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9 votes
Accepted

Aren’t most constraining variable and least constraining value the exact opposite?

Yes, these two heuristics sound like inconsistent. Most Constrained Variable (MCV) (also called MRV for Minimum Remaining Values) tries to reduce the size of the next branch to search while Least ...
John L.'s user avatar
  • 39k
9 votes
Accepted

SAT algorithm for determining if a graph is disjoint

Given a graph $G = (V,E)$, here is a SAT instance which is satisfiable iff the graph is not connected. Pick an arbitrary vertex $v_0 \in V$, and add the following clauses, over the variables $x_v$ ...
Yuval Filmus's user avatar
7 votes
Accepted

Constraint Satisfaction: maximizing total value with no overlaps

You are looking for a maximum weight independent set in an interval graph, which can be solved in linear time (by a deterministic algorithm). By the way, the same is true also for a superclass of ...
Juho's user avatar
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6 votes
Accepted

How could an SMT solver be implemented as simple as possible?

What happens next is that we invoke a SMT solver to try to check whether the formula $$x+y < 20 \quad \land \quad x > 10 \quad \land \quad y > 10$$ is satisfiable. The solver tries to ...
D.W.'s user avatar
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6 votes
Accepted

Efficient algorithm for simple constraint satisfaction problem

There is unlikely to be any efficient algorithm. Your first class of constraints are monotone exactly-1 CNF clauses. Your second class of constraints are monotone CNF clauses. The monotone part ...
D.W.'s user avatar
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5 votes

How do we place $8n$ objects in a grid of size $n \times n$?

Given a set of objects in a cell grid, if no 4-set of it forms a rectangle with sides parallel to the sides of the grid, we will call those objects rectangle-free. The more general problem is to ...
John L.'s user avatar
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5 votes

SAT algorithm for determining if a graph is disjoint

Yuval describes a boolean CNF formula that is satisfiable iff the graph is not connected, using $|V|$ variables; and a boolean CNF formula that is satisfiable iff the graph is connected, using $|V|^2$ ...
D.W.'s user avatar
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4 votes
Accepted

Are finite-domain binary constraint satisfaction problems solvable in polynomial time?

I don't know, but if you manage, make sure to let us know. The reduction seems easy indeed, with $d$ mapping to the number of colors and each constraint to an edge. However, a polynomial-time ...
quicksort's user avatar
  • 4,262
4 votes
Accepted

Binarization of Constraints

Introduce a new variable $Q$, whose domain is $\{0000,0001,0010,\dots,1111\}$. It represents the value of $C1,C2,C3,P$. For instance, if $Q=0001$, that means that $C1=0$, $C2=0$, $C3=1$, $P=0$. ...
D.W.'s user avatar
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4 votes
Accepted

General structure of solutions to 3-SAT circuits

The theory you are after is universal algebra. See the excellent expository article of Hubie Chen, A rendezvous of logic, complexity, and algebra, which contains a streamlined proof of Schaefer’s ...
Yuval Filmus's user avatar
4 votes
Accepted

Is every X3SAT instance with no cycles satisfiable?

The graph below is a positive answer without words. Here is the detailed proof. Definitions Let $X$ be an instance of X3SAT. $X$ is linear if any two clause shares at most one variable. $X$ is ...
John L.'s user avatar
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4 votes
Accepted

Sum of unique integers to cnf constraint

Here's a strategy for solving Kakuro with a SAT solver. Make a nine variables for each cell, each variable indicating whether that cell contains $1$, $2$, etc. Add a exactly-one-out-nine constraint ...
orlp's user avatar
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4 votes

Linear programming over a finite field

Your problem, solving a system of linear equations, can be solved using an ancient algorithm, Gaussian elimination, which works over all fields. Note that linear programming is more general, allowing ...
Yuval Filmus's user avatar
4 votes

Is the following problem NP-Complete?

No. This problem is equivalent to XOR-3SAT, in which we interpret each clause as $x \oplus y \oplus z$, where $\oplus$ is the XOR operator, and ask whether it's possible to find values for all ...
j_random_hacker's user avatar
4 votes
Accepted

Is 2-SAT over Linear Real Arithmetic in P or NP?

