# Tag Info

Accepted

This is a special case of the Travelling Salesman with Neighborhoods (TSPN) problem. In the general version, the neighborhoods need not all be the same. A paper by Dumitrescu and Mitchell, ...
• 4,523
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 ...
• 32.1k
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
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 ...
• 521
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-...
• 5,742

One relevant TSP version is "Group TSP". In this problem, the "cities" are divided into groups and the goal is to find a tour that visits each group at least once. This has also been studied on the ...
• 3,246

### Are there any interesting open questions having to do with submodularity, specially in the intersection of theoretical machine learning?

There are many open algorithmic problems. All problems below (other than the last bullet) are NP-hard, so we are interested in the best approximation ratio we can achieve in polynomial time. The ...
• 14.2k
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 ...
• 790
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 ...
• 7,185
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 ...
• 3,236

### 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 ...
• 2,163
Accepted

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 ...
• 9,380
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.
• 387

### The Goemans-Williamson algorithm in the $SOS$ framework

Looking at the Goemans–Williamson algorithm in the SOS framework yields no technical advantages: it is exactly the same algorithm and the same ideas are used in the analysis. The only advantages in ...
• 3,741
Accepted

### What is the intuition behind "hardness of approximation"?

One important reason why problems that look equally hard to compute exactly might be very different to approximate relates to the fragility of NP-completeness reductions. The simplest example I can ...
• 31.9k
Accepted

### Set cover in which some pairs of sets are forbidden

This problem is way harder than set cover. Here is why... Intuitively, you can encode independent set as a problem of this type. Indeed, you are given an instance of independent set - a graph $G$ ...
• 9,586

### 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

### What are some of the most ingenious linear programs developed for tackling hard combinatorial problems?

The relaxation of Calinescu, Karloff and Rabani for the undirected Multiway Cut problem is one my favorites. Had a big influence on subsequent work. http://www.sciencedirect.com/science/article/pii/...

### Coreset and VC dimension

Samples provides you only with statistical guarantees. For a fixed $\epsilon$ whatever you compute would hold for everything "except" $(1-\epsilon)$ fraction (I am being very informal here). Thus, ...
• 9,586
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 ...
• 8,871