A Simple Gap-producing Reduction for the Parameterized Set Cover Problem Bingkai Lin National Institute of Informatics [email protected]

April 11, 2018 Abstract Given an n-vertex bipartite graph I = (S, U, E), the goal of set cover problem is to find a minimum sized subset of S such that every vertex in U is adjacent to some vertex of this subset. It is NP-hard to approximate set cover to within a (1 βˆ’ o(1)) ln n factor [14]. If we use the sized of the optimum solution k as the parameter, then it can be solved in nk+o(1) time [16]. A natural question is: can we approximate set cover to within an o(ln n) factor in nkβˆ’ time? In [25], Karthik, Bundit and Pasin showed that assuming the Strong Exponential Time Hypothesis (SETH), for any computable function f, no f(k) Β· nkβˆ’ -time algorithm that can approx1

imate set cover to a factor below (log n) poly(k,e()) for some function e. They also obtained a hardness approximation ratio of (log n)1/poly(k) for f(k) Β· ndk/2eβˆ’ -time algorithms assuming the k-SUM hypothesis and the same ratio for f(k)no(k) -time algorithms assuming the Exponential Time Hypothesis (ETH) and for f(k) Β· nO(1) -time algorithms assuming W[1] 6= FPT . This paper presents a simple reduction which improves these hardness approximation ratios q to (1 βˆ’ o(1)) Β·

1

k

log n . log log n

Introduction

We consider the set cover problem (S ET C OVER): given an n-vertex bipartite graph I = (S, U, E), where U is the underlying universe set and S represents the set family, find a minimum sized subset C of S such that every vertex of U is adjacent to some vertex of C. We use S(I), U(I) and opt(I) to denote the sets S, U and the minimum size of the solution of I respectively. A vertex u ∈ U is covered by a subset C βŠ† S if u is adjacent to some vertex of C. The set cover problem is NP-hard [24]. Unless P = NP, we do not expect to solve it in polynomial time. One way to handle NP-hard problems is to use approximation algorithms. An algorithm of S ET C OVER achieves an rapproximation if for every input instance I, it returns a subset C of S(I) such that C covers U(I) and |C| 6 r Β· opt(I). The polynomial time approximability of S ET C OVER is well-understood: the greedy algorithm can output a solution of size at most opt(I) Β· ln n [11, 22, 27, 32, 33] and it was shown that no polynomial time algorithm can achieve an approximation factor within (1 βˆ’ o(1)) ln n unless P = NP [4, 14, 17, 28, 31]. On the other hand, if we take the optimum solution size k = opt(I) as a parameter, then the simple brute-force searching algorithm can solve this problem in nk+1 time. Assuming the exponential time hypothesis (ETH) [20, 21], i.e., 3-SAT on n variables cannot be solved in 2o(n) time, there is no no(k) time algorithm for S ET C OVER. Under the strong exponential time hypothesis (SETH) [20, 21], which claims that for any  ∈ (0, 1) there exists a d > 3 such that d-SAT on n variables cannot be solved in 2(1βˆ’)n time, we can further rule out nkβˆ’ -time algorithm for set cover for any  > 0. Unlike ETH, SETH is not so widely believed. Some lower bounds based on SETH even appeared as a possible approach to obtaining faster algorithms for SAT [30]. Before seeking an nkβˆ’ -time algorithm for S ET C OVER to refute SETH, one should consider an nkβˆ’ -time approximation algorithm first, i.e., 1

Question 1.1. Is there any o(ln n)-approximation algorithm for the parameterized set cover problem with running time nkβˆ’ ? Exponential time approximation algorithms for the unparameterised version of set cover problem were studied in [7, 13]. It was shown that for any ratio r, there is a (1 + ln r)-approximation algorithm for S ET C OVER with running time 2n/r nO(1) . No nkβˆ’ time algorithm for S ET C OVER achieving an approximation ratio in o(ln n) is known in literature. On the other hand, proving inapproximability for a parameterized problem is not an easy task. In fact, even the constant FPT-approximability, i.e., the existence of f(k) Β· nO(1) -time algorithm for any computable function f (henceforth referred to as FPT-algorithm) with constant approximation, has been open for many years [29]. Lacking techniques like PCP-theorem [5], many results on the parameterized inapproximability of set cover problem have to use strong conjectures [10, 19, 6, 8] to create a gap in the first place. It is of great interest to develop techniques to prove hardness of approximation for parameterized problems using only hypothesis such as SET H, ET H or even weaker assumptions like W[1] 6= FPT or W[2] 6= FPT [15, 18] from the parameterized complexity theory. The success of this quest might extend the arsenal of methods for proving hardness of approximation and lead to PCP-like theorems for Fine-Grained Complexity [3]. The first constant FPT-inapproximability result for parameterized S ET C OVER based on W[1] 6= FPT was given by [9] using the one-sided gap of B ICLIQUE from [26]. In fact, [9] deals with dominating set problem, which is essentially the same as S ET C OVER. Recently, Karthik, Bundit O(1) and Pasin [25] significantly improved the FPT-inapproximation factor to (log n)1/k under the O(1) 1/k hypothesis W[1] 6= FPT . They also rule out the existence of (log n) -approximation algorithm with running time f(k) Β· no(k) for any computable function f, assuming ETH, and the existence 1

of (log n) (k+e())O(1) -approximation algorithms with running time f(k) Β· nkβˆ’ , assuming SETH. 1 Their approach is to first establish a (log n) Ω(k) gap for M AX C OVER, then reduce M AX C OVER to 1

S ET C OVER and obtain a (log n) Ω(k2 ) -gap. This paper presents a new technique q which allows us to

design simple reductions improving the inapproximation factor to (1 βˆ’ ) Β·

k

log n log log n .

