26
Luca, since a year has passed, you probably have researched your own
answer. I'm answering some of your questions here just for the
record. I review some Lagrangian-relaxation algorithms for the
problems you mention, and sketch the connection to learning (in
particular, following expert advice). I don't comment here on SDP
algorithms.
Note that the ...
22
In some technical sense you are asking whether $P = NP \cap coNP$. Suppose that $L \in NP \cap coNP$, thus there exists poly-time $F$ and $G$ so that $x \in L$ iff $\exists y: F(x,y)$ and $x \not\in L$ iff $\exists y: G(x,y)$. This can be recast as a minmax characterization by
$f_x(y) = 1$ if $F(x,y)$ and $f_x(y) = 0$ otherwise; $g_x(y) = 0$ if $G(x,y)$ ...
22
Your problem is NP-hard, even for polynomials of degree 2.
The crucial reference is
Theodore Motzkin and Ernst Strauss (1965)
"Maxima for graphs and a new proof of a theorem of Turan"
Canadian Journal of Mathematics 17, pp 533-540
Motzkin and Strauss consider an undirected graph $G=(V,E)$
with vertex set $V=\{1,2,\ldots,n\}$.
They show that ...
20
Predecessor versions of this paper have been around for more than 15 years. I remember that there were counter-examples to the first versions, then first revisions, counter-examples to the first revisions, second revisions, new counter-examples, further revisions, further counter-examples, and so on.
It would be much better, if the authors were able to ...
19
I think the difficulty is that this wording slightly misleading; as they state more clearly in the Introduction (1.2), "the expected values of the dual variables constitute a feasible dual solution."
For every fixed setting of the dual variables $X$, we obtain some primal solution of value $f(X)$ and a dual solution of value $\frac{e}{e-1}f(X)$. (The dual ...
19
Per OP's wish, here's the math.SE answer I link to in my comment above.
Maybe it's worthwhile to talk through where the dual comes from on an example problem. This will take a while, but hopefully the dual won't seem so mysterious when we're done.
Suppose with have a primal problem as follows.
$$
Primal =\begin{Bmatrix}
\max \ \ \ \ 5x_1 ...
19
The current fastest algorithm is actually linear in the length of the integer linear program for every fixed value of $n$. In his PhD thesis, Lokshtanov (2009) nicely summarizes the results by Lenstra (1983), Kannan (1987), and Frank & Tardos (1987) by the following theorem.
Integer Linear Programming can be solved using $O(n^{2.5n+o(n)} \cdot L)$
...
18
A typical problem is MaxCut: output a cut in a graph that (approximately) maximizes the number of edges cut. Goemans and Williamson showed an SDP approximates the value of MaxCut to within a factor at least 0.878. Recently, Chan, Lee, Raghavendra, and Steurer showed that for a natural linear encoding of the MaxCut problem, all polynomial size LPs achieve ...
16
A major open problem in mathematical programming is designing a strongly polynomial time linear programming algorithm. A related problem is whether any variant of the simplex algorithm runs in strongly polynomial time. It makes sense to first prove strong polynomial time bounds for variants of simplex applied to problems for which we already know strong ...
16
The ellipsoid method and interior point methods can be extended to solve SDPs as well. You can refer to any standard-texts on SDPs for details. Here's one:
Semidefinite Programming. Vandenberge and Stephen Boyd, 1996.
14
Seymour and Thomas showed a min-max characterization of treewidth. Yet, tree width is NP-hard. This however is not quite the kind of characterization you are asking for, because the dual function $g$ is not a polynomial time computable function of a short certificate. This is most likely unavoidable for NP complete problems, because otherwise we would have ...
14
As you have noted, complementary slackness follows immediately from strong duality, i.e., equality of the primal and dual objective functions at an optimum.
Complementarity slackness can be thought of as a combinatorial optimality condition, where a zero duality gap (equality of the primal and dual objective functions) can be thought of as a numerical ...
13
I strongly recommend the paper by Bixby, the "father" of CPLEX, that surveys not only on implementing aspects of the (revised) simplex algorithm: Robert E. Bixby, Solving Real-World Linear Programs: A Decade and More of Progress, Operations Research (50) 2002, 3-15.
13
Parity games, Mean-payoff games, Discounted games, and Simple Stochastic games fall within this category.
All of them are infinite two-player zero-sum games played on graphs, where players control vertices and choose where a token should go next. All have equilibria in memoryless positional strategies, meaning that each player chooses an edge at each choice ...
13
Complementary slackness is key in designing primal-dual algorithms. The basic idea is:
Start with a feasible dual solution $y$.
Attempt to find primal feasible $x$ such that $(x, y)$ satisfy complementary slackness.
If step 2. succeeded we are done. Otherwise an obstruction to finding $x$ gives a way to modify $y$ so that the dual objective function value ...
