Contents. Typical techniques. Proving hardness. Constructing efficient algorithms

Transcription

1 Contents Typical techniques Proving hardness Constructing efficient algorithms

2 Generating Maximal Independent Sets Consider the generation of all maximal sets of an independence system.

3 Generating Maximal Independent Sets Consider the generation of all maximal sets of an independence system. I 2 V is an independence system if X I, Y X implies Y I. max I = {X I Y X Y I}

4 Generating Maximal Independent Sets Consider the generation of all maximal sets of an independence system. I 2 V is an independence system if X I, Y X implies Y I. max I = {X I Y X Y I} THEOREM: If there exists an algorithm which can generate, in total polynomial time, all the maximal independent sets of an arbitrary independence system I 2 V, represented by a membership oracle, then P=NP. (Lawler, Lenstra and Rinnooy Kan, 1980)

5 Generating Maximal Independent Sets Given a CNF Φ = (X 1 X 3 X 7 ) (X 2 X 3 X 5 ), on n variables, let V = {X 1, X 1, X 2, X 2,, X n, X n }, and let us define a subset I V to be independent if either there is an index j such that I {X j, X j } =, or I has a literal from all clauses of Φ and I {X j, X j } 1 for all j = 1, 2,..., n. Then, the only maximal independent sets are the sets of the form V \ {X j, X j } for j = 1, 2,..., n, and the sets formed by the true literals in the satisfying assignments of Φ. Clearly, Φ serves as a membership oracle for this independence system, and the generation should terminate in O(n) time, whenever Φ 0.

6 Contents Typical techniques Proving hardness Constructing efficient algorithms Closure operation Flashlight principle Supergraph approach Projection approach

7 Strategies for Generation Closure operation Let H be the family of maximal independent sets of an independence system H = max I 2 V represented by an independence oracle Ω, and let [, ] : H H H

8 Strategies for Generation Closure operation Let H be the family of maximal independent sets of an independence system H = max I 2 V represented by an independence oracle Ω, and let [, ] : H H H Call a subfamily C H closed if [A, B] C for all A, B C. Clearly, the intersection of any two closed subfamilies is also closed. Let C denote the smallest closed subfamily of H containing C (called the closure of C).

9 Strategies for Generation Closure operation Let H be the family of maximal independent sets of an independence system H = max I 2 V represented by an independence oracle Ω, and let [, ] : H H H Call a subfamily C H closed if [A, B] C for all A, B C. Clearly, the intersection of any two closed subfamilies is also closed. Let C denote the smallest closed subfamily of H containing C (called the closure of C). Theorem. If, by using the oracle Ω, a subfamily C H can be generated in polynomial time, for which C = H, and [A, B] can be computed in polynomial time for any A, B H, then H can be generated in incremental polynomial time.

10 Strategies for Generation Closure operation Let H be the family of maximal independent sets of an independence system H = max I 2 V represented by an independence oracle Ω, and let [, ] : H H H Call a subfamily C H closed if [A, B] C for all A, B C. Clearly, the intersection of any two closed subfamilies is also closed. Let C denote the smallest closed subfamily of H containing C (called the closure of C). Theorem. If, by using the oracle Ω, a subfamily C H can be generated in polynomial time, for which C = H, and [A, B] can be computed in polynomial time for any A, B H, then H can be generated in incremental polynomial time. Examples include: cycles of a graph (Dr. Folklore, Late-Pleistocene) prime implicants/implicates of a DNF/CNF (Blake, 1937; Quine, 1952; Robinson, 1965)

11 An Example for Applying a Closure Operation Consider a graph G = (V, E) and call a subset F E of the edges independent, if E \ F contains a cycle. This defines an independence system I for which the graph G serves as an oracle. The family H of maximal independent sets are the complements of simple cycles of G.

12 An Example for Applying a Closure Operation Consider a graph G = (V, E) and call a subset F E of the edges independent, if E \ F contains a cycle. This defines an independence system I for which the graph G serves as an oracle. The family H of maximal independent sets are the complements of simple cycles of G. Given cycles C, C E, for every edge e C C there exists a cycle C (C C ) \ {e}. Let [C, C ] = C.

