Polynomial-delay Enumeration Algorithms in Set Systems

04/16/2020
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by   Kazuya Haraguchi, et al.
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We consider a set system (V, 𝒞⊆ 2^V) on a finite set V of elements, where we call a set C∈𝒞 a component. We assume that two oracles L_1 and L_2 are available, where given two subsets X,Y⊆ V, L_1 returns a maximal component C∈𝒞 with X⊆ C⊆ Y; and given a set Y⊆ V, L_2 returns all maximal components C∈𝒞 with C⊆ Y. Given a set I of attributes and a function σ:V→ 2^I in a transitive system, a component C∈𝒞 is called a solution if the set of common attributes in C is inclusively maximal; i.e., ⋂_v∈ Cσ(v)⊋⋂_v∈ Xσ(v) for any component X∈𝒞 with C⊊ X. We prove that there exists an algorithm of enumerating all solutions (or all components) in delay bounded by a polynomial with respect to the input size and the running times of the oracles.

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1 Introduction

Let be a finite set of elements. A set system on a set of elements is defined to be a pair of of elements and a family , where a set in is called a component. For a subset in a system , a component with is called -maximal if no other component satisfies , and let denote the family of all -maximal components. For two subsets , let denote the family of components such that . We call a set function from to the set of reals a volume function if for any subsets . A subset is called -positive if . To discuss the computational complexities for solving a problem in a system, we assume that a system is implicitly given as two oracles and such that

  • given non-empty subsets , returns a component (or if no such exists) in time and space; and

  • given a non-empty subset , returns in time and space.

Given a volume function , we assume that whether holds or not can be tested in time and space. We also denote by an upper bound on , where we assume that is a non-decreasing function in the sense that holds for any subsets .

We define an instance to be a tuple of a set of elements, a family , a set of items and a function . Let be an instance. The common item set over a subset is defined to be . A solution to instance is defined to be a component such that

every component with satisfies .

Let denote the family of all solutions to instance . Our aim is to design an efficient algorithm for enumerating all solutions in .

We call an enumeration algorithm
 - output-polynomial if the overall computation time is polynomial with respect to
 the input and output size;
 - incremental-polynomial if the computation time between the -th output and
 the -st output is bounded by a polynomial with respect to
 the input size and ; and
 - polynomial-delay if the delay (i.e., the time between any two consecutive outputs),
 preprocessing time and postprocessing time are all bounded by a polynomial
 with respect to the input size.
In this paper, we design an algorithm that enumerates all solutions in by traversing a family tree over the solutions in , where the family tree is a tree structure that represents a parent-child relationship among solutions. The following theorem summarizes our main result.

Theorem 1

Let be an instance on a set system with a volume function , where and . All -positive solutions in to the instance can be enumerated in delay and in space.

The problem is motivated by enumeration of solutions in an instance such that is transitive. We call a system transitive if any tuple of components with implies . For such an instance, we proposed an algorithm in [9] that enumerates all solutions such that the delay is bounded by a polynomial with respect to the input size and the running times of oracles. The proposed algorithm yields the first polynomial-delay algorithms for enumerating connectors in an attributed graph [1, 6, 7, 8, 9, 12, 13, 14, 15, 16] and for enumerating all subgraphs with various types of connectivities such as all -edge/vertex-connected induced subgraphs and all -edge/vertex-connected spanning subgraphs in a given undirected/directed graph for a fixed .

It is natural to ask whether the result in [9] is extensible to an instance with a general set system. This paper gives an affirmative answer to the question; even when we have no assumption on the system of a given instance , there is an algorithm that enumerates all solutions in polynomial-delay with respect to the input size and the running times of oracles.

The paper is organized as follows. We prepare notations and terminologies in Section 2. In Section 3, we present a polynomial-delay algorithm that enumerates all solutions in an instance such that is an arbitrary set system. We also show that all components are enumerable in polynomial-delay, using the algorithm. Finally we conclude the paper in Section 4.

2 Preliminaries

Let (resp., ) denote the set of reals (resp., non-negative reals). For a function for a finite subset and a subset , we let denote .

