the motif from the
solution is minimized. We also study a decision version of Graph Motif where
the connectivity constraint is replaced by the well known notion of graph
modularity. While the problem remains NP-complete, it allows algorithms in FPT
for biologically relevant parameterizations
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