Theorem 1.2. Assuming SETH, for every , Ξ΄ ∈ (0, 1) and computable function f : N β†’ N, there is no f(k) Β· Nkβˆ’ time algorithm that can, given an N-vertex set cover instance I, distinguish between β€’ opt(I) 6 k, β€’ opt(I) >

1 1+Ξ΄



log N log log N

 k1

.

Theorem 1.3. Assuming ETH, there is a constant  ∈ (0, 1) such that for every Ξ΄ ∈ (0, 1) and computable function f : N β†’ N, no f(k) Β· Nk time algorithm that can, given an N-vertex set cover instance I , distinguish between β€’ opt(I) 6 k, β€’ opt(I) >

1 1+Ξ΄

Β·



log N log log N

 k1

.

Behind these results is a reduction which, given an integer k, an n-vertex set cover instance k I and an integer h 6 O(log n/ log log n), produces an nO(1) Β· (|U(I)|)O(h ) -vertex instance I 0 k in nO(1) Β· |U(I)|O(h ) time such that if opt(I) 6 k then opt(I 0 ) 6 k, otherwise opt(I 0 ) > h. Therefore, to prove the parameterized inapproximability of S ET C OVER, it suffices to show the hardness of S ET C OVER when the input instances have logarithmic sized universe set. Note that the standard reduction for SETH-hardness of set cover parameterized by the solution size k produces instances I with |U(I)| = O(k log |S(I)|). With our reduction, this immediately yields the above theorems. Let us not fail to mention that the results of [25] also imply the hardness of S ET C OVER with logarithmic sized universe set assuming the k-SUM hypothesis and W[1] 6= FPT hypothesis respectively. Similarly, we can obtain the corresponding inapproximability for set cover based

2

on each of these hypotheses as well. In particular, using a simple trick, we can even rule out (log N)1/(k) -approximation FPT-algorithm of set cover for any unbounded computable function  under W[1] 6= FPT . Theorem 1.4. Assuming k-SUM hypothesis for any Ξ΄,  ∈ (0, 1) and computable function f : N β†’ N, there is no f(k) Β· Ndk/2eβˆ’ time algorithm that can, given an N-vertex set cover instance I, distinguish between β€’ opt(I) 6 k, β€’ opt(I) >

1 1+Ξ΄



log N log log N

 k1

.

Theorem 1.5. Assuming W[1] 6= FPT , for and computable function f : N β†’ N and unbounded computable function  : N β†’ N, there is no f(k) Β· NO(1) -time algorithm that can, given an N-vertex set cover instance I, distinguish between β€’ opt(I) 6 k, β€’ opt(I) > log N1/(k) . The main technique contribution of this paper is to introduce a gadget that can be used to design gap-producing reductions for the parameterized set cover problem and provide a construction of this gadget using universal sets.

2

Preliminaries

For n, k ∈ N, an (n, k)-universal set is a set of binary strings with length n, such that the restriction to any k indices contains all the 2k possible binary configurations. √ Lemma 2.1. [See Sections 10.5 and 10.6 of [23]] For k2k 6 n, (n, k)-universal sets of size n can be computed in n3 time. Hypotheses

Below is a list of hardness hypotheses we will use in this paper.

β€’ W[1] 6= FPT : for any computable function f : N β†’ N, no algorithm can, given an n-vertex graph G and an integer k, decide if G contains a k-clique in f(k) Β· nO(1) time. β€’ W[2] 6= FPT : for any computable function f : N β†’ N, there is no algorithm which, given an n-vertex set cover instance I and an integer k, decides if opt(I) 6 k in f(k) Β· nO(1) time. β€’ Exponential Time Hypothesis (ETH)[20, 21]: there exists a Ξ΄ ∈ (0, 1) such that 3-SAT on n variables cannot be solved in 2Ξ΄n time. β€’ Strong Exponential Time Hypothesis (SETH)[20, 21] for any  ∈ (0, 1) there exists d > 3 such that d-SAT on n variables cannot be solved in 2(1βˆ’)n time. β€’ k-SUM hypothesis (k-SUM) [1]: for every k > 2 and  > 0, no O(ndk/2eβˆ’ ) time algorithm 2k 2k can, given k sets S1 , . . . , Sk each Pwith n integers in [βˆ’n , n ], decide if there are k integers x1 ∈ S1 , . . . , xk ∈ Sk such that i∈[k] xi = 0. We refer the reader to [18, 15] for more information about the parameterized complexity hypotheses. Using the Sparsification lemma [21], we can assume that the instances of 3-SAT in ETH have Cn clauses for some constant C and the instances of k-SAT in SETH have Ck, n clauses where Ck, depends on k and .

