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 β 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 nkβ 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· nkβ -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/2eβ -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 β 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 β U is covered by a subset C β 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 β 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 β (0, 1) there exists a d > 3 such that d-SAT on n variables cannot be solved in 2(1β)n time, we can further rule out nkβ -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 nkβ -time algorithm for S ET C OVER to refute SETH, one should consider an nkβ -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 nkβ ? 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 nkβ 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· nkβ , assuming SETH. 1 Their approach is to first establish a (log n) β¦(k) gap for M AX C OVER, then reduce M AX C OVER to 1

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

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

k

log n log log n .

Theorem 1.2. Assuming SETH, for every , Ξ΄ β (0, 1) and computable function f : N β N, there is no f(k) Β· Nkβ 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 β (0, 1) such that for every Ξ΄ β (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΄, β (0, 1) and computable function f : N β N, there is no f(k) Β· Ndk/2eβ 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 β 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. β 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΄ β (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 β (0, 1) there exists d > 3 such that d-SAT on n variables cannot be solved in 2(1β)n time. β’ k-SUM hypothesis (k-SUM) [1]: for every k > 2 and > 0, no O(ndk/2eβ ) time algorithm 2k 2k can, given k sets S1 , . . . , Sk each Pwith n integers in [βn , n ], decide if there are k integers x1 β S1 , . . . , xk β Sk such that iβ[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| = β¦(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 β [m], |Ai | = `. (G2) B is partitioned into (B1 , B2 , . . . , Bk ). For every j β [k], |Bj | = n. (G3) For all b1 β B1 , b2 β B2 , . . . bk β Bk , there exist a1 β A1 , . . . , am β Am such that for all i β [m] and j β [k], ai is adjacent to bj . (G4) For all X β B and a1 β A1 , . . . , am β 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 βͺ S2 . . . βͺ Sk and |Si | = n for all i β [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 β S1 , . . . , sk β 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 β [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 = ( iβ[m] UAi ). β’ For all s β S 0 and f β UAi where i β [m], E 0 contains {s, f} if there exists an a β Ai such that (Eβ1) {s, f(a)} β E, (Eβ2) {a, s} β ET . Completeness. If opt(I) 6 k, then there exist s1 β S1 , . . . , sk β S that can cover the whole set U. We will show that for every f β U 0 , f is covered by some vertex in {s1 , s2 , Β· Β· Β· , sk }. Firstly, by (G3), there exist a1 β A1 , . . . , am β Am such that ai sj β ET for all i β [m] and j β [k]. Assume that f β UAi for some i β [m]. Observe that f(ai ) β U must be covered by some sj with j β [k], i.e., {sj , f(ai )} β E. Since {ai , sj } β ET and {sj , f(ai )} β E, according to the definition of E 0 , we must have {sj , f} β E 0 . Soundness. Suppose opt(I) > k. Let X β S 0 be a set covering U 0 . For every a β A, let NT (a) be the set of neighbors of a in T . We have the following claim. Claim 1. For every i β [m] there exists ai β Ai such that |NT (ai ) β© X| > k + 1. Proof of Claim 1. Suppose there exists an i β [m] such that for all a β Ai , |NT (a) β© X| 6 k. Since opt(I) > k, every solution of I has size at least k + 1. It follows that for every a β Ai , there exists some ua β U such that ua is not covered by NT (a)β©X in the set cover instance I. Define a function f β UAi such that f(a) = ua for every a β Ai . We claim that f is not covered by X. Otherwise, suppose there exists an s β X that can cover f. According to the definition of E 0 , there must exists an a β Ai such that (Eβ1) and (Eβ2) hold. However, if s β NT (a) β© X, then {s, f(a)} = {s, ua } β / E. On the other hand, if s β / NT (a) β© X, then {a, s} β / ET . In both cases, we obtain contradictions. 4