13 An Example for Applying a Closure Operation Consider a graph G = (V, E) and call a subset F E of the edges independent, if E \ F contains a cycle. This defines an independence system I for which the graph G serves as an oracle. The family H of maximal independent sets are the complements of simple cycles of G. Given cycles C, C E, for every edge e C C there exists a cycle C (C C ) \ {e}. Let [C, C ] = C. Let F E be a spanning tree, and let C e denote the unique cycle contained in F {e} for e F. Let C = {C e e E \ F }.

14 An Example for Applying a Closure Operation Consider a graph G = (V, E) and call a subset F E of the edges independent, if E \ F contains a cycle. This defines an independence system I for which the graph G serves as an oracle. The family H of maximal independent sets are the complements of simple cycles of G. Given cycles C, C E, for every edge e C C there exists a cycle C (C C ) \ {e}. Let [C, C ] = C. Let F E be a spanning tree, and let C e denote the unique cycle contained in F {e} for e F. Let C = {C e e E \ F }. Then C = H.

15 An Example for Applying a Closure Operation Consider a graph G = (V, E) and call a subset F E of the edges independent, if E \ F contains a cycle. This defines an independence system I for which the graph G serves as an oracle. The family H of maximal independent sets are the complements of simple cycles of G. Given cycles C, C E, for every edge e C C there exists a cycle C (C C ) \ {e}. Let [C, C ] = C. Let F E be a spanning tree, and let C e denote the unique cycle contained in F {e} for e F. Let C = {C e e E \ F }. Then C = H. Therefore, H can be generated in incremental polynomial time.

16 Contents Typical techniques Proving hardness Constructing efficient algorithms Closure operation Flashlight principle Supergraph approach Projection approach

17 Strategies for Generation Flashlight Principle Let H be the family of maximal independent sets of an independence system H = max I 2 V represented by an independence oracle Ω. PeekingAhead(I, O) Given subsets I, O V, decide if there exists a maximal independent set H H for which H I and H O =.

18 Strategies for Generation Flashlight Principle Let H be the family of maximal independent sets of an independence system H = max I 2 V represented by an independence oracle Ω. PeekingAhead(I, O) Given subsets I, O V, decide if there exists a maximal independent set H H for which H I and H O =. Theorem. If, by using the oracle Ω, problem PeekingAhead(I, O) can be solved in polynomial time for all subsets I, O V, then H can be generated with polynomial delay (by depth-firstsearch).

19 Strategies for Generation Flashlight Principle Let H be the family of maximal independent sets of an independence system H = max I 2 V represented by an independence oracle Ω. PeekingAhead(I, O) Given subsets I, O V, decide if there exists a maximal independent set H H for which H I and H O =. Theorem. If, by using the oracle Ω, problem PeekingAhead(I, O) can be solved in polynomial time for all subsets I, O V, then H can be generated with polynomial delay (by depth-firstsearch). Examples include: maximal independent sets of a matroid, (Dr. Folklore, Late-Pleistocene) spanning trees, s t paths, s t cuts, simple cycles, etc., in graphs. (Reed and Tarjan, 1975)

20 An Example for Applying the Flashlight Principle Consider a graph G = (V, E) and call a subset F E of the edges independent, if F is a subset of the edge set of a complete bipartite subgraph (not necessarily induced) of G. This defines an independence system for which the graph G serves as an oracle.

21 An Example for Applying the Flashlight Principle Consider a graph G = (V, E) and call a subset F E of the edges independent, if F is a subset of the edge set of a complete bipartite subgraph (not necessarily induced) of G. This defines an independence system for which the graph G serves as an oracle. Problem PeekingAhead(I, O) can be solved in O( V 2 ) time for all subsets I, O E. (In fact, it would be enough to show this for subsets I which form a connected subgraph.) This yields an O( V 4 )-delay generation of all maximal bipartite subgraphs of G.