12
The premise of the question is a little flawed: there are many who would argue that quadratics are the real "boundary" for tractability and modelling, since least-squares problems are almost as 'easy' as linear problems. There are others who'd argue that convexity (or even submodularity in certain cases) is the boundary for tractability.
Perhaps what is ...
12
For many combinatorial optimization problems (for instance Max-Cut), semidefinite programming yields much stronger relaxations than the LP relaxation of IP formulations. This allows the design of approximation algorithms, and of exact algorithms which are more efficient than their linear counterparts due to the better quality of the bounds. Examples can be ...
12
I cannot say for sure if you will consider the following approach as better, but from a complexity-theoretic point of view there is a more efficient solution. The idea is to rephrase your question in the first-order theory of the rationals with addition and order. You have that $P_1$ is included in $P_2$ if and only if
\begin{align*}
\Phi :=
\forall \vec{x}....
12
Consider the problem $\text{MAX-LIN}(R)$ of maximizing the number of satisfied linear equations over some ring $R$, which is often NP-hard, for example in the case $R=\mathbb{Z}$
Take an instance of this problem, $Ax=b$ where $A$ is a $n\times m$ matrix. Let $k=m+1$. Construct a new linear system $\tilde{A}\tilde{x} = \tilde{b}$, where $\tilde{A}$ is a $kn ...
11
Yup. Everything follows from duality. (I am only half joking).
A partial list:
Boosting
The Hard-Core Lemma
Online Learning
The ability to actually solve LPs efficiently
A large fraction of approximation algorithms results.
Much more
To develop algorithms, you often need a constructive or algorithmic version of the duality theorem (which is ...
10
In the geometric setting, where $C(x,y) = \|x - y \|$, this formulation is called the bottleneck matching problem. It's possible that this is the generic term for it (I've seen this formulation used in the Kleinberg-Tardos algorithms book for MSTs).
10
For 0/1-polytopes (all vertex coordinates are 0 or 1), this is not known to be true. There is a conjecture by Mihail and Vazirani that the edge expansion of the graph of a 0/1-polytope is at least one. More information is described in a paper by Volker Kaibel.
I should note two things. (1) For 0/1-polytopes, the Hirsch conjecture is true. (2) When ...
10
ILP is NP-complete. This has a few implications to your question.
Moreover, I believe that there are inherently nonlinear problems that cannot be expressed in the form of LP nor ILP.
As long as you consider problems in NP, no, all problems in NP can be rewritten as ILP in polynomial time. If you consider problems not in NP, then they cannot be ...
10
Fixed Point Logic $+$ Counting (FPC) is believed to capture most of the $P$ solvable problems. Anderson, Dawar and Holm 2015 [1]showed that optimization of linear programs is expressible in FPC. Dawar and Wang 2016 [2]showed that The FPC implementation of the ellipsoid method extends to semidefinite programs (subject to some technical conditions).
9
Are you aware of the work on positive LPs/SDPs? There are a bunch of results in the area, mostly along the lines of "if the constraints of the LP/SDP are positive, then the problem can be solved in NC."
Some important references in this line of work are Luby-Nisan 93 and Jain-Yao 11. Another excellent resource is this slide of Rahul Jain's talk at the "...
9
How can this system be solved without using linear programming?
It cannot.
Your problem is the linear feasibility problem in the standard form, with an extra condition that the constraint matrix is sparse. Now note:
(1) It is well-known that linear programming is equivalent to the linear feasibility problem in the standard form.
(2) You can easily ...
9
For your first question, without the total order, the answer to your question is that it's essentially as hard as linear programming. Here's an outline of a proof.
First, let's establish a variable $x_1>0$, which we call $\epsilon$. Now, let's choose another variable $x_i$, which we will call $1$. We want to make sure that $$\epsilon \ll 1\, .$$
To do ...
9
Two years ago, when researching the (unweighted) blossom algorithm, I found two great sets of notes, one by Tarjan and one by Zwick. They made the unweighted case seem quite straightforward and I was able to implement it in Mathematica using recursion. It works quite well.
The notes that I found useful are
http://www.cs.tau.ac.il/~zwick/grad-algo-06/match....
9
In general it is not true: Consider two dual- to cyclic d-polytopes with n facets each and merge them along a vertex. (This is the dual operation of gluing two polutops). The number of vertices will be like $n^{[d/2]}$ and the spectrual gap will be roughly 1 over this. (You can use d edges to separete the graph into two parts.
I proved 1/poly(n) seperation ...
8
The new version of the problem is indeed NP-hard.
Reduction from set cover: Subsets are $S_1, S_2, ..., S_m$ and the universe is $U = \{ a_1, a_2, ..., a_n \}$. Construct a node-weighted directed graph $(N,S,c)$ as follows:
Nodes are $N = \{ a_1, a_2, ..., a_n, S_1, S_2, ..., S_m, S'_1, S'_2, ..., S'_m \}$.
Edges are: $(a_j,S_i)$ iff $a_j \in ...
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