a By Claim 1, we can pick ai β Ai for each i β [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 β 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 = β (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 β 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 β [m] and c β [n], Mr,c β [h]. (M2) For any set C β [n] with |C| 6 h, there exists a row r β [m] such that |{Mr,c : c β C}| = |C|. Condition (M1) is obvious. To see why (M2) holds, for each C β [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 β S such that each block in C takes distinct value. It follows that |{Mr,c : c β C}| = |C|. For each i β [m], let Ai = {(a1 , a2 , . . . , ak ) : for all j β [k], aj β [h]}. Note that |Ai | = hk . For each j β [k], let Bj = [n]. Let T = (A, B, E) be a bipartite graph with S β’ A = iβ[m] Ai . S β’ B = jβ[k] Bj . β’ E = {{~ a, b} : a ~ β Ai , b β Bj and Mi,b = a ~ [j] for all j β [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 β B1 , b2 β B2 , . . . , bk β Bk . We define a ~ i β Ai by setting a ~ i = (Mi,b1 , Mi,b2 , . . . , Mi,bk ). It is routine to check that {~ ai , bj } β E for all i β [m] and j β [k]. T satisfies (G4). Let X β B and a ~ 1 β A1 , a ~ 2 β A2 , . . . , a ~ m β Am . Suppose for every i β [m], a ~ i has at least k + 1 neighbors in X and |X| 6 h. By (M2), there exists an r β [m] such that |{Mr,c : c β X}| = |X|. Since a ~ r has at least k + 1 neighbors in X, there exists an j β [k] such that a ~r has two neighbors b, b 0 in X β© 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 β 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 β N, Ξ΄ β (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 β [k], let Si be the set of assignments to the i-th part. Let S 0 = S1 βͺ Β· Β· Β· βͺ Sk . Let U 0 be the set consisting of all the clauses of Ο and k additional nodes u1 , u2 , . . . , uk . For every i β [k] and assignment s β Si , we add an edge between s and ui . If the assignment s β S 0 satisfies a clause u β 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 β S1 , Β· Β· Β· , sk β 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 β k, n β M, ` β

log M ,h (1+Ξ΄/2)k log log M 4 5n/k

β

1 1+Ξ΄/2

Β·

k

log M log log M

and

m β 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· Nkβ -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 Ξ΄ β (0, 1), choose

3 1/k

k β 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 β /k + 1/k , by SETH, there exists an integer d such that d-SAT with n variables 0 cannot be solved in 2n(1β ) -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 )(kβ) n(1β/k+1/k2 )

opt(I) >

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

62 = 2n(1β ) , 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 β N satisfy k < p, pd > kM + 1, and s = (k + 1)dβ1 . There is a collection of mappings f1 , . . . , fs : [0, M] Γ [0, kM] β [βkp, kp]d , each computable in time O(poly log M + kd ), such that for all numbers x1 , . . . , xk β [0, M] and targets t β [0, kM], k k X X xj = t β βi β [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 [βf(k), f(k)]g(k) log n for some computable functions f and g, outputs a set cover instance I = (S, U, E) kβ1 kβ1 with |U| 6 k(2f(k)) g(k) log n and S = S1 βͺ S2 βͺ . . . βͺ Sk in k(2f(k)) g(k)nO(1) -time such that P (i) if there exist ~x1 β S1 , . . . , ~xk β Sk such that ~xi = ~0, then {~x1 , . . . , ~xk } covers U; iβ[k]

(ii) if the sum of any k vectors ~x1 β S1 , . . . ~xk β Sk is not zero, then opt(I) > k. P kβ1 Proof. Let D = {(d1 , . . . , dk ) β [βf(k), f(k)]k : . iβ[k] di = 0}. Note that |D| 6 (2f(k)) |D| Suppose D = {~ a1 , . . . , a ~ |D| }. For every j β [g(k) log n], let Uj = [k] . We define the target set cover instance I = (S, U, E) as follows. β’ S = S1 βͺ Β· Β· Β· βͺ Sk . S β’ U = iβ[g(k) log n] Ui . β’ For every ~x β Si and every ~u β Uj , we add an edge {~x, ~u} into E if there exists ` β [|D|] such that ~u[`] = i and ~x[j] = a ~ ` [i]. Completeness. Suppose there exist ~x1 β S1 , . . . , ~xk β Sk such that j β [g(k) log n] we have ~x1 [j] + ~x2 [j] + . . . + ~xk [j] = 0, i.e.,

P

xi iβ[k] ~

= ~0. Then for all

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

(1)

Soundness. Suppose the sum of any k vectors in S1 βͺ Β· Β· Β· βͺ 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 β© Si = β for some i β [k], then the vector ~u = (i, i, . . . , i) β [k]|D| is not covered by any vector in X. Now assume that P X = {~x1 , ~x2 , . . . , ~xk } and ~xi β Si for all i β [k]. Since iβ[k] ~xi 6= ~0, there exists a j β [g(k) log n] such that X ~xi [j] 6= 0. iβ[k]

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

(2)

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

(3)

