Computer Science 6743, Winter '10 Guest Lectures (Wareham)

Tuesday, March 2 (Lecture #1)

• Overview: Computational Complexity and (In)tractability
(or: You Can't Always Get What You Want (But Sometimes You Get What You Need))
• Relative to a particular type of algorithm, a problem P is (in)tractable if there is (not) an algorithm of that type for P.
• Computational complexity techniques can be used to assess (in)tractability of computational problems. Requires the following 4 components:

• Description of problems of interest (implicit in this is class U, which is the set of problems)
• Description of tractable-algorithm type (implicit in this is class T, which is the set of tractable problems).
• A reducibility on pairs of problems in U that preserves tractability, i.e., a reducibility such that the following two properties hold:

• If P reduces to Q and Q is in T then P is in T.
• If P reduces to Q and Q reduces to R them P reduces to R.

Any such reducibility by definition (in particular, by the first property above) also has the following third property:

• If P reduces to Q and P is not in T (modulo conjecture X) then Q is not in T (modulo conjecture X).

• A subset C of U such that T is a (proper) subset of C.

• Recall that relative to a class C, a problem P is C-hard if every problem in C reduces to P; if P is also in C then P is C-complete.
• Relative to the framework above, can use C-hardness to show that a problem P is not in T and hence not tractable (possibly modulo some conjecture X)!
• Example: Theory of NP-completeness
• Relative to the framework above, U = decision problems, T = P (the class of all decision problems solvable in deterministic polynomial time), the reducibility is polynomial-time many-one reducibility, and C = NP (the class of all decision problems whose candidate solutions can be verified in nondeterministic polynomial time).
• As NP is only known to contain (rather than properly contain) P, an NP-hardness result for a problem only implies polynomial-intractability if P != NP.
• Have to read the fine print for any intractability result for a problem P; P may in fact be tractable under different but nonetheless acceptable forms of tractability.
• Example: What an NP-hardness result really means:

if a problem is NP-hard
then there is no deterministic algorithm
that gives optimal solutions
for all instances of the problem
in polynomial time
unless P = NP.

An NP-hard problem may still have usable algorithms that violate one or more of the italicized clauses above, e.g., polynomial-time randomized optimal-solution algorithms, polynomial-time deterministic approximate-solution algorithms, exponential-time deterministic optimal-solution algorithms that are effectively polynomial-time for problem instances of interest.
• Over the next three lectures, we'll look at two kinds of tractability that may be options for NP-hard problems -- namely, value / structure approximability and fixed-parameter tractability.
• Value / Structure Approximability
• Motivation
• Our universe U will be the set of all optimization problems.
• Definition: Optimization problem [ACG+99, Definition 1.1.6]
• Tractability phrased in terms of particular types of approximation algorithms.
• Definition: Approximation algorithm [ACG+99, Definition 3.1]
• Two types of approximation criteria: solution value and solution structure
• Value Approximability ([ACG+99] and references; see also [GJ79, Chapter 6; Gon07; HS78, Chapter 12; Vaz03])
• Motivation
• Our approximability criterion is the closeness of the value of the approximation-algorithm solution to optimal-solution value.
• Definition: Additive approximation (v-a-approx) [ACG+99, Definition 3.3]
• Definition: Ratio approximation (v-r-approx) [ACG+99, Definition 3.7]
• Definition: Polynomial-time approximation scheme (PTAS) [ACG+99, Definition 3.10]
• Definition: Fully polynomial-time approximation scheme (FPTAS) [ACG+99, Definition 3.12]
• Can prove v-approximability by giving an algorithm with the appropriate properties.
• Example: Min Coloring for Planar Graphs is 3 v-a-approximable [ACG+99, Example 3.1]
• Example: Min Vertex Cover is 2 v-r-approximable [CLRS01, Theorem 35.1]
• Prior to the late 1980's, most proofs of v-inapproximability were done on a problem-by-problem basis.
• Example: Max Clique is not c v-a-approximable for any constant c > 0 unless P = NP [ACG+99, p. 89]
• In the next lecture, we will look at the two class-based frameworks for proving v-inapproximability.

