Class NP and NP-Completeness

Class NP

The class of decision problems for which there is a polynomially bounded nondeterministic algorithm.

The class of problems that are "slightly harder" than P.

In general, an exponential number of subproblems, each of which can be solved or verified in polynomial time.

Being in class NP constitutes an upper bound on problem complexity.

Nondeterministic Turing Machine (NDTM)

Model a problem as a tree in which the nodes represent states, with time flowing from the root to the children. A leaf represents a solution.

In an NDTM, the computation splits whenever a guess is needed.

The tree has polynomial depth but is exponentially bushy.

"Guess" nodes are like OR nodes: if either of the children returns T, the "guess" node returns T.

NP-hard

A problem Q is NP-hard if every problem in NP is reducible to Q.

A lower bound on a problem: "at least as hard as any problem in NP."

NP-Complete

The hardest decision problems in NP.

If there were a polynomially bounded algorithm for an NP-complete problem, then there would be a polynomially bounded algorithm for each problem in NP.

Establishes both a lower and an upper bound (although it may be that P=NP).

If any NP-complete problem is in P, then P=NP.

Satisfiability (SAT)

Instance: Given a set U = {u₁, ..., u_m} of variables and a collection C = {c₁, ..., c_n} of clauses over U.

Question: Is there a truth assignment to U that satisfies C?

A logical formula in conjunctive normal form (CNF): a conjunction of disjunctions
e.g., (x₁ + x₂ + ¬x₃) · (x₁ + x₃) · (¬x₂)

easy to falsify: make all terms in one clause fail.

hard to satisfy: terms in clauses interact.

Cook's Theorem: SAT is NP-complete.

uses SAT to simulate a machine

Once the primordial NP-complete problem is found, we can prove other problems NP-complete without reference to machines:

To prove that problem Q is NP-complete, reduce SAT (or any other NP-complete problem) to Q:
SAT <=_P Q

E.g., 3SAT: like SAT, but | c_i | = 3 for 1 <= i <= n.

Need to show that 3SAT is NP-hard:
Since X <=_P SAT for any problem X in NP, we need only show SAT <=_P 3SAT.

Proof:
General form of clauses in SAT is c_j = (z₁ + z₂ + ... + z_k), where each z_i is either an input or the negation of an input.

k = 1
{z₁ + z₁ + z₁}

k = 2
{z₁ + z₂ + z₂}

k = 3
{z₁ + z₂ + z₃}

k >= 4
Add auxiliary variables that transmit information between pieces.
E.g., {z₁ + z₂ + z₃ + z₄ + z₅} becomes {(z₁ + z₂ + y₁) (¬y₁ + z₃ + y₂) (¬y₂ + z₄ + z₅)}

The transformation is polynomial in the length of the SAT instance.

Note: 2SAT is solvable in polynomial time.

Other NP-Complete Problems

Graph coloring

A coloring of a graph is the assignment of a "color" to each vertex of the graph such that adjacent vertices are not assigned the same color. The chromatic number of a graph G, X(G) [Chi(G)], is the smallest number of colors needed to color G.

Optimization problem: Given an undirected graph G, determine X(G) (and produce an optimal coloring).

Decision problem: Given G and a positive integer k, is there a coloring of G using at most k colors (i.e., is G k-colorable)?

Job scheduling with penalties

Suppose we are given n jobs to be executed one at a time and, for each job, an execution time, deadline, and penalty for missing the deadline.

Optimization problem: Determine the minimum possible penalty (and find an optimal schedule).

Decision problem: Given a nonnegative integer k, is there a schedule with penalty < k?

Note: Job scheduling without penalties such that at most k jobs miss their deadlines is in P.

Multiprocessor scheduling

Suppose we are given a deadline and a set of tasks of varying length to be performed on two identical processors.

Optimization problem: Determine an allocation of the tasks to the two processors such that the deadline can be met.

Decision problem: Can the tasks be arranged so that the deadline is met?