Suppose ~u is covered by xi β X, then by the definition, there exists ` β [|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 [βn2k , 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 [βkp, kp]d in O(poly log M + kd ) time such that P β’ there exist x1 β S1 , . . . , xk β Sk with jβ[k] xj = 0 if and only if there exist i β [s] such that P 2k 2k ~ jβ[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 β [s], and j β [k], let Sj = {fi (x + n , kn2k ) : x β 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 βͺ 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 β S1 , . . . , xk β Sk with jβ[k] xj = 0, then there exist i β [s] and y1 = fi (x1 + n2k , n2k ) β Si1 . . . yk = fi (xk + n2k , n2k ) β Sik such that y1 , . . . , yk cover U(Ii ). P β’ If there are no x1 β S1 , . . . , xk β Sk with jβ[k] xj = 0, then for all i β [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/2eβ 6 ndk/2eβ+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 β [s], we apply the f(k) Β· Ndk/2eβ -time algorithm to decide if opt(Ii ) 6 k or opt(Ii ) > 1 1/k . If for some i β [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/2eβ 6 O(poly log M + kd ) + s Β· ndk/2eβ+1/k 6 ndk/2eβ/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 βͺ V2 βͺ Β· Β· Β· βͺ Vk such that G[Vi ] is an independent set S for all i β [k], outputs a set cover instance I = (S, U, E) with |U| = kO(1) log n and S = E(G) = {i,j}β([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} β [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 β V(G), denote by encode(v) β {0, 1} the binary string representation of v. For every ` β [log n], the `th bit of encode(v) is encode(v)[`]. For every i β [k], let Οi : [k] \ {i} β [k β 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}β([k]) S{i,j} , where S{i,j} = {{vi , vj } : vi β Vi , vj β Vj , {vi , vj } β E(G)}. 2 β’ U = [k] Γ [k β 1]{0,1} Γ [log n]. β’ For s = {vi , vj } β S and u = (i, f, `) β U we add {s, u} into E if vi β Vi , vj β Vj and f(encode(vi )[`]) = Οi (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} β [k] U. Suppose that v1 β V1 , . . . , vk β Vk 2 ) that can cover [k] induce a k-clique. Let X = {{vi , vj } : {i, j} β 2 }. We will show that X covers the whole set U. For any (i, f, `) β U, let b = encode(vi )[`]. Since f(b) β [k β 1], there must exist a j β [k] \ {i} such that Οi (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 β S be a set such that |X| 6 k2 and X covers U. For each {i, j} β [k] 2 , define X{i,j} = {{vi , vj } : vi β Vi , vj β Vj , {vi , vj } β X}. We claim that for every {i, j} β [k] 2 , |X{i,j} | > 0. Otherwise let f(0) = f(1) = Οi (j) and consider the vertex (i, f, 1) β U. According to the definition of E, if a vertex {v, u} β S covers (i, f, 1), then either v or u must be in Vi . Let us assume v β Vi and u β Vj 0 for some j 0 β [k] \ {i}. We must have f(encode(vi )[1]) = Οi (j 0 ). However, if j 6= j 0 , then f(0) = f(1) = Οi (j) 6= Οi (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}β([k] 2 ) {i, j} β [k] 2 . For every i β [k] and distinct j, j 0 β [k] \ {i}, let {{v, vj }} = X{i,j} and {{v 0 , vj 0 }} = Xi,j 0 , where v, v 0 β Vi , we claim that v = v 0 . Otherwise, since v 6= v 0 there exists ` β [log n] such that encode(v)[`] 6= encode(v 0 )[`]. Now consider a function f with f(encode(v 0 )[`]) = Οi (j) and f(encode(v)[`]) = Οi (j 0 ). The vertex (i, f, `) must be covered by some {x, y} with x β Vi and y β Vh such that Οi (h) = f(encode(v)[`]) β {Οi (j), Οi (j 0 )}. We must have y β Vj or y β 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 Οi (j) = f(encode(v)[`]) = Οi (j 0 ) 6= Οi (j), a contradiction. Similarly, if {x, y} = {v 0 , vj 0 }, then Οi (j 0 ) = f(encode(v 0 )[`]) = Οi (j) 6= Οi (j 0 ). We conclude that the vertex (i, f, `) can not be covered by X. Now we have for every i β [k], there exists a vi β Vi such that \ {vi } = e. jβ[k]\{i},eβX{i,j}

Obviously, for every {i, j} β 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/Ο(k) -approximation algorithm for S ET C OVER with running time nkβ ? What is the best approximation ratio we can achieve for parameterized set cover in nkβ time?

