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Algorithms for hard problems Parameterized complexity – definitions, sample algorithms. Juris Viksna, 2013. Parameterized complexity. Combinatorial "explosion" for NP -hard problems. [Adapted from R.Downey and M.Fellows]. Parameterized complexity.
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Algorithms for hard problems Parameterized complexity – definitions, sample algorithms Juris Viksna, 2013
Parameterized complexity Combinatorial "explosion" for NP-hard problems [Adapted from R.Downey and M.Fellows]
Parameterized complexity Parameterized complexity attempts to confine combinatorial "explosion" [Adapted from R.Downey and M.Fellows]
Parameterized complexity - Definitions Definition (FPT) A parametrized problemL ** is Fixed Parameter Tractable if there is an algorithm that for input (x,y) ** with |x| = k and |y| = n decides whether (x,y) L in time f(k) n , where f is an arbitrary function and is a constant. Definition does not change if f(k) n is replaced by f(k) + n (!)
Parameterized complexity - Definitions • Mkfor every n solves the problem in timef(k) n • For each k there is a constant ck, such that • f(k) n > n + 1 for n ck • M'k: - simulates Mk for n ck • - looks up value from the table for n ck • M'k solves the problem in time f(k) ck + n a+1
Some parameterized problems VERTEX COVER Instance: A graph G=(V,E) Parameter: A positive integer k Question: Is there a subset SV, such that |S|=k and for all {x,y}E either xS or yS? [Adapted from R.Downey and M.Fellows]
Some parameterized problems GRAPH GENUS Instance: A graph G=(V,E) Parameter: A positive integer k Question: Does graph G have genus k? [Adapted from R.Downey and M.Fellows]
Some parameterized problems GRAPH LINKING NUMBER Instance: A graph G=(V,E) Parameter: A positive integer k Question: Can G be embedded in 3-space such that the maximum size of a collection of topologically linked disjoint cycles is bounded by k? [Adapted from R.Downey and M.Fellows]
Some parameterized problems • VERTEX COVER: • A single algorithm G running in time 2k |G| for each k • [Downey and Fellows 1992] • GRAPH GENUS: • A single algorithm F, which for each fixed k determines • whether graph G has genus k in time O(|G|3) • [Fellows and Langston 1988] • GRAPH LINKING NUMBER: • For each k there is an algorithm k, which determines • whether graph G has linking number k in time O(|G|3) • [Fellows and Langston 1988]
Parameterized complexity - Definitions A problem L is uniformly FPT, if there is an algorithm F, a constant c and a function f, such that: - the running time of F(‹x,k›)is at most f(k)|x|c - ‹x,k›A iff F(‹x,k›) = 1. A problem L is strongly uniformly FPT, if L is uniformly FPT via some F and f, such that f is recursive. A problem L is nonuniformly FPT if there is a constant c, a function f and a collection of algorithms {Fk}kN, such that for each k: - the running time of Fk(‹x,k›)is at most f(k)|x|c - ‹x,k›A iff Fk(‹x,k›) = 1.
"klam" values Tower of 2's of height described by tower of 2's of height described by tower of 2's... (impractical even for k = 1 :) • Algorithms for VERTEX COVER: • O(f(k) n3) [Fellows, Langston 1986] • O(f(k) n2) [Johnson, 1987] 22500k FPT if solvable in time f(k)+nc klam value - the largest k for which f(k) is less that some universal "speed limit" U (e.g. U= 1020) [Adapted from R.Downey and M.Fellows]
Parameterized complexity • Parametrized tractability • (methods for constructing FPT • algorithms) • Parametrized intractability • (how to "prove" that there are no • FPT algorithm for a given problem)
Bounded search tree methods • First, construct a search space (tree), such that the size • search space depends only from parameter k • Then run some relatively efficient algorithm (say DFS) • on each branch of the tree For fixed k search space is of constant size, thus we obtain a FPT algorithm
Bounded search tree methods - Vertex cover , G {u1,v1} {u1}, G-{u1} {v1}, G-{v1} k+1 {u2,v2} {v1,u2}, G-{v1,u2} {v1,v2}, G-{v1,v2} By considering only graphs with vertices of degree 3 or higher, it is possible to shrink search space from 2k to (51/4)k [Balasubramanian et al 1997] Theorem [Downey and Fellows 1992] VERTEX COVER is solvable in time O(2k |V(G)|).
Kernel methods The main idea of the reduction to problem kernel is to reduce a problem instance I to an equivalent instance I', where the size of I' is bounded by some function of the parameter k. The instance I' is then exhaustively analyzed, and a solution for I' lifted to a solution to I. Usually this will give and additive rather that multiplicative f(k) exponential factor
Kernel methods - Vertex cover Theorem [Buss 1989] VERTEX COVER is solvable in time O((2k)k + k |V(G)|). Observation Any vertex of degree greater than k must belong to every k-element vertex cover. Step 1 Locate all vertices in G of degree greater than k; let p equal the number of such vertices. If p > k, there is no k-vertex cover. Otherwise let k' = k – p. Step 2 Discard all vertices found in step 1 and the edges incident to them. If the resulting graph G' has more than k'(k+1) vertices, reject.
Kernel methods - Vertex cover Observation Any vertex of degree greater than k must belong to every k-element vertex cover. Step 1 Locate all vertices in G of degree greater than k; let p equal the number of such vertices. If p > k, there is no k-vertex cover. Otherwise let k' = k – p. Step 2 Discard all vertices found in step 1 and the edges incident to them. If the resulting graph G' has more than k'(k+1) vertices, reject. Step 3 If G' has no k'-vertex cover, reject. Otherwise, any k'-vertex cover of G' plus the p vertices from step 1 constitutes a k-vertex cover for G. Graph with k' vertex cover and vertex degree bounded by k has up to k'(k+1) vertices Doable in O((2k)k) time
Kernel methods - Vertex cover Search tree method O(2k |V(G)|) O((51/4)k |V(G)|) Kernel method O((2k)k + k |V(G)|) If we use reduction to problem kernel first, and then apply search tree method to problem kernel, we improve additive bounds to: O(kn + 2kk2) [Balasubramanian et al 1992] O(kn + ((51/4)kk2) [Balasubramanian et al 1992]
