20
votes
Accepted
Is there a counterexample to this work?
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 ...
- 5,772
11
votes
Accepted
Anti bin packing
This problem is called bin covering. It's NP-hard and hard to approximate to better than a factor of two by an easy reduction from subset sum but has an asymptotic approximation scheme (i.e. one that ...
- 50.9k
8
votes
Accepted
What are some example problems for integer programming that are *not binary*
Here are a few examples:
(Most Common) Cutting Stock Problem - determine patterns for which to cut boards in order to meet demand.
(Kinda same problem) - Knapsack with general integer variables - ...
7
votes
Accepted
Is finding an optimal solution to this Knapsack-like problem NP-hard?
NOTE: My original reduction didn't work. Fixed now.
Can't subset-sum be reduced to this problem fairly easily? Suppose all the profits $v_i$ are the same, say $v_i = 1$, and all the profits $p_i$ are ...
- 24.3k
7
votes
Accepted
Consequences of faster parameterized integer programming
An instance of CNF-SAT with $k$ variables can easily be written as a 0/1 integer linear program over the same variable set, since a clause such as $x_1 \vee x_3 \vee \neg x_4 \vee \neg x_6$ naturally ...
- 5,255
6
votes
Accepted
Fixed parameter tractable Integer Programming and $FPP$
You're confusing decision problems (in the classical sense) with parameterized decision problems. Classical decision problems are subsets of $\Sigma^*$, whereas parameterized decision problems are ...
- 396
6
votes
Accepted
Characterization of integral polyhedra
Here's a proof (sketch) that doesn't explicitly use duality. More precisely, it replaces duality by a seemingly weaker (and hopefully easily believable) geometric fact, in Step 3 below.
EDIT: But, ...
- 9,595
5
votes
Is that edge orientation optimization problem NP-hard?
Summary
OP's problem has a polynomial-time algorithm via reduction to min-cost bipartite matching. (Lemma 1, below.)
Alternatively, one can strengthen OP's relaxation QP directly, by modeling the ...
- 9,595
5
votes
Have fixed parameter integer program algorithms ever been implemented for research use?
The original algorithm of Lenstra (from 1983) has not been implemented AFAIK. Certainly, no open-source code is known to be available.
Lovasz and Scarf proposed (in 1992) a generalized basis ...
- 316
5
votes
Accepted
NP completeness of linear $0-1$ assignment problem
If I understood it well, (1) is also NP-complete, a possible reduction is from SUBSET SUM:
Given a set of $m$ positive integers $A = \{a_1, ..., a_m\}$, and a positive integer $B$, is there a subset ...
- 22.6k
5
votes
On integer programming
It's NP-hard. Given an integer programming problem $P$, add an irrelevant variable $z$ with no constraints; call the resulting problem $P'$. Now if $P$ has no solutions, then $P'$ has no solutions; ...
- 11.7k
5
votes
Accepted
On integer programming
(1) As finding a second satisfying assignment to a 3SAT formula is still $\mathsf{FNP}$-complete (indeed, it is $\mathsf{ASP}$-complete, see Theorem 3.5 of [1]), and we can encode 3SAT as an integer ...
- 36.9k
4
votes
Accepted
Which is the approximation class of Minimum 0-1 integer programming if only non-negative integers are allowed?
Kolloiopoulos and Young give an $O(\log m)$ approximation for general covering integer programs. See the paper below.
http://www.sciencedirect.com/science/article/pii/S0022000005000656
- 6,979
4
votes
NP completeness of linear $0-1$ assignment problem
It seems that we can reduce Subset Sum to your problem (2). Hence, your problem (2) is NP-complete.
Consider the following formulation of Subset Sum.
Instance: A multi-set consisting of $n$ ...
- 5,127
4
votes
Accepted
Is this edge orientation optimization problem NP-hard?
I believe that this problem is NP-hard, here is a sketch proof (don't hesitate to ask for more details if needed).