3

3

Reductions

We start with the definition of (k, n, m, `, h)-gap-gadgets. In Lemma 3.2, we show how to use theses gadgets to create an (h/k)-gap for the set cover problem. Lemma 3.3 gives a polynomial time construction of gap-gadgets with h 6 O(log n/ log log n) and ` = hk . Since for every input instance I = (U, S, E) of set cover, our reduction runs in time |S|O(1) |U|` . If |U| = Ω(n), we can not afford such running time. Our next step is to prove the hardness of set cover with U = f(k) Β· (log n)O(1) based on each of the aforementioned hypotheses. Definition 3.1 ( (k, n, m, `, h)-Gap-Gadget). A (k, n, m, `, h)-Gap-Gadget is a bipartite graph T = (A, B, E) satisfying the following conditions. (G1) A is partitioned into (A1 , A2 , . . . , Am ). For every i ∈ [m], |Ai | = `. (G2) B is partitioned into (B1 , B2 , . . . , Bk ). For every j ∈ [k], |Bj | = n. (G3) For all b1 ∈ B1 , b2 ∈ B2 , . . . bk ∈ Bk , there exist a1 ∈ A1 , . . . , am ∈ Am such that for all i ∈ [m] and j ∈ [k], ai is adjacent to bj . (G4) For all X βŠ† B and a1 ∈ A1 , . . . , am ∈ Am , if every ai has at least k + 1 neighbors in X, then |X| > h. Lemma 3.2. There is an algorithm which, given an integer k, an instance I = (S, U, E) of S ET C OVER, where S = S1 βˆͺ S2 . . . βˆͺ Sk and |Si | = n for all i ∈ [k], and a (k, n, m, `, h)-Gap-Gadget, outputs a set cover instance I 0 = (S 0 , U 0 , E 0 ) with S 0 = S and U 0 = m|U|` in |U|` Β· nO(1) time such that β€’ if there exist s1 ∈ S1 , . . . , sk ∈ Sk that can cover U, then opt(I 0 ) 6 k; β€’ if opt(I) > k, then opt(I 0 ) > h. Proof. Let T = (A, B, ET ) be the (k, n, m, `, h)-Gap-Gadget. Without loss of generality, assume that for all i ∈ [k] Bi = Si . The new instance I 0 = (S 0 , U 0 , E 0 ) is defined as follows. β€’ S 0 = S. S β€’ U 0 = ( i∈[m] UAi ). β€’ For all s ∈ S 0 and f ∈ UAi where i ∈ [m], E 0 contains {s, f} if there exists an a ∈ Ai such that (E’1) {s, f(a)} ∈ E, (E’2) {a, s} ∈ ET . Completeness. If opt(I) 6 k, then there exist s1 ∈ S1 , . . . , sk ∈ S that can cover the whole set U. We will show that for every f ∈ U 0 , f is covered by some vertex in {s1 , s2 , Β· Β· Β· , sk }. Firstly, by (G3), there exist a1 ∈ A1 , . . . , am ∈ Am such that ai sj ∈ ET for all i ∈ [m] and j ∈ [k]. Assume that f ∈ UAi for some i ∈ [m]. Observe that f(ai ) ∈ U must be covered by some sj with j ∈ [k], i.e., {sj , f(ai )} ∈ E. Since {ai , sj } ∈ ET and {sj , f(ai )} ∈ E, according to the definition of E 0 , we must have {sj , f} ∈ E 0 . Soundness. Suppose opt(I) > k. Let X βŠ† S 0 be a set covering U 0 . For every a ∈ A, let NT (a) be the set of neighbors of a in T . We have the following claim. Claim 1. For every i ∈ [m] there exists ai ∈ Ai such that |NT (ai ) ∩ X| > k + 1. Proof of Claim 1. Suppose there exists an i ∈ [m] such that for all a ∈ Ai , |NT (a) ∩ X| 6 k. Since opt(I) > k, every solution of I has size at least k + 1. It follows that for every a ∈ Ai , there exists some ua ∈ U such that ua is not covered by NT (a)∩X in the set cover instance I. Define a function f ∈ UAi such that f(a) = ua for every a ∈ Ai . We claim that f is not covered by X. Otherwise, suppose there exists an s ∈ X that can cover f. According to the definition of E 0 , there must exists an a ∈ Ai such that (E’1) and (E’2) hold. However, if s ∈ NT (a) ∩ X, then {s, f(a)} = {s, ua } ∈ / E. On the other hand, if s ∈ / NT (a) ∩ X, then {a, s} ∈ / ET . In both cases, we obtain contradictions. 4

a By Claim 1, we can pick ai ∈ Ai for each i ∈ [m] such that every ai has at least k + 1 neighbors in X. By the property of Gap-Gadget, |X| > h.