References [1] Amir Abboud and Kevin Lewi. Exact weight subgraphs and the k-sum conjecture. In International Colloquium on Automata, Languages, and Programming, pages 1β12. Springer, 2013. [2] Amir Abboud, Kevin Lewi, and Ryan Williams. Losing weight by gaining edges. In European Symposium on Algorithms, pages 1β12. Springer, 2014. [3] Amir Abboud, Aviad Rubinstein, and Ryan Williams. Distributed PCP theorems for hardness of approximation in P. In Foundations of Computer Science (FOCS), 2017 IEEE 58th Annual Symposium on, pages 25β36. IEEE, 2017. [4] Noga Alon, Dana Moshkovitz, and Shmuel Safra. Algorithmic construction of sets for krestrictions. ACM Transactions on Algorithms (TALG), 2(2):153β177, 2006. [5] Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM (JACM), 45(3):501β555, 1998. [6] E. Bonnet, B. Escoffier, E. Kim, and V. Th. Paschos. On subexponential and FPT-time inapproximability. In Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, pages 54β65, 2013. [7] Nicolas Bourgeois, Bruno Escoffier, and Vangelis Paschos. Efficient approximation of min set cover by βlow-complexityβ exponential algorithms. 2008. [8] Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From gap-ETH to FPT-inapproximability: Clique, dominating set, and more. In Foundations of Computer Science (FOCS), 2017 IEEE 58th Annual Symposium on, pages 743β754. IEEE, 2017. [9] Yijia Chen and Bingkai Lin. The constant inapproximability of the parameterized dominating set problem. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, pages 505β514. IEEE, 2016. [10] R. Chitnis, M. T. Hajiaghayi, and G. Kortsarz. Fixed-parameter and approximation algorithms: A new look. In Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, pages 110β122, 2013. [11] V. ChvΒ΄ atal. A greedy heuristic for the set-covering problem. Mathematics of Operations, 4(3):233 β 235, 1979. [12] Marek Cygan, Fedor V Fomin, Εukasz Kowalik, Daniel Lokshtanov, DΒ΄ aniel Marx, Marcin Pilipczuk, MichaΕ Pilipczuk, and Saket Saurabh. Parameterized algorithms, volume 4. Springer, 2015. 11

[13] Marek Cygan, Εukasz Kowalik, and Mateusz Wykurz. Exponential-time approximation of weighted set cover. Information Processing Letters, 109(16):957β961, 2009. [14] I. Dinur and D. Steurer. Analytical approach to parallel repetition. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 624β633, 2014. [15] R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer-Verlag, 1999. [16] Friedrich Eisenbrand and Fabrizio Grandoni. On the complexity of fixed parameter clique and dominating set. Theoretical Computer Science, 326(1-3):57β67, 2004. [17] Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634β652, 1998. [18] J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, 2006. [19] M. T. Hajiaghayi, R. Khandekar, and G. Kortsarz. Fixed parameter inapproximability for Clique and SetCover in time super-exponential in OPT. CoRR, abs/1310.2711, 2013. [20] Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. Journal of Computer and System Sciences, 62:367β375, 2001. [21] Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512β530, 2001. [22] D. S. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 9(3):256β278, 1974. [23] Stasys Jukna. Extremal combinatorics: with applications in computer science. Springer Science & Business Media, 2011. [24] R. M. Karp. Reducibility among combinatorial problems. In Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York., pages 85β103, 1972. [25] CS Karthik, Bundit Laekhanukit, and Pasin Manurangsi. On the parameterized complexity of approximating dominating set. In STOC, 2018. [26] Bingkai Lin. The parameterized complexity of k-biclique. In Proceedings of the TwentySixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 605β615, 2015. [27] L. LovΒ΄ asz. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13(4):383 β 390, 1975. [28] C. Lund and M. Yannakakis. On the hardness of approximating minimization problems. Journal of the ACM, 41(5):960β981, 1994. [29] D. Marx. Parameterized complexity and approximation algorithms. The Computer Journal, 51(1):60β78, 2008. [30] Mihai PΛ atrasΒΈcu and Ryan Williams. On the possibility of faster SAT algorithms. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 1065β1075. SIAM, 2010. [31] R. Raz and S. Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computing, STOC 1997, El Paso, Texas, USA, May 4-6, 1997, pages 475β484, 1997.

12

[32] P. SlavΒ΄Δ±k. A tight analysis of the greedy algorithm for set cover. Journal of Algorithms, 25(2):237 β 254, 1997. [33] S. K. Stein. Two combinatorial covering theorems. Journal of Combinatorial Theory, Series A, 16(3):391β397, 1974.

13