For two integers and , let denote the set of integers with . For a set with a total order over the elements in , we define a total order over the subsets of as follows. For two subsets , we denote by if the minimum element in belongs to . We denote if or . Note that holds whenever . Let denote the maximum element in . Then holds for , and , , if and only if the sequence of length with is lexicographically smaller than the sequence of length with . Hence we see that is a total order on .

Suppose that an instance is given. To facilitate our aim, we introduce a total order over the items in by representing as a set of integers. We define subsets and for each item . For each non-empty subset , define subset . For , define . For each subset , let denote the minimum item in , where for . For each , define a family of solutions in ,

Note that is a disjoint union of , . In Section 3.5, we will design an algorithm that enumerates all solutions in for any specified integer .

3 Enumerating Solutions

For a notational convenience, let for each item denote the family of components and let for each subset denote the family of components.

We can test whether a given component is a solution or not as follows.

Lemma 1

Let be an instance, be a component in and .

  1. if and only if ; and

  2. Whether is a solution or not can be tested in delay and in space.

Proof: (i) Note that . By definition, if and only if there is a component such that and , where a maximal one of such components belongs to . Hence if no such component exists then . Conversely, if then no such component exists.

(ii) Let be a subset such that . We claim that holds if and only if L returns the component . The necessity is obvious. For the sufficiency, if there is such that , would be a superset of , contradicting the -maximality of . By (i), to identify whether or not, it suffices to see whether L returns . We can compute in time and in space, and can decide whether the oracle returns in time and in space.  

3.1 Defining Family Tree

To generate all solutions in efficiently, we use the idea of family tree, where we first introduce a parent-child relationship among solutions, which defines a rooted tree (or a set of rooted trees), and we traverse each tree starting from the root and generating the children of a solution recursively. Our tasks to establish such an enumeration algorithm are as follows:

  • Select some solutions from the set of solutions as the roots, called “bases;”

  • Define the “parent” of each non-base solution , where the solution is called a “child” of the solution ;

  • Design an algorithm A that, given a solution , returns its parent ; and

  • Design an algorithm B that, given a solution , generates a set of components such that contains all children of . We can test whether each component is a child of by constructing by algorithm A and checking if is equal to .

Starting from each base, we recursively generate the children of a solution. The complexity of delay-time of the entire algorithm depends on the time complexity of algorithms A and B, where is bounded from above by the time complexity of algorithm B.

3.2 Defining Base

For each integer , define a set of components

,

and . We call each component in a base.

Lemma 2

Let be an instance.

  1. For each non-empty set or , it holds that ;

  2. For each , any solution is contained in a base in ; and

  3. and .

Proof: (i) Let be a component in . Note that holds. When (i.e., ), no proper superset of is a component, and is a solution. Consider the case of . To derive a contradiction, assume that is not a solution; i.e., there is a proper superset of such that . Since , we see that . This, however, contradicts the -maximality of . This proves that is a solution.

(ii) We prove that each solution is contained in a base in . Note that holds. By definition, it holds that . Let be a solution. Note that holds. Since for (resp., for ), we see that . This proves that is a base in . Therefore is contained in a base .

(iii) Let . We see from (i) that , which implies that . We prove that any solution is a base in . By (ii), there is a base such that , which implies that and . We see that , since for , and for . Hence would contradict that is a solution. Therefore , as required.  

Lemma 2(iii) tells that all solutions in can be found by calling oracle for and . In the following, we consider how to generate all solutions in for each item .

3.3 Defining Parent

This subsection defines the “parent” of a non-base solution.

For two subsets , we denote if “” or “ and ” and let mean or .

Let be a subset such that . We call a solution a superset solution of if and . A superset solution of is called minimal if no proper subset is a superset solution of . We call a minimal superset solution of the lex-min solution of if for all minimal superset solutions of . For each item , we define the parent of a non-base solution to be the lex-min solution of , and define a child of a solution to be a non-base solution such that .

The next lemma tells us how to find the item set of the parent of a given solution .

