Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Integrality Gap of the Subtour LP for the 1,2-TSP   In this paper, we study the integrality gap of the subtour LP relaxation for
the traveling salesman problem in the special case when all edge costs are
either 1 or 2. For the general case of symmetric costs that obey triangle
inequality, a famous conjecture is that the integrality gap is 4/3. Little
progress towards resolving this conjecture has been made in thirty years. We
conjecture that when all edge costs $c_{ij}\in \{1,2\}$, the integrality gap is
$10/9$. We show that this conjecture is true when the optimal subtour LP
solution has a certain structure. Under a weaker assumption, which is an analog
of a recent conjecture by Schalekamp, Williamson and van Zuylen, we show that
the integrality gap is at most $7/6$. When we do not make any assumptions on
the structure of the optimal subtour LP solution, we can show that integrality
gap is at most $5/4$; this is the first bound on the integrality gap of the
subtour LP strictly less than $4/3$ known for an interesting special case of
the TSP. We show computationally that the integrality gap is at most $10/9$ for
all instances with at most 12 cities.

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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph