Algorithms for hard problems Parameterized complexity – definitions, sample algorithms Juris Viksna, 2015

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 problem L   *  * 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 M k for every n solves the problem in time f(k) n  For each k there is a constant c k, such that f(k) n  > n  + 1 for n  c k M' k :- simulates M k for n  c k - looks up value from the table for n  c k M' k solves the problem in time f(k) c k + n a+1

Some parameterized problems VERTEX COVER Instance:A graph G=(V,E) Parameter:A positive integer k Question:Is there a subset S  V, such that |S|=k and for all {x,y}  E either x  S or y  S? [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  running in time 2 k |G| for each k [Downey and Fellows 1992] GRAPH GENUS: A single algorithm  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 , a constant c and a function f, such that: - the running time of  (‹x,k › ) is at most f(k)|x| c - ‹x,k›  A iff  (‹x,k›) = 1. A problem L is strongly uniformly FPT, if L is uniformly FPT via some  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 {  k } k  N, such that for each k: - the running time of  k (‹x,k›) is at most f(k)|x| c - ‹x,k›  A iff  k (‹x,k›) = 1.

"klam" values Algorithms for VERTEX COVER: O(f(k) n 3 ) [Fellows, Langston 1986] O(f(k) n 2 ) [Johnson, 1987] 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 :) k FPT if solvable in time f(k)+n c klam value - the largest k for which f(k) is less that some universal "speed limit" U (e.g. U= ) [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 {v 1 }, G-{v 1 } {u 1 }, G-{u 1 } {v 1,v 2 }, G-{v 1,v 2 }{v 1,u 2 }, G-{v 1,u 2 } {u 2,v 2 } {u 1,v 1 } k+1 Theorem [Downey and Fellows 1992] VERTEX COVER is solvable in time O(2 k |V(G)|). By considering only graphs with vertices of degree 3 or higher, it is possible to shrink search space from 2 k to (5 1/4 ) k [Balasubramanian et al 1997]

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 Kernel method O((2k) k + k |V(G)|) Search tree method O(2 k |V(G)|) O((5 1/4 ) 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 + 2 k k 2 )[Balasubramanian et al 1992] O(kn + ((5 1/4 ) k k 2 )[Balasubramanian et al 1992]

Closest string problem