Inapproximability of Unique Games in Fixed-Point Logic with Counting
We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational...
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| Published in: | Logical methods in computer science Vol. 20, Issue 2 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science e.V
10.04.2024
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| Subjects: | |
| ISSN: | 1860-5974, 1860-5974 |
| Online Access: | Get full text |
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| Abstract | We study the extent to which it is possible to approximate the optimal value
of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally,
we prove lower bounds against the accuracy of FPC-interpretations that map
Unique Games instances (encoded as relational structures) to rational numbers
giving the approximate fraction of constraints that can be satisfied. We prove
two new FPC-inexpressibility results for Unique Games: the existence of a
$(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within
any constant factor. Previous recent work has established similar
FPC-inapproximability results for a small handful of other problems. Our
construction builds upon some of these ideas, but contains a novel technique.
While most FPC-inexpressibility results are based on variants of the
CFI-construction, ours is significantly different. We start with a graph of
very large girth and label the edges with random affine vector spaces over
$\mathbb{F}_2$ that determine the constraints in the two structures.
Duplicator's strategy involves maintaining a partial isomorphism over a minimal
tree that spans the pebbled vertices of the graph. |
|---|---|
| AbstractList | We study the extent to which it is possible to approximate the optimal value
of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally,
we prove lower bounds against the accuracy of FPC-interpretations that map
Unique Games instances (encoded as relational structures) to rational numbers
giving the approximate fraction of constraints that can be satisfied. We prove
two new FPC-inexpressibility results for Unique Games: the existence of a
$(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within
any constant factor. Previous recent work has established similar
FPC-inapproximability results for a small handful of other problems. Our
construction builds upon some of these ideas, but contains a novel technique.
While most FPC-inexpressibility results are based on variants of the
CFI-construction, ours is significantly different. We start with a graph of
very large girth and label the edges with random affine vector spaces over
$\mathbb{F}_2$ that determine the constraints in the two structures.
Duplicator's strategy involves maintaining a partial isomorphism over a minimal
tree that spans the pebbled vertices of the graph. We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a $(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\mathbb{F}_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph. |
| Author | Tucker-Foltz, Jamie |
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| Snippet | We study the extent to which it is possible to approximate the optimal value
of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally,
we... We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we... |
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| Title | Inapproximability of Unique Games in Fixed-Point Logic with Counting |
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