3.1

Construction of Gap-Gadgets

Lemma 3.3. There is an algorithm that can, for every k, h, n ∈ N with k log log n 6 log n and h 6 (2+)loglogn log n , compute a (k, n, n log h, hk , h)-Gap-Gadget in n4 time. Proof. Let m = n log h and K = h log h. Note that (log m)/2 = √ (log n + log log h)/2 > (2 + )h log h/2 > log h + log log h + h log h = log K + K, i.e., K2K 6 m. By Lemma 2.1, an (m, K)universal set S = {s1 , s2 , . . . , sm } can be constructed in m3 6 n4 time. Partition every s ∈ S into m n = log h blocks so that each block has length log h. Interpret the values of blocks as integers in [h]. We obtain an m Γ— n matrix M by setting the value Mr,c equal to the value of the c-th block of sr . The matrix M satisfies the following conditions. (M1) For all r ∈ [m] and c ∈ [n], Mr,c ∈ [h]. (M2) For any set C βŠ† [n] with |C| 6 h, there exists a row r ∈ [m] such that |{Mr,c : c ∈ C}| = |C|. Condition (M1) is obvious. To see why (M2) holds, for each C βŠ† [n] with |C| 6 h, let C 0 be the set of indices corresponding to the blocks in C. Note that |C 0 | = |C| log h 6 h log h = K. By the property of (m, K)-universal set, there exists an sr ∈ S such that each block in C takes distinct value. It follows that |{Mr,c : c ∈ C}| = |C|. For each i ∈ [m], let Ai = {(a1 , a2 , . . . , ak ) : for all j ∈ [k], aj ∈ [h]}. Note that |Ai | = hk . For each j ∈ [k], let Bj = [n]. Let T = (A, B, E) be a bipartite graph with S β€’ A = i∈[m] Ai . S β€’ B = j∈[k] Bj . β€’ E = {{~ a, b} : a ~ ∈ Ai , b ∈ Bj and Mi,b = a ~ [j] for all j ∈ [k]}. We will show that T is an (k, n, m, hk , h)-gap-gadget. Obviously, T satisfies (G1) and (G2). T satisfies (G3). For any b1 ∈ B1 , b2 ∈ B2 , . . . , bk ∈ Bk . We define a ~ i ∈ Ai by setting a ~ i = (Mi,b1 , Mi,b2 , . . . , Mi,bk ). It is routine to check that {~ ai , bj } ∈ E for all i ∈ [m] and j ∈ [k]. T satisfies (G4). Let X βŠ† B and a ~ 1 ∈ A1 , a ~ 2 ∈ A2 , . . . , a ~ m ∈ Am . Suppose for every i ∈ [m], a ~ i has at least k + 1 neighbors in X and |X| 6 h. By (M2), there exists an r ∈ [m] such that |{Mr,c : c ∈ X}| = |X|. Since a ~ r has at least k + 1 neighbors in X, there exists an j ∈ [k] such that a ~r has two neighbors b, b 0 in X ∩ Bj . According to the definition of E, we must have Mr,b = Mr,b 0 = a ~ r [j]. This contradicts the fact that |{Mr,c : c ∈ X}| = |X|. The construction above produces gap-gadgets with ` = hk . Note that the parameter h is related to the inapproximation factor we will get for the set cover problem and the running time of our reduction is nO(1) |U|` . We want to set h as large as possible while keeping the running time of reduction in f(k) Β· nO(1) . Assuming |U| = g(k) Β· (log n)O(1) , the best we can achieve is h = (log n/ log log n)1/k . 5

3.2

Proofs of Theorem 1.2 and Theorem 1.3

Lemma 3.4. There is an algorithm, which given k ∈ N, Ξ΄ ∈ (0, 1) with (1 + 1/k3 )1/k 6 (1 + Ξ΄)/(1 + Ξ΄/2) and (1 + Ξ΄/2)k > 2k4 and a SAT instance Ο† with n variables and Cn clauses, where n is much 3 larger than k and C, outputs an integer N 6 2n/k+n/k and a set cover instance I satisfying the 5n/k following conditions in 2 time. β€’ |S(I)| + |U(I)| 6 N. β€’ If Ο† is satisfiable, then opt(I) 6 k. β€’ If Ο† is not satisfiable, then opt(I) >

1 1+Ξ΄

Β·

q k

log N log log N .

Proof. Let k be a positive integer and Ο† be a CNF with n variables and Cn clauses. We first construct a set cover instance I 0 = (S 0 , U 0 , E 0 ) as follows. Partition the variable set into k parts, each having at most dn/ke variables. For each i ∈ [k], let Si be the set of assignments to the i-th part. Let S 0 = S1 βˆͺ Β· Β· Β· βˆͺ Sk . Let U 0 be the set consisting of all the clauses of Ο† and k additional nodes u1 , u2 , . . . , uk . For every i ∈ [k] and assignment s ∈ Si , we add an edge between s and ui . If the assignment s ∈ S 0 satisfies a clause u ∈ U 0 , we also add an edge between u and s. The set cover instance I 0 has the following properties. β€’ If Ο† is satisfiable, then opt(I 0 ) = k. Moreover, there exist k vertices s1 ∈ S1 , Β· Β· Β· , sk ∈ Sk that can cover the whole set U 0 . β€’ If Ο† is not satisfiable, then opt(I 0 ) > k. β€’ |U 0 | = k + Cn. β€’ |S 0 | 6 k2n/k . 3

