15
votes
Accepted
Proof assistant usage in complexity theory research?
A general rule of thumb is that the more abstract/exotic the mathematics you want to mechanise, the easier it gets. Conversely, the more concrete/familiar the mathematics is, the harder it will be. So ...
- 32.4k
14
votes
Accepted
Is the following graph optimization problem approximable within a constant factor?
The problem is very similar to Min Uncut. In Min Uncut, given a graph $G = (V, E)$, we need to find a subset of edges $E'$ s.t. $G - E'$ is bipartite; the objective is to minimize the size of $|E'|$. ...
- 3,899
11
votes
Accepted
Can we approximate the number of words accepted by an NFA?
There exists a FPRAS (Fully Polynomial Randomized Approximation Scheme) for the problem of counting the words of length $n$ accepted by a NFA in the general case (without restricting to the acyclic ...
- 541
11
votes
Accepted
Does Max Planar 3-SAT admit a PTAS?
Yes, a PTAS for Max-Planar-3-SAT can be constructed by using Brenda Baker's approach.
This has been observed, for instance, in Theorem 17 in
Pierluigi Crescenzi and LucaTrevisan:
"Max NP-...
- 5,772
10
votes
Accepted
Is the current best approximation ratio for Vertex Cover problem also a lower bound?
Cormen, Leiserson, Rivest and Stein say that this algorithm that achieves ratio of 2 is tight. They did not exclude the possibility of another algorithm achieving better.
Vertex Cover is NP-hard to ...
- 810
10
votes
Accepted
Additive versus multiplicative accuracy
Why is it sometimes easier to achieve $\epsilon$-additive accuracy than $\epsilon$-multiplicative?
Consider a problem where the objective value OPT is guaranteed to lie in a constant non-negative ...
- 9,595
10
votes
Accepted
Find research partner (profession and beginner)
I don't know of such a page. Most researchers have specific problems that they are interested in working on, and would only want to collaborate on those. If you pick a particular area and focus on ...
- 7,595
9
votes
Accepted
W[1]-hard problems with FPT time approximation algorithms
In the Directed Odd Cycle Transversal problem the input is a graph $G$ and the task is to find a smallest set $S$ of vertices such that $G-S$ has no (directed) cycles of odd length. In the ...
- 3,256
9
votes
Why is the Greedy Conjecture so difficult?
Let me first try to summarize what is known about the Greedy Conjecture.
Blum, Jiang, Li, Tromp, Yannakakis prove that the Greedy Algorithm gives a 4-approximation, and Kaplan and Shafrir show that ...
- 2,247
8
votes
Accepted
What is the fastest known algorithm for computing a 1.99-approximation of Vertex Cover?
I think for getting 1.99-approximation algorithm this paper by Manurangsi and Trevisan, has the current fastest algorithm.
- 397
6
votes
Accepted
The complexity of decomposing a bi-stochastic matrix
Summary. Using your favourite $O(n^d)$ algorithm for finding a matching in graphs on $O(n)$ vertices, there is a simple algorithm using $O(\max\{n^{d+2},n^4\})$ operations over the reals for ...
- 8,821
6
votes
Accepted
How can we bound the optimal solution of the dual bin packing when we solve the knapsack problem for each bin?
The "dual bin packing problem" is more commonly referred to as the Multiple Knapsack problem. One can show that $ALG \ge (1-1/e) OPT_b$ assuming an optimal algorithm for the Knapsack problem is used ...
- 6,979
6
votes
Does k-PATH admit a constant approximation?
Karger, Motwani and Ramkumar (1997) discuss this question.
They show that if any polynomial-time algorithm can approximate the longest path to a ratio of $2^{O(\log^{1−\epsilon} n)}$, for any $\...
- 5,772
6
votes
Is there a sensible notion of an approximation algorithm for an undecidable problem?
This is answering the title of the question more than its content, but you can also consider "approximations" of the halting problem as algorithms which will give you a correct answer on "almost all" ...
- 329
6
votes
Proof assistant usage in complexity theory research?
One very prominent example is of course Gonthiers Coq formalization of the 4 color theorem in Coq which uses a lot of combinatorics.
My colleague Uli Schöpp used the ssreflect library developed by ...