Bin packing

Optimization problem: Given an unlimited number of bins, each of size 1, and n objects with sizes s₁, ..., s_n where 0 < s_i < 1, determine the smallest number of bins into which the objects can be packed (and find an optimal packing).

Decision problem: Given, in addition to the inputs described above, an integer k, do the objects fit in k bins?

Knapsack problem

Optimization problem: Given a knapsack of capacity C and n objects with sizes s₁, ..., s_n and "profits" p₁, ..., p_n, find the largest total profit of any subset of the objects that fits in the knapsack ( and find a subset that achieves the maximum profit).

Decision problem: Given k, is there a subset of the objects that fits in the knapsack and has total profit at least k?

Subset sum problem

Optimization problem: Given a positive integer C and n objects with positive sizes s₁, ..., s_n, among subsets of the objects with sum at most C, what is the largest subset sum?

Decision problem: Is there a subset of the objects whose sizes add up to exactly C?

The subset sum problem is a simpler version of the knapsack problem, in which the profit for each object is the same as its size.

Partition

Suppose we are given a set of integers.

Decision problem: Can the integers be partitioned into two sets whose sum is equal?

Hamiltonian cycles and Hamiltonian paths

A Hamiltonian cycle (path) is a simple cycle (path) that passes through every vertex of a graph exactly once.
Decision problem: Does a given undirected graph have a Hamiltonian cycle (path)?

Note: An Euler tour -- a cycle that traverses each edge of a graph exactly once -- can be found in O(e) time.

Traveling salesperson problem (TSP) or minimum tour problem

Optimization problem: Given a complete, weighted graph, find a minimum-weight Hamiltonian cycle.
Decision problem: Given a complete, weighted graph and an integer k, is there a Hamiltonian cycle with total weight at most k?

A complete graph has an edge between each pair of vertices.

Vertex cover

A vertex cover for an undirected graph G is a subset V' of vertices such that each edge in G is incident upon some vertex in V'.

Optimization problem: Find a vertex cover for G with as few vertices as possible.

Decision problem: Given an integer k, does G have a vertex cover consisting of k vertices?

Note: The edge cover problem is in P.

Clique

A clique is a subset V' of vertices in an undirected graph G such that every pair of distinct vertices in V' is joined by an edge in G (i.e., the subgraph induced by V' is complete). A clique with k vertices is called a k-clique.

Optimization problem: Find a clique with as many vertices as possible.

Decision problem: Given an integer k, does G have a k-clique?

Independent set

An independent set is a subset V' of vertices in an undirected graph G such that no pair of vertices in V' is joined by an edge of G.

Optimization problem: Find an independent set with as many vertices as possible.

Decision problem: Given an integer k, does G have an independent set consisting of k vertices?

Feedback edge set

A feedback edge set in a digraph G is a subset E' of edges such that every cycle in G has an edge in E'.

Optimization problem: Find a feedback edge set with as few edges as possible.

Decision problem: Given an integer k, does G have a feedback edge set consisting of k edges?

Note: The feedback edge set for undirected graphs is in P.

Longest simple path problem

Optimization problem: What is the longest simple path between any two vertices in graph G?

Decision problem: Given an integer k, is there a path in graph G longer than k?

Note: The shortest path between any two vertices can be found in O(ne) time.

Subgraph isomorphism

Decision problem: Given two graphs G₁ and G₂, determine if G₁ is isomorphic to a subgraph of G₂.

Integer linear programming

Linear programming is a method of analyzing systems of linear inequalities to arrive at optimal solutions to problems

Decision problem: Given a linear program, is there an integral solution?

References

Baase and Van Gelder, Computer Algorithms, 3e (Addison Wesley Longman, 2000)
Sedgewick, Algorithms (Addison Wesley, 1983)
Cormen, Leiserson, Rivest, and Stein, Introduction to Algorithms, 2e (MIT Press/McGraw Hill, 2001) The authors provide NP-completeness proofs reducing 3SAT to CLIQUE to VERTEX-COVER to HAMILTONIAN-CYCLE to TSP, and 3SAT to SUBSET-SUM.