Thursday, March 4 (Lecture #2)

• Value / Structure Approximability (Cont'd)
• Value Approximability (Cont'd)
• General v-inapproximability: v-approximability class hardness/-completeness
• v-approximability tractability classes: FPTAS, PTAS, APX, NPO ([CP91]; see also [ACG+99, Section 3]) (note absence of v-a-approx class)
• There is also a reducibility that preserves v-r-approximability (and hence PTAS and FPTAS qapproximability as well).
• Definition: L-reduction [PY91, p. 427]
• Theorem: FPTAS is properly contained in PTAS is properly contained in APX is properly contained in NPO unless P = NP ([CP91, Theorem 6]; see also [ACG+99, Section 3])
• Complete problems relative to L-reducibility are known for classes APX and NPO (see [ACG+99, Section 8] and references).
• Short-cut to PTAS inapproximability: MAX SNP-hardness
• Class MAX SNP is a class of optimization problems whose optimal solutions can be described by a particular type of logic formula. Every problem in this class can be v-r-approximated within a fixed ratio [PY91, Theorem 1].
• [PY91] gave a number of MAX SNP-complete problems and, given the above, noted that if any such problem had a PTAS, all problems in MAX SNP had a PTAS. However, the converse was shown the following year, when [ALM+92] showed that a particular MAX SNP-complete problem (Max 3SAT) could not have a PTAS unless P = NP [ALMS+92, Theorem 3]. Hence no MAX SNP-hard problem can have a PTAS unless P = NP.
• Though the Ausiello et al framework deals with many more kinds of v-inapproximability than MAX SNP-hardness, the latter is easier to use; this may be because (1) there are a lot more MAX SNP-hardness results known (and there have been since the beginning [PY91]), (2) these results are for problems that are the basis of many known NP-hardness reductions (and hence, if one is lucky, can be adapted to your problem of interest), and (3) it is a bit easier to find MAX SNP-hardness results, e.g., []PY91], than PTAS- or APX-hardness results (as the [ACG+99] approximability problem compendium is organized by problem name rather than by class-hardness).
• A general caveat that can be derived from this is, to make a new theory of intractability useful, you need to both have simple usable core definitions of U, T, tractability-preserving reducibility, and C and a good core set of C-hardness/completeness results for basic problems of interest from which new results can be derived.
• Failing wrt either of these characteristics can cause problems for an intractability theory -- we will see several examples of this over the next two lectures.
• Structure Approximability [HMRW07]
• Motivation
• Focus again on optimization problems and approximation algorithms; however, this time, our criterion is closeness of approximation-algorithm solution structure to optimal-solution structure.
• Assuming a particular representation of solutions and a distance function d(x, y) on pairs of solutions, measure closeness of structure by value of d(A(x), y), where y is the closest optimal solution to A(x) relative to d() [HMRW07, p. 6].
• Definition: Additive approximation (s-a-approx) [HMRW07, Definition 8]
• Definition: Ratio approximation (s-r-approx) [HMRW07, Definition 8]
• Definition: Polynomial-time structure approximation scheme (sPTAS) [HMRW07, Definition 9]
• Definition: Fully polynomial-time structure approximation scheme (sFPTAS) [HMRW07, Definition 10]
• Once again, can prove approximability by giving an algorithm with the appropriate properties.
• Example: Weighted Max Cut is 0.5|x| / d s-a-approximable and 0.5 / d s-r-approximable [HMRW07, Theorem 12 and Corollary 13]
• All known proofs of s-inapproximability done on a problem-by-problem basis.
• Example: Max Clique is not c / d s-a-approximable for any constant c . 0 unless P = NP (see [HMRW07, pp. 16-18])
• Have defined structural analogues of classical v-approximability classes as well as a structural L-reducibility [HMRW07, pp. 19-21] -- however, to date, no known complete or hard problems for these classes.
• This is a research opportunity (hint, hint) ...
• Wrt structural approximability, maybe part of the reason we do not yet have complete/hard problems is that we don't yet have the right definitions for some (or all) of the core concepts. This happens a lot in the early stages of developing a new theory of (in)tractability. We will see one such example of how people initially got it wrong (and perhaps now have it right) when we look at fixed-parameter tractability.
• Motivation
• All forms of exponential time are not created equal; sometimes, if the exponential term in the running time arises from an aspect of the problem that is of restricted or constant value in practice, such algorithms are useful, e.g., O(n^k) and O(2^kn) algorithms are OK if k is small (with the latter being particularly attractive)[GJ79, Section 4.3].
• Lemma [GJ79, Lemma 3.1]: For any graph G and a subset V' of the vertices V of G, the following statements are equivalent:

• V' is a vertex cover for G;
• V - V' is an independent set for G; and
• V - V' is a clique in the complement of G.