22 An Example for Applying the Flashlight Principle Consider a graph G = (V, E) and call a subset F E of the edges independent, if F is a subset of the edge set of a complete bipartite subgraph (not necessarily induced) of G. This defines an independence system for which the graph G serves as an oracle. Problem PeekingAhead(I, O) can be solved in O( V 2 ) time for all subsets I, O E. (In fact, it would be enough to show this for subsets I which form a connected subgraph.) This yields an O( V 4 )-delay generation of all maximal bipartite subgraphs of G. For other approaches see (Epstein, 1994) (Alexe, Alexe, Crama, Foldes, Hammer, and Simeone, 2002)

23 An Example for Applying the Flashlight Principle Consider a graph G = (V, E) and call a subset F E of the edges independent, if F is a subset of the edge set of a complete bipartite subgraph (not necessarily induced) of G. This defines an independence system for which the graph G serves as an oracle. Problem PeekingAhead(I, O) can be solved in O( V 2 ) time for all subsets I, O E. (In fact, it would be enough to show this for subsets I which form a connected subgraph.) This yields an O( V 4 )-delay generation of all maximal bipartite subgraphs of G. For other approaches see (Epstein, 1994) (Alexe, Alexe, Crama, Foldes, Hammer, and Simeone, 2002) Counting maximal bipartite subgraphs of a graph is a #P -hard problem. (Kuznetsov, 1989)

24 Contents Typical techniques Proving hardness Constructing efficient algorithms Closure operation Flashlight principle Supergraph approach Projection approach

25 Strategies for Generation Supergraph Approach Let H be the family of maximal independent sets of an independence system H = max I 2 V represented by an independence oracle Ω. Let us define a super-graph Ĝ with V (Ĝ) = H such that (i) for every H H the neighborhood N(H) in Ĝ can be generated, by using the oracle Ω, in incremental polynomial time, and (ii) the graph Ĝ is (strongly) connected.

26 Strategies for Generation Supergraph Approach Let H be the family of maximal independent sets of an independence system H = max I 2 V represented by an independence oracle Ω. Let us define a super-graph Ĝ with V (Ĝ) = H such that (i) for every H H the neighborhood N(H) in Ĝ can be generated, by using the oracle Ω, in incremental polynomial time, and (ii) the graph Ĝ is (strongly) connected. Theorem. If Ĝ satisfies the above conditions, then H can be generated in incremental polynomial time (by breadth-first-search). Furthermore, if N(H) can be generated in polynomial time for all H H, then H can be generated with polynomial delay.

27 Strategies for Generation Supergraph Approach Let H be the family of maximal independent sets of an independence system H = max I 2 V represented by an independence oracle Ω. Let us define a super-graph Ĝ with V (Ĝ) = H such that (i) for every H H the neighborhood N(H) in Ĝ can be generated, by using the oracle Ω, in incremental polynomial time, and (ii) the graph Ĝ is (strongly) connected. Theorem. If Ĝ satisfies the above conditions, then H can be generated in incremental polynomial time (by breadth-first-search). Furthermore, if N(H) can be generated in polynomial time for all H H, then H can be generated with polynomial delay. Examples include feedback arc and vertex sets in graphs and directed graphs (Schwikowski and Speckenmeyer, 1997)

28 An Example for Applying the Supergraph Approach Consider a directed graph G = (V, E) and call a subset F E of the arcs independent, if G F = (V, F ) is an acyclic graph. This defines an independence system for which the graph G serves as an oracle. Let H denote the family of all maximal independent sets.

29 An Example for Applying the Supergraph Approach Consider a directed graph G = (V, E) and call a subset F E of the arcs independent, if G F = (V, F ) is an acyclic graph. This defines an independence system for which the graph G serves as an oracle. Let H denote the family of all maximal independent sets. Let us define a directed graph Ĝ on vertex set H, in which ( F, F ) is an arc if F can be obtained from F by (i) adding an arc ( u, v) E \ F, (ii) deleting all arcs of F leaving vertex v, and (iii) adding additional arcs from E \ F in an arbitrary way until a maximal acyclic arc set F is obtained.

