On the Hardness of Approximating Stopping and Trapping Sets in LDPC Codes
aa r X i v : . [ c s . I T ] A ug On the Hardness of Approximating Stopping and Trapping Sets ∗ Andrew McGregor † Olgica Milenkovic ‡ Abstract
We prove that approximating the size of stopping and trapping sets in Tanner graphs of linearblock codes, and more restrictively, the class of low-density parity-check (LDPC) codes, is NP-hard.The ramifications of our findings are that methods used for estimating the height of the error-floor ofmoderate- and long-length LDPC codes based on stopping and trapping set enumeration cannot provideaccurate worst-case performance predictions.
In the past decade, the search for efficient and near-optimal decoding algorithms for linear block codes cul-minated with the rediscovery and generalization of the notion of sparse codes and iterative message passingalgorithms. Although Maximum Likelihood (ML) decoding of linear block codes is NP-hard [5], iterativedecoders can approach the Shannon limit of reliable communication with polynomial time complexity, pro-vided that they operate on codes with long length that have sparse parity-check matrices, also known asLDPC codes [17]. Decoding is achieved via message passing on the
Tanner graph of the code, a suitablychosen bipartite graphical representation of the code which contains a very small number of edges. On suchgraphs, probabilistic inference of the form of iterative message passing is known to have linear complexityin the code length.The performance of linear block codes under iterative decoding, and the performance of LDPC codes inparticular, depends on the structural properties of their chosen Tanner graphs. For each channel-decoderpair, there exist vertex configurations in the code graph on which the given iterative decoder fails. Forsome frequently encountered Discrete Memoryless Channels (DMCs), such configurations are known as near-codewords [26], trapping and stopping sets [12, 31], pseudocodewords [41, 22], and instantons [36].It is known that ML decoders fail when transmission errors are confined to Tanner graph configurationscontaining codewords, while iterative decoders usually fail to make correct decisions on (strictly) larger setsof configurations. For example, iterative edge-removal (ER) decoders for signalling over the Binary ErasureChannel (BEC) fail on stopping sets [12], a subset of which are the codewords themselves. For the AdditiveWhite Gaussian Noise (AWGN) channel and sum-product decoding, failures arise due to subsets of verticesin the code graph that have similar structural properties as codewords, and are consequently termed near-codewords [26]. As a result, iterative decoders exhibit sub-optimal performance compared to ML decoders,and this performance loss most frequently manifests itself in terms of the emergence of error-floors in theBit-Error-Rate (BER) curve of the code.The error-floor phenomena is a problem of focal importance in the theory of iterative decoding, sincemany practical applications of codes on graphs require extremely low operational BERs. Since such low BERsare well beyond the scope of current Monte-Carlo simulation techniques, several methods were proposed forestimating the height of the error-floor through enumerating small stopping and small trapping sets [34],and exploring dominant instantons [31, 36]. These techniques operate fairly accurately for codes of very ∗ Part of the results were presented at the 2007 Information Theory Workshop, Lake Tahoe. The work was supported in partby the NSF Grant CCF 0644427, the NSF Career Award, and the DARPA Young Faculty Award of the second author. † Microsoft Research, Silicon Valley Campus. Email: [email protected] . ‡ Dept. of Electrical and Computer Engineering, University of Illinois, Urbana-Champaign. Email: [email protected] . a) Stopping Set. (b) ZP Trapping Set. Figure 1: Examples of Stopping and ZP-Trapping Sets. (a) AWGN ( a, b )-Trapping Set. (b) AWGN Elementary ( a, b )-Trapping Set.