3

Let M = k2n/k > |S| and N = M1+1/k 6 2n/k+n/k . Note that log M/ log log M >q n/(k log n) >

k. Applying Lemma 3.3 with k ← k, n ← M, ` ←

log M ,h (1+Ξ΄/2)k log log M 4 5n/k

←

1 1+Ξ΄/2

Β·

k

log M log log M

and

m ← M log h 6 M log log M, we obtain a gap-gadget T in M 6 2 time. Using Lemma 3.2 on I 0 and T , we obtain our target set cover instance I = (S, U, E) satisfying the following properties. β€’ If Ο† is a yes-instance, then opt(I) 6 k. 1 1+Ξ΄/2

β€’ If Ο† is a no-instance, then opt(I) > (1 + Ξ΄)/(1 + Ξ΄/2), we get opt(I) >

1 1+Ξ΄

Β·

p k

Β·

p k

log M/ log log M. Using (1 + 1/k3 )1/k 6

log N/ log log N.

β€’ |S| = |S| 6 k2n/k . log M

log M

β€’ |U| 6 M log log M Β· |U| (1+Ξ΄/2)k log log M = M log log M Β· (k + Cn) (1+Ξ΄/2)k log log M . The number of vertices in I is log M

|S(I)| + |U(I)| 6 M + M log log M Β· (k + Cn) (1+Ξ΄/2)k log log M log M

6 M + M log log M Β· (2Ck log M) (1+Ξ΄/2)k log log M 2 log log M

6 M + M log log M Β· (log M) (1+Ξ΄/2)k log log M 6 M + M log log M Β· M

4

6 M + M log log M Β· M1/k 3

6 M1+1/k

(using log M > 2Ck for large n)

2 (1+Ξ΄/2)k

(using (1 + Ξ΄/2)k > 2k4 )

3

4

(using M1/k > 1 + M1/k log log M for large n)

= N.

6

Now we are ready to prove Theorem 1.2. Suppose for some computable function f, there is an f(k) Β· Nkβˆ’ -time algorithm that can, for every N-vertex q set cover instance I and every integer 1 1+Ξ΄

k, distinguish between opt(I) 6 k and opt(I) >

Β·

k

log N log log N .

For every δ ∈ (0, 1), choose

3 1/k

k ∈ N large enough so that (1 + 1/k ) 6 (1 + Ξ΄)/(1 + Ξ΄/2) and (1 + Ξ΄/2)k > 2k4 hold. 0 2 Let  = 1 βˆ’ /k + 1/k , by SETH, there exists an integer d such that d-SAT with n variables 0 cannot be solved in 2n(1βˆ’ ) -time. Given an instance Ο† of d-SAT with n variables and m clauses. By the sparsification lemma [21], we can assume that m = Cd, 0 Β· n for some constant Cd, 0 depending on d and  0 . Without loss of generality, assume that n is much larger than k. Applying 3 Lemma 3.4 on Ο† and k, we obtain a set cover instance I with N 6 2n/k+n/k vertices in time 5n/k n 2 6 2 forq k > 5/. Then we use the approximation algorithm to decide if opt(I) 6 k or log N 1 k 1+Ξ΄ Β· log log N . Thus we (n/k+n/k3 )(kβˆ’) n(1βˆ’/k+1/k2 )

opt(I) >

can solve d-SAT in time 2n + f(k) Β· Nkβˆ’ 6 2n + f(k) Β· 0

62 = 2n(1βˆ’ ) , which contradicts SETH. Theorem 1.3 can be proved similarly. By ETH, there exists  > 0 such that 3-SAT on n variables cannot be solved in 2n time. Let  0 = /2. For every 3-SAT instance Ο† with n variable and Cn clause, where n is much larger than k, apply Lemma 3.4 to obtain a set cover instance I with 3 0 0k N = 2n/k+n/k vertices in 25n/k 6 2 n time. If there q is an f(k) Β· N -time algorithm that can

2

distinguish between opt(I) 6 k and opt(I) > 0n

satisfiable in time 2

3.3

(n/k+n/k3 )Β· 0 k

+ f(k) Β· 2

1 1+Ξ΄

Β·

k

n

.

62

log N log log N ,

then we can decide whether Ο† is

Proof of Theorem 1.4

We use a lemma in [2] to reduce k-SUM to k-VECTOR-SUM over small numbers. Then we present a reduction from k-VECTOR-SUM to set cover. Lemma 3.5 (Lemma 3.1 of [2]). Let k, p, d, s, M ∈ N satisfy k < p, pd > kM + 1, and s = (k + 1)dβˆ’1 . There is a collection of mappings f1 , . . . , fs : [0, M] Γ— [0, kM] β†’ [βˆ’kp, kp]d , each computable in time O(poly log M + kd ), such that for all numbers x1 , . . . , xk ∈ [0, M] and targets t ∈ [0, kM], k k X X xj = t ⇔ βˆƒi ∈ [s] such that fi (xj , t) = ~0. j=1

j=1

Lemma 3.6. There is an algorithm which, given k sets S1 , S2 , . . . , Sk where Si is a set of n vectors in [βˆ’f(k), f(k)]g(k) log n for some computable functions f and g, outputs a set cover instance I = (S, U, E) kβˆ’1 kβˆ’1 with |U| 6 k(2f(k)) g(k) log n and S = S1 βˆͺ S2 βˆͺ . . . βˆͺ Sk in k(2f(k)) g(k)nO(1) -time such that P (i) if there exist ~x1 ∈ S1 , . . . , ~xk ∈ Sk such that ~xi = ~0, then {~x1 , . . . , ~xk } covers U; i∈[k]