• Theorem [GJ79, Section 3.1.3]: Vertex Cover, Clique, and Independent Set are NP-complete.
• Example: Vertex Cover is solvable in O(2^kn^2) time.
• How do we show that such algorithms do not exist for a particular problem?
• Example: Clique and Independent Set do not appear to have brute-force algorithms that run in O(n^k) time.

Tuesday, March 9 (Lecture #3)

• Fixed-Parameter Tractability (Cont'd)
• Fixed-Parameter Tractability, Take 1: The FL framework
(Nonuniform fp-tractability [Fel89,FL88])
• Based on Robertson-Seymour graph minor theory (see [FL88] and references)
• Graph minors.
• Theorem (Kuratowski): A graph is planar iff it does not have K_5 or K_3,3 as a minor.
• Minor-closed families of graphs and obstruction sets (= the minor-minimal set of graphs in the complement of a graph family).
• For a minor-closed family of graphs F, a graph G is in F iff there exists no H in the obstruction set of F such that H is a minor of G!
• Theorem (formerly known as Wagner's Conjecture) [RS04]: Any set of finite graphs contains only a finite number of minor-minimal elements.
• Theorem [RS95]: Determining if a fixed graph H is a minor of a given graph G can be done in polynomial time.
• These theorems in conjunction with the preceding observation yield nonconstructive polynomial-time algorithms for NP-complete problems whose "yes" or "no" instances can be characterized as minor-closed graph families [FL88].
• Theorem [FL88, p. 734]: The NP-complete k-Longest Path problem is nonconstructively solvable in polynomial time.
• This gives a nonuniform fp-tractability, but no way of proving fp-intractability.
• Fixed-Parameter Tractability, Take 2: The AEFM framework (Nonuniform fp-(in)tractability ([AEFM89]; see also [DF99, p. 251])
• U = generator-tester pairs ((indexed) relations)
• T = nonuniform fp-tractability
• Reducibility = many-one reducibility on relations + Turing reductions
• C = IP (set of P-bounded / - indexed / -checkable indexed relations)
• Both too powerful (like DEXPTIME prior to NP) and somewhat arcane wrt definitions and proof-techniques; also, is based on nonuniform fp-tractability, which allows tractable problems to be nonrecursive and hence uncomputable.
• Fixed-Parameter Tractability, Take 3: The DF framework (Uniform fp-(in)tractability (see [DF99] and references))
• Focus on simple case: decision problems and uniform fp-tractability.
• U = parameterized problems (languages (sort of))
• Example: The parameterized problems k-Vertex Cover, k-Clique, and k-Independent Set
• T = (uniform) fp-tractability / class FPT
• Are many techniques for deriving fixed-parameter algorithms (see [Nie06] and references).
• Example: k-Vertex Cover is in FPT
• reducibility = fp-reducibility
• Example: k-Clique reduces to k-Independent Set
• C = {W[1], W[2], ..., W[P], ..., ParaNP} (W-hierarchy)
• The "W" in W-hierarchy stands for (circuit) weft.
• W-classes defined in terms of circuits which recognize "yes"-instances of parameterized problems; classes differ by circuit weft.
• Example: k-Vertex Cover, k-Clique, and k-Independent Set are in W[1].
• Theorem: k-Independent Set and k-Clique are W[1]-complete [DF99, Theorem 10.8].
• Inclusions in the W-hierarchy are not known to be proper; hence, the FPT != W[1] conjecture is invoked. This conjecture seems likely, but you never know ...
• New formulations and versions of parameterized complexity are emerging (see [Fel09,FG06] and references), so this may not yet be the end of the story -- stay tuned ...

References

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