Figure 2: Examples of AWGN Trapping Sets.short and moderate length and small minimum pseudoweight, but they are time consuming, and no rigorousanalytical study of the performance of these search procedures is known.Recently, it was shown that the problem of finding the smallest stopping set in an arbitrary code graphis NP-hard to approximate up to a constant term [27]. In [40], it was shown that finding the smallest k -outset , which represents a straightforward generalization of the notion of a stopping set, is NP-hard as well.Despite the fact that k -out sets may lead to decoding failures similar to those caused by trapping sets, theresults in [40] do not capture the fact that trapping sets are usually characterized in terms of two parameters.Furthermore, the notion of a trapping set is meaningful only in conjunction with a fixed decoding method.Finally, no hardness results for approximating k -out sets or more general trapping sets are currently known.The main contributions of our work are three-fold. First, we improve upon the hardness results forapproximating stopping sets, presented in [27]. Furthermore, we introduce the notion of a cover stoppingset , and show that the problem of finding such a set of smallest cardinality in an arbitrary Tanner graph isNP-hard. Second, we provide a set of new results regarding the hardness of finding trapping sets for GallagerA decoder (GA) [4], the Zyablov-Pinsker (ZP) decoder [43, 42], and the product-sum decoder. The third,and most important finding presented in the paper is that these hardness results carry over to the case ofLDPC code graphs (provided that the notion of “low-density” is properly defined). We discuss the impactof these findings on the accuracy of estimating the error-floor based on trapping set enumeration techniques.In addition, we give a brief overview of the theory of fixed parameter tractability (FPT), and show that theminimum cover stopping set problem is FPT.The paper is organized as follows. Section 2 introduces the trapping set structures under investigation,as well as their corresponding decoding algorithms. Section 3 provides a brief overview of a class of NP-hardproblems that are used in the reduction proofs of our main results. Section 4 contains theorems regardingthe hardness of approximating classes of trapping sets, while Section 5 specializes these results for the classof sparse code graphs and short code lengths. In Section 6 we briefly comment on the accuracy of error-floorestimation procedures relying on exhaustive trapping set enumeration techniques. In Section 7, we describethe notion of fixed parameter tractability and its implications for stopping and trapping set size estimation.Concluding remarks are given in Section 8. A binary, linear [ n, k, d ] code C is a k -dimensional vector subspace of an n -dimensional vector space F n . Thegenerator matrix M of the code C is a k × n matrix of full row-rank, with rows that correspond to basisvectors of the subspace. The parity-check matrix H of C is the generator matrix of the null-space of thecode. The matrix H defines a bipartite graph G = ( L ∪ R, E ), with columns of H indexing the variablenodes in L , and the rows of H indexing the check nodes in R . For i ∈ L and j ∈ R , ( i, j ) ∈ E if and only2f H i,j = 1. The graph G is called the Tanner graph of C with parity-check matrix H . If the parity-checkmatrix of a code contains only a “small” number of non-zero entries, i.e., it is sparse, then the correspondingcode is called a Low-Density Parity-Check (LDPC) code. A precise definition of the notion “small” will begiven in Section 5.For the remaining definitions in this section we need to introduce the following notation and definitions.For S ⊂ L , the notation Γ( S ) is reserved for the set of neighbors of S in R . G S denotes the induced subgraphfor S ⊂ L which is defined as the graph on nodes S ∪ Γ( S ) with edges { ( u, v ) : u ∈ S, v ∈ Γ( S ) } . Equivalently, G S is the Tanner graph of the punctured parity-check matrix of the code, consisting of the columns indexedby S . For any graph G ′ , V ( G ′ ) denotes the set of nodes of G ′ and E ( G ′ )Iterative decoders are a class of inference algorithms that operate on Tanner graphs of codes. Thesedecoders are known to compute the maximum likelihood estimates of variables only on Tanner graphs freeof cycles. Nevertheless, when applied to LDPC codes that contain cycles, they can approach the Shannonlimit on optimal performance with complexity linear in the length of the code.The messages passed between vertices of the Tanner graphs during iterative decoding depend on thecharacteristics of the transmission