The idea is based on a reduction from the Not-all-equal 3-SAT. For $\varphi$ a 3-SAT ...
- 775
4
votes
Is that edge orientation optimization problem NP-hard?
Definition: Given an undirected graph $G$ and an edge orientation $\vec{G}$, an unstable path is a directed path that goes from a node $s$ to a node $t$, such that the out-degree of node $s$ is ...
- 775
3
votes
Accepted
How hard is this combinatorial optimisation problem?
This problem can be solved with dynamic programming in pseudo-polynomial time (proof below). Therefore, it is not possible to show that this problem is strongly NP-hard (unless P=NP).
First, let's ...
- 2,758
3
votes
Accepted
Minimum Union-Sum Cost Path
EDIT (Jan 2019): Lemma 2 as currently stated below is wrong. (Indeed, given any instance, adding a single edge with a single type of very large cost will not change the instance but will yield $N(I)=1$...
- 9,595
2
votes
FPT algorithm for mixed integer program
The complexity of Lenstra's algorithm for mixed-integer programming in his paper runs as $2^{O(n^3)}*poly(d, \phi)$ where there are $n$ integer variables, $d$ continuos variables, and $\phi$ is the ...
2
votes
Which Integer Linear Programs are easy?
An integer program with only equalities can be solved by linear program.
- 12.8k
2
votes
Accepted
Integer programming: enforce the constraint that a subgraph contains at most $k$ connected components?
Base on Komus's constraint, we add another constraint which ensures a Steiner Tree on $G^{'}=(V, E^{'})$, where $E'=\{(i,j): i,j \in V\}$:
$$\sum_{e \in cut(U,V)}x_e \ge 1, \forall u,v \in T, \forall ...
- 461
2
votes
Accepted
Ensuring integral maximizer from integral linear program
Given an integral LP, you can use a LP algorithm to compute in polynomial time
the optimal value of the LP,
a vertex of the feasible region (polytope) of the LP where the optimal value is attained.
...
- 16.9k
2
votes
Name of (and solution to) this generalization of linear assignment
It looks to me like this is a special case of minimum cost flow; introduce one vertex per row and one per column, with an edge for each entry whose cost is the negative of the value of that entry and ...
- 11.7k
2
votes
Accepted
Do there exist two equivalent objective functions one of which can be approximated but another one cannot?
The answer to Question (1) is no. The answer to Question (2) is yes. Here are the details.
I'll work with the following equivalent problem formulations. For the input, we are given $n$ pairs of ...
- 9,595
2
votes
Accepted
Hardness of computing the dimension of an integral polytope?
Yes, MORE-SAT, defined below, is one example of a combinatorial optimization problem for which the natural 0/1 integer linear program (ILP) is guaranteed to be feasible, and determining the dimension ...
- 9,595
1
vote
Is that edge orientation optimization problem NP-hard?
Note that the related problem pointed in the motivation of the original post is NP-hard. The proof is available in the Annex B of this paper: https://arxiv.org/pdf/2203.04774.pdf
- 121
1
vote
How do minimally violated k-mod cuts work (intuitive explanation)?
Suppose you have an integer linear program, and it contains the following constraints:
x1 + x2 + x3 >= 1,
x1 + x2 + x4 >= 1,
x1 + x3 + x4 >= 1,
x2 + x3 + x4 >= 1.
If we set x1, ..., x4 to 1/3, we ...
1
vote
Max weight travel on a graph with deadline
Since you specifically ask later about the version of the problem in which the given path is restricted in how many times it visits each vertex/edge, I assume that the original version of the problem ...
- 2,758
1
vote
Accepted
Min cost set of edges to connect 2 subgraphs s.t dist of nodes between subgraphs <= K
When the costs are all equal, you are looking for the fewest number of edges to bring your graph to being of at most diameter $k$. This is the graph diameter augmentation problem. You are looking for ...
- 1,276
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