You can express the fact that a variable $x_i$ is Boolean as follows: $$ (0 \leq x_i \leq 1) \land ((x_i \leq 0) \lor (x_i \geq 1)). $$ You can express the condition $x_i \lor x_j \lor x_k$ as $$ x_i +...
Yuval Filmus's user avatar
4 votes

what is the background theory that Z3 uses to prove constraints unsat

There are basically two components: the underlying SAT solver procedure (DPLL), and the additional theory-specific procedures. For the underlying SAT solver: considering your example of ...
Caleb Stanford's user avatar
3 votes

How could an SMT solver be implemented as simple as possible?

You may not need a general SMT solver. In this case, you have a bunch of inequalities conjoined together, which you can solve as a linear program. A simpler alternative to an SMT solver is to ...
Pseudonym's user avatar
  • 22.1k
3 votes
Accepted

Theoretical CSPs where (in)equality constraints can be expressed as a single constraint?

In complexity theory, CSPs are usually specified as a set of allowed predicates. If the (finite) domain is $D$, a predicate of arity $d$ is an arbitrary subset of $D^d$ of allowed values. In ...
Yuval Filmus's user avatar
3 votes

What is Least-Constraining-Value?

see this link: https://people.cs.pitt.edu/~wiebe/courses/CS2710/lectures/constraintSat.example.txt It first picks variable "O" and then tests "O" with all of it's legal values "i" to see the number ...
HsnVahedi's user avatar
  • 181
3 votes

Algorithm for solving planar constraint problem ("Pokemon Go monster finding")

I think you could use a "spatial join". I haven't played the game, but I assume $d_{max}$ is rather small, i.e. there are in the order of 10 or so $n$ and $m$ in the neighborhood of each $m$. I ...
TilmannZ's user avatar
  • 764
3 votes

what does it mean to extend an assignment?

It means that you have a partial solution, and you extend that partial solution by assigning at least one more variable to some value, thus producing either a new partial solution or a solution where ...
Juho's user avatar
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3 votes
Accepted

Is this problem that's similar to integer linear programming also an NP-complete problem?

Your problem is solvable in polynomial time. Your problem is equivalent to ILP with difference constraints, i.e., every linear inequality has the form $x_j - x_i \ge c_{i,j}$. (Why? Set $x_i = v_1 +...
D.W.'s user avatar
  • 160k
3 votes

Selection over combinatorics that satisfies a distribution

Consider the following special case where for each element $i$ the table contains the constraint $\#i \geq (1/l) \cdot l$. This means we need to select the sets in such a way that each element appears ...
Narek Bojikian's user avatar
3 votes

Algorithm to create dense style crossword puzzles

There may simply be no solution to some of these problem instances. And the fact that the problem is NP-hard means that you cannot expect to find any efficient algorithm to find solutions for large ...
j_random_hacker's user avatar
3 votes

Automated reasoning with real numbers

In your case, the simplest solution may be to use SAT. Your first clause includes $x \le 0.25$ and $x > 0.91$. This means that there are five regions of interest for the variable $x$, which we ...
Pseudonym's user avatar
  • 22.1k
3 votes
Accepted

Efficient Algorithm to Find the Closest Integer Representation, in the Form $A\times\frac{N}{D}$ for a Value

Generally speaking, this problem is called diophantine approximation. However, your inputs are not just real numbers but double-precision floating point (aka double)...
Vincenzo's user avatar
  • 3,302
3 votes

Bipartite matching with constraints on one part

The problem is not approximable in polynomial-time within a factor of $n^{1-\varepsilon}$ for any constant $\varepsilon>0$, unless $\mathsf{NP} = \mathsf{ZPP}$. To see this you can reduce from ...
Steven's user avatar
  • 29.5k
3 votes
Accepted

How fast can we make generalized k-SAT?

Worst-case bounds on running time are usually not very informative for SAT solvers. In particular, on problem instances that arise in practice, often SAT solvers run a lot faster than the bounds ...
D.W.'s user avatar
  • 160k

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