channel, and there usually exist many different iterative decoding methodsthat can be used for the same channel. For various decoder architectures specialized for the BEC, BSC,and AWGN channel, the interested reader is referred to [32]. For clarity of the future exposition, we brieflydescribe three of these procedures: the edge-removal (ER) algorithm, the Zyablov-Pinsker (ZP) bit-flippingmethod [12, 43, 42], and the regular Gallager A algorithm [7]. The first algorithm operates on outputs of theBEC, while the second two are designed for the BSC. A detailed description of different decoding proceduresfor signalling over the AWGN channel can be found in [31].The ER algorithm is used for codes transmitted over the BEC channel, where the input to the channelis a vector c c . . . c n ∈ C , and the output is a vector v v . . . v n over the symbol alphabet { , , e } . For aBEC channel with erasure probability p , one has Pr[ v i = c i ] = 1 − p , and Pr[ v i = e ] = p . The ER algorithmassigns to each vertex i in L of the Tanner graph of C the symbol v i . It then iteratively searches for verticesin R adjacent only to one e symbol in L . Due to the even-parity restriction, the corresponding c i value forsuch a symbol can be uniquely determined. The decoder terminates either when the correct codeword isrecovered or if every every parity-check vertex connected to one e symbol is connected to at least two suchsymbols. In the latter case, we say that the decoder failed on a stopping set . Definition 1 (BEC Stopping Sets) . Given a bipartite graph G = ( L ∪ R, E ) , we say that S ⊂ L is astopping-set if the degree of each vertex in Γ( S ) in the induced subgraph G S is at least two. Of independent interest is the problem of determining the size of the smallest stopping set S such thatΓ( S ) = R , i.e., the smallest set of vertices that covers each check node in R at least twice. We refer to such aset as the cover stopping set. If symbols corresponding to a cover stopping set are erased, then the decodingprocess terminates before proceeding with the first iteration, and no erasure can be corrected.Assume next that the Tanner graph of C is left-regular, with degree ℓ . For a BSC channel with errorprobability p , the word v v . . . v n ∈ { , } n and Pr[ v i = c i ] = 1 − p , and Pr[ v i = ¯ c i ] = p . In the first iterationof ZP-decoding, the decoder scans for received symbols v i that are connected to ℓ unsatisfied parity-checkequations. If symbols with such a property are encountered, the decoder flips their values sequentially. Theprocedure is repeated for vertices with ℓ − ℓ −
2, ..., ℓ −⌊ ( ℓ − / ⌋ unsatisfied check-equations. The decoderterminates by either recovering the correct codeword or by encountering a word for which each symbol isincluded in less than ℓ − ⌊ ( ℓ − / ⌋ unsatisfied check-equations. In the latter case, we say that the decoderfailed on a ZP trapping set . Definition 2 (BSC ZP-Trapping Sets) . Let G = ( L ∪ R, E ) be a left-regular bipartite graph with degree ℓ . We say that S ⊂ L is a ZP-trapping set if the induced subgraph G S is such that all vertices in S areconnected to less than ℓ − ⌊ ( ℓ − / ⌋ odd degree vertices in G S . Another frequently used iterative decoding algorithm for signaling over the BSC that has a completecharacterization of trapping sets is the Gallager A algorithm for regular codes with left vertex degree ℓ = 3.The decoding rule is straightforward: unless all incoming massages to a variable node are identical, the3igure 3: Example of Majority Trapping Set.variable node transmits its received symbol. Otherwise, the node transmits the consensus vote. On theother hand, the check-nodes pass on their parity estimates to their neighboring variable nodes. Definition 3 (BSC GA-Trapping Sets) . Let G = ( L ∪ R, E ) be a bipartite graph with left-degree three,such that all vertices in R have degree r > . Let T ⊂ L and let G T be the subgraph of G induced by T .Let O = { v ∈ Γ( T ) : deg G T ( v ) odd } . We say T is a GA-trapping set with parameter a if | O | = a and if | Γ( u ) ∩ O | ≤ for each u ∈ T and no two checks in O have a common neighbor in L \ T . For the AWGN channel, and message-passing algorithms, no precise analytic characterization of failingconfigurations is known. Extensive computer simulations [31, 24] show that errors are usually confinedto near codewords , also known as trapping sets or instantons. Roughly speaking, trapping sets resemblecodewords in so far that they result in a very small number of unsatisfied check equations (for codewords,this number equals zero). We focus our attention on three such configurations, defined below.