(ii) if the sum of any k vectors ~x1 ∈ S1 , . . . ~xk ∈ Sk is not zero, then opt(I) > k. P kβˆ’1 Proof. Let D = {(d1 , . . . , dk ) ∈ [βˆ’f(k), f(k)]k : . i∈[k] di = 0}. Note that |D| 6 (2f(k)) |D| Suppose D = {~ a1 , . . . , a ~ |D| }. For every j ∈ [g(k) log n], let Uj = [k] . We define the target set cover instance I = (S, U, E) as follows. β€’ S = S1 βˆͺ Β· Β· Β· βˆͺ Sk . S β€’ U = i∈[g(k) log n] Ui . β€’ For every ~x ∈ Si and every ~u ∈ Uj , we add an edge {~x, ~u} into E if there exists ` ∈ [|D|] such that ~u[`] = i and ~x[j] = a ~ ` [i]. Completeness. Suppose there exist ~x1 ∈ S1 , . . . , ~xk ∈ Sk such that j ∈ [g(k) log n] we have ~x1 [j] + ~x2 [j] + . . . + ~xk [j] = 0, i.e.,

P

xi i∈[k] ~

= ~0. Then for all

(~x1 [j], ~x2 [j], . . . , ~xk [j]) = a ~ ` ∈ D for some ` ∈ [|D|]. For all ~u ∈ Uj , let i = ~u[`] ∈ [k]. Then by (1), ~xi [j] = a ~ ` [i]. It follows that {~xi , ~u} ∈ E. 7

(1)

Soundness. Suppose the sum of any k vectors in S1 βˆͺ Β· Β· Β· βˆͺ Sk is not zero. Let X be a subset of S with |X| 6 k, we need to show that X does not cover U. Firstly, we note that if X ∩ Si = βˆ… for some i ∈ [k], then the vector ~u = (i, i, . . . , i) ∈ [k]|D| is not covered by any vector in X. Now assume that P X = {~x1 , ~x2 , . . . , ~xk } and ~xi ∈ Si for all i ∈ [k]. Since i∈[k] ~xi 6= ~0, there exists a j ∈ [g(k) log n] such that X ~xi [j] 6= 0. i∈[k]

We deduce that (~x1 [j], ~x2 [j], . . . , ~xk [j]) ∈ / D. In other word, for all ` ∈ [|D|], there exists an i` ∈ [k] such that ~xi` [j] 6= a ~ ` [i` ].

(2)

Define a vector ~u ∈ Uj such that for all ` ∈ [|D|], ~u[`] = i` .

(3)

Suppose ~u is covered by xi ∈ X, then by the definition, there exists ` ∈ [|D|] such that i = ~u[`] = i` and ~xi` [j] = a ~ ` [i` ], which contradicts (2) and (3). c+1

Proof of Theorem 1.4. Given k sets S1 , . . . , Sk of integers in [βˆ’n2k , n2k ]. Let p = k4k , M = 2n2k and d = log n/kc . Without loss of generality, assume that k is large and n is much larger than k, we have pd = k4k log n > n4k > 2kn2k + 1. On the other hand, for any  > 0, we can pick c such c that s = (k + 1)d = nlog(k+1)/k 6 n/4 . Applying Lemma 3.5, we obtain a collection of mappings f1 , . . . , fs : [0, M] Γ— [0, kM] β†’ [βˆ’kp, kp]d in O(poly log M + kd ) time such that P β€’ there exist x1 ∈ S1 , . . . , xk ∈ Sk with j∈[k] xj = 0 if and only if there exist i ∈ [s] such that P 2k 2k ~ j∈[k] fi (xj + n , kn ) = 0. log n 1/k log n 1 )-gapUsing Lemma 3.3, we construct a (k, n, O(n log log n), (1+Ξ΄/2) k log log n , (1+Ξ΄/2) Β· ( log log n ) i 2k gadget T for some small Ξ΄ > 0. For every i ∈ [s], and j ∈ [k], let Sj = {fi (x + n , kn2k ) : x ∈ Sj }. Applying Lemma 3.2 to Si1 , Si2 , . . . , Sik and T , we obtain a set cover instance Ii with log n

