815 research outputs found
A Stronger LP Bound for Formula Size Lower Bounds via Clique Constraints
We introduce a new technique proving formula size lower bounds based on the linear programming bound originally introduced by Karchmer, Kushilevitz and Nisan (1995) and the theory of stable set polytope. We apply it to majority functions and prove their formula size lower bounds improved from the classical result of Khrapchenko (1971). Moreover, we introduce a notion of unbalanced recursive ternary majority functions motivated by a decomposition theory of monotone self-dual functions and give integrally matching upper and lower bounds of their formula size. We also show monotone formula size lower bounds of balanced recursive ternary majority functions improved from the quantum adversary bound of Laplante, Lee and Szegedy (2006)
A Computational Comparison of Optimization Methods for the Golomb Ruler Problem
The Golomb ruler problem is defined as follows: Given a positive integer n,
locate n marks on a ruler such that the distance between any two distinct pair
of marks are different from each other and the total length of the ruler is
minimized. The Golomb ruler problem has applications in information theory,
astronomy and communications, and it can be seen as a challenge for
combinatorial optimization algorithms. Although constructing high quality
rulers is well-studied, proving optimality is a far more challenging task. In
this paper, we provide a computational comparison of different optimization
paradigms, each using a different model (linear integer, constraint programming
and quadratic integer) to certify that a given Golomb ruler is optimal. We
propose several enhancements to improve the computational performance of each
method by exploring bound tightening, valid inequalities, cutting planes and
branching strategies. We conclude that a certain quadratic integer programming
model solved through a Benders decomposition and strengthened by two types of
valid inequalities performs the best in terms of solution time for small-sized
Golomb ruler problem instances. On the other hand, a constraint programming
model improved by range reduction and a particular branching strategy could
have more potential to solve larger size instances due to its promising
parallelization features
Tight Size-Degree Bounds for Sums-of-Squares Proofs
We exhibit families of -CNF formulas over variables that have
sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank)
but require SOS proofs of size for values of from
constant all the way up to for some universal constant.
This shows that the running time obtained by using the Lasserre
semidefinite programming relaxations to find degree- SOS proofs is optimal
up to constant factors in the exponent. We establish this result by combining
-reductions expressible as low-degree SOS derivations with the
idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and
Riis'03], and then applying a restriction argument as in [Atserias, M\"uller,
and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a
generic method of amplifying SOS degree lower bounds to size lower bounds, and
also generalizes the approach in [ALN14] to obtain size lower bounds for the
proof systems resolution, polynomial calculus, and Sherali-Adams from lower
bounds on width, degree, and rank, respectively
Approximation Limits of Linear Programs (Beyond Hierarchies)
We develop a framework for approximation limits of polynomial-size linear
programs from lower bounds on the nonnegative ranks of suitably defined
matrices. This framework yields unconditional impossibility results that are
applicable to any linear program as opposed to only programs generated by
hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations
for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound
applies to linear programs using a certain encoding of CLIQUE as a linear
optimization problem.) Moreover, we establish a similar result for
approximations of semidefinite programs by linear programs. Our main ingredient
is a quantitative improvement of Razborov's rectangle corruption lemma for the
high error regime, which gives strong lower bounds on the nonnegative rank of
certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure
Decorous lower bounds for minimum linear arrangement
Minimum Linear Arrangement is a classical basic combinatorial optimization problem from the 1960s, which turns out to be extremely challenging in practice. In particular, for most of its benchmark instances, even the order of magnitude of the optimal solution value is unknown, as testified by the surveys on the problem that contain tables in which the best known solution value often has one more digit than the best known lower bound value. In this paper, we propose a linear-programming based approach to compute lower bounds on the optimum. This allows us, for the first time, to show that the best known solutions are indeed not far from optimal for most of the benchmark instances
From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More
We consider questions that arise from the intersection between the areas of
polynomial-time approximation algorithms, subexponential-time algorithms, and
fixed-parameter tractable algorithms. The questions, which have been asked
several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a
non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and
Minimum Dominating Set (DomSet) problems parameterized by the size of the
optimal solution. In particular, letting be the optimum and be
the size of the input, is there an algorithm that runs in
time and outputs a solution of size
, for any functions and that are independent of (for
Clique, we want )?
In this paper, we show that both Clique and DomSet admit no non-trivial
FPT-approximation algorithm, i.e., there is no
-FPT-approximation algorithm for Clique and no
-FPT-approximation algorithm for DomSet, for any function
(e.g., this holds even if is the Ackermann function). In fact, our results
imply something even stronger: The best way to solve Clique and DomSet, even
approximately, is to essentially enumerate all possibilities. Our results hold
under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which
states that no -time algorithm can distinguish between a satisfiable
3SAT formula and one which is not even -satisfiable for some
constant .
Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for
Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and
Maximum Induced Matching in bipartite graphs. Additionally, we rule out
-FPT-approximation algorithm for Densest -Subgraph although this
ratio does not yet match the trivial -approximation algorithm.Comment: 43 pages. To appear in FOCS'1
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