Definition 4 (AWGN ( a, b )-Trapping Sets) . Given a bipartite graph G = ( L ∪ R, E ) , we say that S ⊂ L is an ( a, b )-trapping set if | S | = a and the induced subgraph is such that Γ( S ) has exactly b vertices of odddegree. Similarly, we say that S ⊂ L is an elementary ( a, b )-trapping set if b vertices in Γ( S ) have degreeone, and | Γ( S ) | − b vertices have degree two. Definition 5 (AWGN Majority Trapping Set) . Given a bipartite graph G = ( L ∪ R, E ) we say S ⊂ L is good if the induced subgraph G S is such that the majority of vertices of G S in Γ( S ) have even degree. T isa majority trapping set if T and L \ T are both good. Examples of Tanner graphs including stopping sets, ZP-trapping sets, as well as AWGN trapping setsare shown in Figures 1, 2, and 3, respectively. Circles denote variable nodes in L , while squares denote checknodes in R of the Tanner graph G ( L ∪ R, E ). Complexity Theory:
A problem belongs to the class NP if it can be solved in polynomial time by a non-deterministic Turing machine. Alternatively, the complexity category of decision problems for which answerscan be checked for correctness using a certificate and an algorithm with polynomial running time in the sizeof the input is known as the NP class. A problem is NP-hard if the existence of a deterministic polynomialtime algorithm for the problem would imply the existence of deterministic polynomial time algorithms forevery problem in NP. This consequence is widely believed to be false, and hence determining that a problemis NP-hard is a very strong indicator that the problem in computational intractable, i.e., no deterministic,polynomial time algorithm exists for the problem.For optimization problems, there exists a large body of work that considers approximate solutions ratherthan exact solutions [39]. When minimizing a function subject to constraints, we say an algorithm is an α -approximation algorithm if it always returns a solution whose value is at most a factor α greater than thevalue for the optimal solution. For some NP-hard problems, it is possible to show that it is also NP-hardto α -approximate the problem. For a more thorough treatment of these and other subjects in complexitytheory, the interested reader is referred to [20]. 4 ur Results: We are concerned with the worst-case computational complexity of the following problems.1.
MinStop : Find a stopping set of minimum cardinality.2.
MinCStop : Find a cover stopping set of minimum cardinality .3.
MinTrap ZP : Find a ZP-trapping set of minimum cardinality.4. MinTrap GA : Given a , find a GA-trapping set of minimum cardinality.5. MinTrap
AWGN : Given a , find an ( a, b )-trapping set with minimum parameter b .6. MinTrap
AWGN − elem : Given b , find an ( a, b )-elementary trapping set with minimum parameter a .7. MinTrap
AWGN − maj : Find a majority trapping set of minimum cardinality.We show that there are no polynomial time algorithms for any of the above problems under standardcomplexity assumptions. Furthermore, there are no polynomial time algorithms that even approximate theoptimal solutions to a guaranteed precision. Many of these hardness results also apply when we restrict ourattention to Tanner graphs that correspond to LDPC codes. Our proofs can all be cast as reductions from theNP-hard Minimum Set Cover , Minimum Distance [37, 38, 15], and
Maximum Three-Dimensional Matching problems [5]. These, and some other relevant problems subsequently referred to, are briefly described in thefollowing section.
For completeness, we provide known NP hardness and approximation results for a class of combinatorialoptimization problems that will be used in the proofs of Sections 3, 4, and 5. Most of the results presentedin this section are available at [21].1.
The Minimum Set Cover Problem, MinSetCov:
Given a set of sets S = { S , . . . , S a } of [ b ],find S ′ ⊂ S of minimum cardinality such that ∪ S ∈S ′ S = [ b ]. It is NP-hard to c log N -approximate MinSetCov [30] for some c where N is the description length of the problem. Even in the case that | S i ∩ S j | ≤
1, for 1 ≤ i < j ≤ a , it can be shown that there exists no polynomial time c log N -approximation algorithm unless N P ⊂ ZT IM E ( N O (log log N ) ) [23] where ZT IM E ( t ) denotes the classof problems that have a probabilistic algorithm with expected running time t and with zero errorprobability.2. The Minimum Hitting Set Problem, MinHitSet:
Given a set of subsets S = { S , . . . , S b } of [ a ],find a set S ′ of smallest cardinality, such that | S ′ ∩ S i | ≥
1, for all i = 1 , , . . . , b . The MinHitSet problem is equivalent to the
MinSetCov problem [2] and as a consequence it is also NP-hard to( c log N )-approximate MinHitSet [30] for some c >
0. In the case when | S i | = 2 for all i ∈ [ b ] theproblem is often called the vertex cover problem MinVertCov . The vertex cover problem, even whenwe have |{ i : j ∈ S i }| ≤ α > The Maximum Three-Dimensional Matching Problem, MaxThreeDimMatch:
Given a set T ⊂ X × X × X , determine if a set S ⊂ T of size | X | exists such that no elements in S agree in anycoordinate. This decision problem is NP-hard even if no element of X appears more than 3 times inthe same coordinate of sets from T [20].4. The Maximum Likelihood Decoding Problem, MaxLikeDecode:
Given a code C specified byan m × n parity-check matrix H (we may assume H has linearly independent rows), a vector s ∈ F m ,and an integer ω >
0, determine if there is a vector x ∈ F n with weight bounded from above by ω andsuch that H x T = s . The MaxLikeDecode problem is NP-hard to approximate within any constantfactor [1]. 5.