3

S(Ii ) = Si1 βˆͺ Si2 . . . Sik and |U(Ii )| 6 n log log n Β· (g(k) log n) (1+Ξ΄/2)k log log n 6 n1+1/k . The set cover instances I1 , . . . , Is satisfy the following properties. P β€’ If there exist x1 ∈ S1 , . . . , xk ∈ Sk with j∈[k] xj = 0, then there exist i ∈ [s] and y1 = fi (x1 + n2k , n2k ) ∈ Si1 . . . yk = fi (xk + n2k , n2k ) ∈ Sik such that y1 , . . . , yk cover U(Ii ). P β€’ If there are no x1 ∈ S1 , . . . , xk ∈ Sk with j∈[k] xj = 0, then for all i ∈ [s], opt(Ii ) >  1/k log n 1 . 1+Ξ΄/2 Β· log log n 2

Let N = n1+1/k . We have 3

|S(Ii )| + |U(Ii )| 6 kn + n1+1/k 6 N, f(k) Β· Ndk/2eβˆ’ 6 ndk/2eβˆ’+1/k , and 1 (1 + Ξ΄)



log N log log N

1/k 6

1 (1 + Ξ΄/2)



log n log log n

1/k .

For every i ∈ [s], we apply the f(k) Β· Ndk/2eβˆ’ -time algorithm to decide if opt(Ii ) 6 k or opt(Ii ) > 1 1/k . If for some i ∈ [s], it found that opt(Ii ) 6 k, then we know that 1+Ξ΄ Β· (log N/ log log N) the input instance of k-SUM is a yes-instance. The running time is O(poly log M + kd ) + f(k) Β· Ndk/2eβˆ’ 6 O(poly log M + kd ) + s Β· ndk/2eβˆ’+1/k 6 ndk/2eβˆ’/2 for large k. 8

3.4

Proof of Theorem 1.5

Firstly, we give a reduction from C LIQUE to S ET-C OVER which produces instances with logarithmic sized universe set. The main idea of this reduction is due to Karthik et al [25]. Lemma 3.7. There is an nO(1) -time algorithm which, given an integer k, an n-vertex graph G with V(G) = V1 βˆͺ V2 βˆͺ Β· Β· Β· βˆͺ Vk such that G[Vi ] is an independent set S for all i ∈ [k], outputs a set cover instance I = (S, U, E) with |U| = kO(1) log n and S = E(G) = {i,j}∈([k]) S{i,j} , where each S{i,j} is 2 the set of edges between Vi and Vj , such that   (i) if G contains a k-clique, then opt(I) 6 k2 . Moreover, there exists a k2 -sized subset of S, which  contains exactly one vertex from each S{i,j} ({i, j} ∈ [k] 2 ), that can cover U;  (ii) if G contains no k-clique, then opt(I) > k2 . Proof. We will construct a set cover instance I such that if G has a k-clique, then we can select its  k log n 2 edges to cover the whole universe set. For every v ∈ V(G), denote by encode(v) ∈ {0, 1} the binary string representation of v. For every ` ∈ [log n], the `th bit of encode(v) is encode(v)[`]. For every i ∈ [k], let Οƒi : [k] \ {i} β†’ [k βˆ’ 1] be an arbitrary bijection. Our target set cover instance I = (S, U, E) is defined as follows. S β€’ S = E(G) = {i,j}∈([k]) S{i,j} , where S{i,j} = {{vi , vj } : vi ∈ Vi , vj ∈ Vj , {vi , vj } ∈ E(G)}. 2 β€’ U = [k] Γ— [k βˆ’ 1]{0,1} Γ— [log n]. β€’ For s = {vi , vj } ∈ S and u = (i, f, `) ∈ U we add {s, u} into E if vi ∈ Vi , vj ∈ Vj and f(encode(vi )[`]) = Οƒi (j). The set cover instance I satisfies the following conditions.  β€’ If G contains a k-clique, then there exists a k2 -sized subset of S which contains exactly one  vertex from each S{i,j} ({i, j} ∈ [k] U. Suppose that v1 ∈ V1 , . . . , vk ∈ Vk 2 ) that can cover  [k] induce a k-clique. Let X = {{vi , vj } : {i, j} ∈ 2 }. We will show that X covers the whole set U. For any (i, f, `) ∈ U, let b = encode(vi )[`]. Since f(b) ∈ [k βˆ’ 1], there must exist a j ∈ [k] \ {i} such that Οƒi (j) = f(b). By the definition of E, {vi , vj } is adjacent to (i, f, `) .   β€’ If G does not contain a k-clique, then opt(I) > k2 . Let X βŠ† S be a set such that |X| 6 k2 and X covers U.  For each {i, j} ∈ [k] 2 , define X{i,j} = {{vi , vj } : vi ∈ Vi , vj ∈ Vj , {vi , vj } ∈ X}.  We claim that for every {i, j} ∈ [k] 2 , |X{i,j} | > 0. Otherwise let f(0) = f(1) = Οƒi (j) and consider the vertex (i, f, 1) ∈ U. According to the definition of E, if a vertex {v, u} ∈ S covers (i, f, 1), then either v or u must be in Vi . Let us assume v ∈ Vi and u ∈ Vj 0 for some j 0 ∈ [k] \ {i}. We must have f(encode(vi )[1]) = Οƒi (j 0 ). However, if j 6= j 0 , then f(0) = f(1) = Οƒi (j) 6= Οƒi (j 0 ).  P Since k2 > |X| = |X{i,j} | and |X{i,j} | > 0, we conclude that |X{i,j} | = 1 for all {i,j}∈([k] 2 )  {i, j} ∈ [k] 2 . For every i ∈ [k] and distinct j, j 0 ∈ [k] \ {i}, let {{v, vj }} = X{i,j} and {{v 0 , vj 0 }} = Xi,j 0 , where v, v 0 ∈ Vi , we claim that v = v 0 . Otherwise, since v 6= v 0 there exists ` ∈ [log n] such that encode(v)[`] 6= encode(v 0 )[`]. Now consider a function f with f(encode(v 0 )[`]) = Οƒi (j) and f(encode(v)[`]) = Οƒi (j 0 ). The vertex (i, f, `) must be covered by some {x, y} with x ∈ Vi and y ∈ Vh such that Οƒi (h) = f(encode(v)[`]) ∈ {Οƒi (j), Οƒi (j 0 )}. We must have y ∈ Vj or y ∈ Vj 0 . 9