30 An Example for Applying the Supergraph Approach Consider a directed graph G = (V, E) and call a subset F E of the arcs independent, if G F = (V, F ) is an acyclic graph. This defines an independence system for which the graph G serves as an oracle. Let H denote the family of all maximal independent sets. Let us define a directed graph Ĝ on vertex set H, in which ( F, F ) is an arc if F can be obtained from F by (i) adding an arc ( u, v) E \ F, (ii) deleting all arcs of F leaving vertex v, and (iii) adding additional arcs from E \ F in an arbitrary way until a maximal acyclic arc set F is obtained. The directed supergraph Ĝ is strongly connected, N(F ) E for all F H, and N(F ) can be generated in polynomial time, yielding an O( V E ( V + E ))-delay generation of H.

31 Contents Typical techniques Proving hardness Constructing efficient algorithms Closure operation Flashlight principle Supergraph approach Projection approach

32 Strategies for Generation Projection Approach Let H = max I be the family of maximal independent sets of an independence system I 2 V represented by an independence oracle Ω. Let V = {1, 2,..., n}, and for each j V let H j denote the family of independent sets which are maximal within {1, 2,..., j}.

33 Strategies for Generation Projection Approach Let H = max I be the family of maximal independent sets of an independence system I 2 V represented by an independence oracle Ω. Let V = {1, 2,..., n}, and for each j V let H j denote the family of independent sets which are maximal within {1, 2,..., j}. Lemma. H 1 H 2 H n 1 H n and H n = H. Proof. For all I H j 1 we have either I H j or I {j} H j.

34 Strategies for Generation Projection Approach Let H = max I be the family of maximal independent sets of an independence system I 2 V represented by an independence oracle Ω. Let V = {1, 2,..., n}, and for each j V let H j denote the family of independent sets which are maximal within {1, 2,..., j}. Lemma. H 1 H 2 H n 1 H n and H n = H. Proof. For all I H j 1 we have either I H j or I {j} H j. Theorem. H can be generated in incremental polynomial time, if all independent sets, which are maximal within I {j}, can be generated in polynomial time, for all j V and I H j 1.

35 Strategies for Generation Projection Approach Let H = max I be the family of maximal independent sets of an independence system I 2 V represented by an independence oracle Ω. Let V = {1, 2,..., n}, and for each j V let H j denote the family of independent sets which are maximal within {1, 2,..., j}. Lemma. H 1 H 2 H n 1 H n and H n = H. Proof. For all I H j 1 we have either I H j or I {j} H j. Theorem. H can be generated in incremental polynomial time, if all independent sets, which are maximal within I {j}, can be generated in polynomial time, for all j V and I H j 1. Examples include maximal cliques and stable sets of graphs, maximal set packings, maximal feasible solutions to a knapsack inequality, etc. (Lawler,Lenstra and Rinnooy Kan, 1980)

36 An Example for Applying the Projection Approach Consider a directed graph G = (V, E) and a constant t. Let us call a subset I V of the vertices t-independent, if the subgraph induced by I contains no more than t edges. This defines an independence system for which the graph G serves as an oracle. Let H denote the family of all maximal t-independent sets.

37 An Example for Applying the Projection Approach Consider a directed graph G = (V, E) and a constant t. Let us call a subset I V of the vertices t-independent, if the subgraph induced by I contains no more than t edges. This defines an independence system for which the graph G serves as an oracle. Let H denote the family of all maximal t-independent sets. Let V = {1, 2,..., n} and let H j for j V denote the family of all t- independent sets which are maximal within {1, 2,..., j}.

38 An Example for Applying the Projection Approach Consider a directed graph G = (V, E) and a constant t. Let us call a subset I V of the vertices t-independent, if the subgraph induced by I contains no more than t edges. This defines an independence system for which the graph G serves as an oracle. Let H denote the family of all maximal t-independent sets. Let V = {1, 2,..., n} and let H j for j V denote the family of all t- independent sets which are maximal within {1, 2,..., j}. Given I H j 1, all t-independent sets which are maximal within I {j} can be generated in O(n t ) time, yielding an incremental polynomial algorithm to generate H (as long as t is a constant).