The Minimum Weight Codeword Problem, MinCodeword:
Given a code C specified by an n × k generator matrix M of full row-rank, find the smallest weight of a non-zero codeword. The MinCodeword problem is not approximable within any constant factor unless
N P ⊂ RP , where RPis the set of decision problems for which there exists a randomized algorithm that is always correct onno instances and correct with probability 1/2 on yes instances. We start by showing that
MinStop is not approximable within o (log N ), where N denotes the descriptionlength of the problem, unless P = N P . This results improves upon the finding in [27], where the weakerclaim that
MinStop cannot be approximated within any positive constant was proved. This improvementis a consequence of the fact that our proof relies on reduction from the
MinSetCov , rather than the
MinVertCov problem [27].
Theorem 1.
There exists a constant c > such that it is NP-hard to ( c log N ) -approximate MinStop .Proof.
The proof is by a reduction from
MinSetCov . Let b = (cid:12)(cid:12) ∪ i ∈ [ a ] S i (cid:12)(cid:12) , and without loss of generality,assume that S ⊂ [ b ], for each S ∈ S . Form a bipartite graph G = ( L ∪ R, E ) with L = { u , ...u a , x, y } , R = { v , ...v b , w , ..., w a , z } , and edges E = { ( u i , v j ) : j ∈ S i } ∪ { ( u i , w i ) : i ∈ [ a ] } ∪ { ( x, v ) : v ∈ R } ∪ { ( y, v ) : v ∈ { w , ..., w a , z }} . An illustration of this graphical structure is given in Figure 4.We show that G has stopping distance 2 + t if and only if the minimum set cover is of size t . Since thereis no polynomial algorithm returning an c log N approximation for MinSetCov unless P = N P (for somesufficiently small c > S be a stopping set. Consequently,1. If ( x ∈ S or y ∈ S ), then ( x ∈ S and y ∈ S ) since otherwise d G S ( z ) = 1.2. If x ∈ S then u i ∈ S for some i since otherwise d G S ( v j ) = 1 for some j ∈ [ b ].3. If u i ∈ S then ( x ∈ S or y ∈ S ) since otherwise d G S ( w i ) = 1.Therefore, if S is non-empty x, y, u i ∈ S for some i ∈ [ a ]. But then d G S ( v j ) ≥ j ∈ S i . However thismeans that for all j ∈ [ b ] , d G S \{ x,y } ( v j ) ≥
1. Therefore, S being a stopping set implies that the included u i nodes correspond to a covering of [ b ]. The nodes corresponding to a covering of [ b ], in addition to x and y ,form a stopping set, since every node on the right hand side ( R ) is in the neighborhood and has degree atleast two. Hence the size of the minimum stopping set of G is exactly 2 plus the size of the minimum setcover. MinCStop:
The proof of Theorem 1 also implies that there exists a c > c log n )-approximate MinCStop . This is a consequence of the fact that the family of hard instancesconsidered all had the property that the neighborhood of all stopping sets was all the check nodes. We nextshow that there exists a deterministic, polynomial-time, O (log n )-approximation algorithm for MinCStop .This follows because we can relate
MinCStop to MinHitSet as follows.For each r ∈ R , create a set of sets S r that consists of all ( | Γ( r ) | − r ). For example, ifΓ( r ) = { a, b, c, d } , then S r = { ( a, b, c ) , ( a, b, d ) , ( a, c, d ) , ( b, c, d ) } . Let S = { S r : r ∈ R } . Then Q ⊂ L is ahitting set for S iff it is a cover stopping set of L . This claim can be proved in a straightforward manner: if S contains at least one element, say a , from Γ( r ), then it must contain at least two elements from the sameset since otherwise, the ( | Γ( r ) | −
1) set that does not contain a will not be hit.6 U U U U a-2 U a-1 U a Y … … W W W W a-2 W a-1 W a Z …… V V V V b-1 V b Incidence matrix: S1
Figure 4: Reduction from
MinSetCov to MinStop .Consequently, any α -approximation algorithm for MinHitSet can also be used to obtain an α -approximationalgorithm for MinCStop . For example the following simple greedy algorithm can be shown to be an O (log n )-approximation algorithm for MinHitSet : At each step add the element that appears in the most sets from S can remove these sets from S can repeat until all the elements chosen appear in every set from S .The greedy algorithm searches for cover stopping sets by going through the list of variable nodes indecreasing order of their degree, and it is straightforward to see that the algorithm terminates after at most( n − k ) δ max steps, where δ max denotes the largest degree of any check node in the Tanner graph of thecode. As a consequence, this algorithm is especially well suited for LDPC codes, to be formally defined inSection 5. Hardness under Stronger Assumptions:
Under the assumption that
N P DT IM E ( N polylog N ),it was shown in [27] that there exists no polynomial time approximation algorithm for MinStop within2 (log N ) − ǫ , for any ǫ > ZP , MinTrap GA , and MinTrap AWGN
We show next that the problems
MinTrap ZP , MinTrap GA , and MinTrap
AWGN are computationally atleast as hard as the
MinCodeword problem.