Lemma 3

Let be an instance, be a non-base solution for some item , and denote the lex-min solution of . Denote by so that . For each integer , holds if and only if holds for the item set .

Proof: By Lemma 2(i) and , we see that for any integer .
Case 1. : For any subset , the family is equal to and cannot contain any minimal superset solution of . This implies that .
Case 2. : Let be an arbitrary component in . Then is a solution by Lemma 2(i). Observe that and , implying that is a superset solution of . Then contains a minimal superset solution of , where and . If or , then would hold, contradicting that is the lex-min solution of . Hence and .  

The next lemma tells us how to construct the parent of a given solution .

Lemma 4

Let be an instance, be a non-base solution for some item , and denote the lex-min solution of . Let . Let be a set such that , where is denoted by such that . Then:

  1. for any vertex ;

  2. Every component with satisfies ;

  3. There is an integer such that for each and all components satisfy ;

  4. For the integer in (iii), holds; and

  5. For the integer in (iii), if then holds.

Proof: (i) Since , there exists a vertex . For such a vertex , is a component such that . If is not a -maximal component, then there would exist a component with and , contradicting that is a solution. Hence for any vertex .

(ii) Let be a component with . Note that and . Since is a component, there is a solution such that and . Since , and are distinct solutions and there must be a minimal superset solution of such that , where we see that and . If , then implies that , contradicting that is the lex-min solution of .

(iii) By (i), for some integer , holds and some component satisfies . Let denote the smallest index such that no component satisfies for each . By (ii), for such , the statement of (iii) holds.

(iv) Since no component satisfies for all integers , no component such that can be the lex-min solution . Since some component satisfies , there is a component such that and . The lex-min solution satisfies for all minimal superset solutions of with . Therefore must contain .

(v) By (iv), . If then is a unique minimal superset solution of such that , implying that .  

Lemma 5

Let be an instance, be a non-base solution for some item . Then Parent in Algorithm 1 correctly delivers the lex-min solution of in time and in space.

Proof: Let denote the lex-min solution of . The item set constructed in the first for-loop (lines 5 to 9) satisfies by Lemma 3. The second for-loop (lines 12 to 19) picks up by Lemma 4(iv), and the termination condition (line 13) is from Lemma 4(v).

The first for-loop is repeated times, where we can decide whether the condition in line 6 holds in time and in space. The time and space complexities of the first for-loop are and .

We can decide the set in time and in space.

The second for-loop is repeated times. We can decide whether the condition of line 13 is satisfied by calling the oracle L, which takes time and space. When the condition of line 13 is satisfied, we can decide whether or not (line 15) in time and in space by Lemma 1(ii). The time and space complexities of the second for-loop are and .

The overall time and space complexities are and .  

1:An instance , an item , and a non-base solution , where .
2:The lex-min solution of .
3:Let , where ;
4:;
5:for each integer  do
6:     if   then
7:         
8:     end if
9:end for; holds
10:Let , where ;
11:;
12:for each integer  do
13:     if  then
14:         ;
15:         if  then
16:              Output and halt
17:         end if
18:     end if
19:end for
Algorithm 1 Parent: Finding the lex-min solution of a solution

3.4 Generating Children

This subsection shows how to construct a family of components for a given solution so that contains all children of .

Lemma 6

Let be an instance and be a solution for some item . Then:

  1. Every child of satisfies and is a component in for any item ;

  2. The family of children of is equal to the disjoint collection of families Parent over all items ; and

  3. The set of all children of can be constructed in time and space.

Proof: (i) Note that since . Since are both solutions, . Hence . Let be an arbitrary item in . We see since and . To show that is a component in , suppose that there is a component such that . Since and , we see that . Then should not be a solution since otherwise it would be a superset solution of such that , contradicting that is a minimal superset solution of . Since is not a solution but a component, there is a solution such that and . Hence . Such a solution contains a minimal superset solution of such that and . Then we have , and thus holds, which contradicts that is the lex-min solution of . Therefore, such does not exist, implying that .

(ii) By (i), the family of children of is contained in the family of -maximal components over all items . Hence