# Tag Info

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 ...
• 5,712
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 ...
• 50.2k
Accepted

### Reduction from SAT to 0,1 integer linear program with zero or one solutions

OptP-complete. Krentel showed that MAX-SAT, finding the lexicographically maximum satisfying assignment, is OptP-complete and the reduction above reduces Max-SAT to ILP. ILP sits in OptP pretty much ...
• 8,546
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 - ...
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 ...
• 23.5k
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### 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 ...
• 5,225
Accepted

I think $S_n$ can be written in terms of inequalities in the obvious way. Let $$Q_n = \{(x, y): x = \sum_{i = 0}^{n-1}{2^i y_i}, \forall i: 0 \leq y_i \leq 1\}.$$ I claim that $Q_n = S_n$. First, ...
Accepted

### Quanitifier Free Presburger Arithmetic: Upper bound on solution size?

You can find an answer in the following paper: Joachim von zur Gathen, Malte Sieveking. A bound on solutions of integer linear equalities and inequalities. Proc. AMS 72(1) (1978) (pdf) A simple ...
• 527
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### 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, ...
• 8,228

### 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
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### 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 ...
• 22.3k
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### 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 ...

### 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; ...
• 10.4k
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 ...
• 35.7k
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### 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
• 6,649

### 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 $n$ ...
• 4,900
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• 451
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### 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. ...
• 14.8k

### 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 ...
• 10.4k
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 ...