Theorem 2.
For any constant α , there is no polynomial-time α -approximation algorithm for MinTrap ZP ,unless RP = N P .Proof.
Recall that unless RP = N P , there is no polynomial time
MinCodeword problem is O (1)-hardto approximate even under the restriction that the Tanner graph of the code is left regular. This followsdirectly from the results in [15].Given a Tanner graph G = ( L ∪ R, E ) that is left regular say with degree ⌊ ( ℓ − / ⌋ + 1, for eachnode u ∈ L create ℓ − ⌊ ( ℓ − / ⌋ − R each connected to u . Call the new Tanner graph G ′ . Then any S ⊂ L is a ZP-trapping set in G ′ iff S is the support of a codeword in G . Hence any α -approximation algorithm for MinTrap ZP yields an α -approximation algorithm for MinCodeword and theresult follows.A very similar argument can be used to prove the following claim.
Theorem 3.
For any constant α , there is no polynomial-time, α -approximation algorithm for MinTrap GA ,unless RP = N P . roof. Similarly as in the proof of Theorem 2, create for each node u ∈ L one new node in R each connectedonly to u . Call the new Tanner graph G ′ . Then any S ⊂ L is a GA-trapping set in G ′ iff S is the support ofa codeword in G . This follows due to the fact that the first condition in the definition of GA-trapping sets isidentical to the ZP-restriction, with ℓ = 3. The second condition in the definition of an GA-trapping set isenforced automatically, since vertices in L \ S cannot be connected to odd-degree check nodes in G S due tothe fact that all such checks have degree one. Hence any α -approximation algorithm for MinTrap ZP yieldsan α -approximation algorithm for MinCodeword and the result follows.
Theorem 4.
For any constant α , there is no polynomial-time, α -approximation algorithm for MinTrap
AWGN ,unless RP = N P .Proof.
The proof is by a reduction from
MinCodeword , and follows along similar lines as the proof ofthe above theorems. To this end, we construct the Tanner graph ( L ∪ R, E ) of the dual code C ⊥ where L = { u , ...u k } , R = { v , ...v n } , and E = { ( u i , v j ) : M i,j = 1 } where M denotes a generator matrix of the codeof full row-rank. Note that for each S ⊂ L , Γ( S ) corresponds to a codeword. Hence, if we have an α -approxto the min-trapping set problem for any a , then this gives an α approximation algorithm to the minimumweight codeword problem by running through all values of a and taking the minimum of the resulting b ’s.But, since it is impossible to O (1)-approximate MinCodeword in polynomial time unless RP = N P [15],it is impossible to O (1)-approximate MinTrap
AWGN in polynomial time unless RP = N P . AWGN − elem Theorem 5.