Since |X{i,j} | = |X{i,j 0 } | = 1, we deduce that either {x, y} = {v, vj } or {x, y} = {v 0 , vj 0 }. However, if {x, y} = {v, vj }, we must have Οƒi (j) = f(encode(v)[`]) = Οƒi (j 0 ) 6= Οƒi (j), a contradiction. Similarly, if {x, y} = {v 0 , vj 0 }, then Οƒi (j 0 ) = f(encode(v 0 )[`]) = Οƒi (j) 6= Οƒi (j 0 ). We conclude that the vertex (i, f, `) can not be covered by X. Now we have for every i ∈ [k], there exists a vi ∈ Vi such that \ {vi } = e. j∈[k]\{i},e∈X{i,j}

Obviously, for every {i, j} ∈ k-clique in G.

[k] 2

 , {{vi , vj }} = X{i,j} . This implies that {v1 , v2 , . . . , vk } is a

Proof of Theorem 1.5. Given an n-vertex graph G and a positive integer k, we invoke Lemma 3.7 to obtain a set cover instance I = (S, U, E) with |S| = |E(G)| and |U| 6 k3 log n satisfying (i) and  1/(k) (ii). Let m = |S|. Then we use Lemma 3.3 to construct a ( k2 , m, nO(1) , logloglogmm , logloglogmm 2 )-gapgadget T in mO(1) = nO(1) time. Applying Lemma 3.2 on I and T , we finally obtain our target set cover instance I 0 = (S 0 , U 0 , E 0 ) with the following properties:  β€’ if G has a k-clique, then opt(I 0 ) = k2 , β€’ if G has no k-clique, then opt(I 0 ) >



log m log log m

1/(k2 )

,

β€’ |S 0 | = |E(G)| = m, β€’ |U 0 | = (k3 log n)log m/ log log m = m1+o(1) . Let N = |U 0 | + |S 0 |. We have N = nO(1) . Since  is an unbounded computable   function, there is a computable function g : N β†’ N such that k 0 = g(k) > k2 and (k 0 ) > k2 . When n is large enough, k 1/(k 2) log N log m 1/( 2 ) 0 > > (log N)1/(k ) . log log m O(log log N) 1

Any f(k 0 )Β·NO(1) time algorithm that can distinguish between opt(I 0 ) 6 k 0 and opt(I 0 ) > (log N) (k 0 ) can be used to decide if an input graph G has k-clique in f(g(k))nO(1) time.

4

Conclusion

We have improved the hardness approximation factor for parameterized set cover problem using a simple reduction. Our result shows that in order to prove inapproximability of parameterized set cover, it suffices to prove the hardness of set cover problem with logarithmic sized universe set. A natural question is: Question 4.1. Is there any algorithm that can, given an n-vertex set cover instance I and an integer k, outputs a new instance I 0 and an integer k 0 in f(k) Β· nO(1) time for some computable function f : N β†’ N such that β€’ k 0 = g(k) for some computable function g : N β†’ N, β€’ opt(I) 6 k if and only if opt(I 0 ) 6 k 0 , β€’ |U(I 0 )| 6 h(k) Β· (log |S(I 0 )|)O(1) for some computable function h : N β†’ N.

10

A positive answer to the above question would imply that S ET C OVER parameterized by the optimum solution size has no (log n)1/(k) -approximation FPT algorithm assuming W[2] 6= FPT . Note that using dynamic programming method, S ET C OVER can be solved in 2|U(I)| (|U(I)| + |S(I)|)O(1) time [12]. We do not expect to reduce the size of universe set below o(k log n) under ETH. Our hardness result is far from matching the ln n approximation ratio of the greedy algorithm in polynomial time. Could it be the case that there exists a (ln n)1/ρ(k) -approximation algorithm for S ET C OVER with running time nkβˆ’ ? What is the best approximation ratio we can achieve for parameterized set cover in nkβˆ’ time?

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13

A Simple Gap-producing Reduction for the ...

Apr 11, 2018 - A Simple Gap-producing Reduction for the. Parameterized Set Cover Problem. Bingkai Lin. National Institute of Informatics [email protected]

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