- 643
6
votes
W[1]-hard problems with FPT time approximation algorithms
In [1], the authors prove that MaxSAT parametrized by the clique-width (resp. neighbor diversity) of the incidence graph of the CNF formula has an FPT-AS (Fixed Parameter Tractable Approximation ...
- 2,164
6
votes
W[1]-hard problems with FPT time approximation algorithms
In Defective Coloring we are given a graph $G$ and an integer $\Delta^*$ and are asked to partition the vertices of $G$ into the minimum possible number of color classes so that each class induces a ...
- 3,416
6
votes
Accepted
Count satisfying assignments of CNF formulas over all possible negation assignments
The quantity $\sum_k|\phi_k|$ can be computed in polynomial time, in fact, in uniform $\mathrm{TC}^0$. By double counting, we have
$$\sum_k|\phi_k|=|\{(a,k):a\models\phi_k\}|=\sum_{a\in\{0,1\}^n}|\{k:...
- 16.9k
6
votes
Accepted
Is an algorithm with an approximation factor of 4000 useful?
A very good question! While I think a feasible solution which is 4000 times worse than the optimum is often not very practical, if you have several approximation algorithms to choose from, I would ...
- 5,840
6
votes
Accepted
kmeans++ for arbitrary metric spaces and general potential function
Here's an example that suggests that the stronger claim in the earlier version (Lemma 5.3) is false. I've only given a cursory look at the papers, so please check this carefully to make sure I am ...
- 9,595
5
votes
Approximating the clique size of the graph
It's not possible. The reason is that, to distinguish an $n$-vertex graph with no edges (clique size 1) from a graph with a single randomly-chosen edge (clique size 2) requires $\Theta(n^2)$ queries. ...
- 50.9k
5
votes
W[1]-hard problems with FPT time approximation algorithms
The k-cut problem is to remove a minimum number of edges to create at least k components. W[1] hard when parameterized by k but admits a 2-approximation for any k.
- 6,979
5
votes
W[1]-hard problems with FPT time approximation algorithms
(This question was asked two years ago, but I'll post the answer for other people who may see this question.)
In the Capacitated $k$-median problem we are given a set $F$ of facilities, each facility ...
- 101
5
votes
Max cut problem between two connected subgraphs
Here is a straightforward reduction from the max-cut problem:
Take any graph and add two new vertices $u,v$ and connect them to every other vertex with weight 0 and connect them to each other by a ...
- 3,440
5
votes
Accepted
Is this greedy algorithm for vertex cover studied before?
Unfortunately, the algorithm can be arbitrarily bad. In the following example, each vertex $u_i$ has $d$ disjoint neighbors, of which only four are drawn. The optimal solution is $\{u_1, \ldots, u_p, ...
- 2,560
5
votes
Accepted
Does an upper bound on the integrality gap imply an approximation algorithm with the same ratio?
See Feige and Jozeph's paper on separation between estimation and approximation.
- 6,979
5
votes
Accepted
Approximation algorithm for minimising $x_i+y_i$ for monotonically increasing sequence $x_i$ and monotonically decreasing sequence $y_i$
Theorem 1. For any $\epsilon>0$, there is a $(1+\epsilon)$-approximation algorithm that makes $O(\epsilon^{-1}\log n)$ queries.
Note that if $\epsilon$ is arbitrarily small but constant,
the ...
- 9,595
4
votes
Accepted
Maximizing a monotone supermodular function s.t. cardinality
I think this is an example showing no kind of approximation is possible except with exponential$(k)$ value queries.
Let $f(S) = 0$ if $|S| \leq k$, otherwise $f(S) = |S| - k$. Now pick a special set $...
- 7,595
4
votes
Accepted
What is the reverse of greedy algorithm for setcover?
The approximation guarantee will be significantly worse.
Assume you want to cover the set $U=\{1,\ldots,2n\}$.
For every $i=1,\ldots,n$ define a set with n+1 elements by $S_i=\{i,n+1,\ldots,2n\}$.
...
- 56
4
votes
Accepted
Is there an approximation algorithm for MAX k DOUBLE SET COVER?
(Comment $\rightarrow$ Answer)
Consider the following algorithms for a hypergraph $(U,\mathcal S)$, with $n=|U|$:
For a set $X\subseteq \mathcal S$ of sets and a set $A \in X$, define $d_X(A)$ as ...
- 1,642
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