For any α , it is NP-hard to α -approximate MinTrap
AWGN − elem .Proof. The proof is based on showing that a polynomial time algorithm for solving
MinTrap
AWGN − elem canbe used for solving the MaxThreeDimMatch problem, and is based on similar arguments as those usedfor showing that
MaxLikeDecode is NP-complete [5]. To this end, let us construct the matching incidencematrix D as follows. Let the collection of ordered triples be T ⊂ X × X × X , where | T | = t , and | X | = n .Then D is a 3 n × t dimensional zero-one matrix, with entries1 ≤ i ≤ n : D i,j = 1 , iff x j = i ; n + 1 ≤ i ≤ n : D i,j = 1 , iff y j = i ;2 n + 1 ≤ i ≤ n : D i,j = 1 , iff z j = i. As an example, the matrix D for the set of triples { (1 , , , (3 , , , (2 , , , (1 , , , (2 , , , (3 , , } over X = { , , } has the form D = . The set of triples { (1 , , , (2 , , , (3 , , } is a maximum three-dimensional matching over the set { , , } .Observe that all rows in the sub-matrix of D induced by the three columns corresponding to these tripleshave Hamming weight one. This is a consequence of the defining constraint of the MaxThreeDimMatch problem that asserts that every element in X appears at a given position of the matching exactly once.8ssume next that there exists a polynomial-time, α -approximation algorithm for the MinTrap
AWGN − elem problem. Construct D for a given matching problem, set b = 3 × n , and run the MinTrap
AWGN − elem al-gorithm on D . If the algorithm the algorithm finds an elementary trapping set then it must have size n .Consider the corresponding set of n columns indexed by a set of n triples from T . Each row in the sub-matrix induced by the triples has weight one, which follows from the definition of an elementary trappingset. Consequently, these triples represent a matching for T . This implies that no polynomial time algorithmfor the MinTrap
AWGN − elem problem exists, unless P=NP. AWGN − maj First we prove a hardness of approximation result for the problem of finding the good set of minimumcardinality. Recall that a set S ⊂ L us good if the majority of nodes in Γ( S ) have even degree in G S .We call this problem MinGood . We will then use this to show a hardness of approximation result for
MinTrap
AWGN − maj .Our proof uses a reduction from MinCodeword . Let H be the n × ( n − k ) parity check of somecode. We may assume that the code specified by H includes at least one codeword in addition to the zerovector. This gives rise to the graph G ′ = ( L ′ ∪ R ′ , E ′ ) where L ′ = { x , . . . , x n } , R ′ = { y , . . . , y m } , and E ′ = { ( x i , y j ) : H i,j = 1 } . We will create a bipartite graph G = ( L ∪ R, E ) by augmenting G ′ with graphicalobjects termed “ ZigZag ”s and “
OrGate ”s. These graphical objects will ensure that the minimum cardinalityof a good set is approximately proportional to the minimum weight of any codeword.
ZigZag
For each x ∈ L ′ we add a ZigZag ( x ) structure. This structure consists of 3( m −
1) nodes, given by L ( ZigZag ( x )) = { v , . . . , v m − } , R ( ZigZag ( x )) = { u , . . . , u m − , w , . . . , w m − } , and edges, E ( ZigZag ( x )) = { ( u i , v i ) , ( v i , w i ) : i ∈ [ m − } ∪ { ( v i , w i +1 : i ∈ [ m − } ∪ { ( x, w ) } The intuition behind the
ZigZag ( x ) structure is that if x is in the trapping set then the nodes L ( ZigZag ( x ))will also be in the trapping set. For a subgraph G ′′ of G , and S ∈ L we defineDisc S ( G ′′ ) = |{ v ∈ Γ( S ) ∩ V ( G ′′ ) : d G S ( v ) even }| − |{ v ∈ Γ( S ) ∩ V ( G ′′ ) : d G S ( v ) odd }| . Lemma 1.
For all x ∈ S , Disc S ( ZigZag ( x )) ≤ and Disc S ( ZigZag ( x )) = 0 iff ZigZag ( x ) ∩ L ⊂ S .Proof. Note that |{ v ∈ Γ( S ) ∩ V ( ZigZag ( x )) : d G S ( v ) odd }| ≥ |{ v ∈ S ∩ V ( ZigZag ( x )) | with equality iff L ′ ∩ V ( ZigZag ( x )) ⊂ S because each v i ∈ S is connected to w i which has degree 1. But forany S , |{ v ∈ Γ( S ) ∩ V ( ZigZag ( x )) : d G S ( v ) even }| ≤ |{ v ∈ S ∩ V ( ZigZag ( x )) | with equality iff L ′ ∩ V ( ZigZag ( x )) ⊂ S . OrGate
For each y ∈ R ′ we add OrGate ( y ), and let Γ( y ) ∩ L ′ = { u , . . . , u k ′ } . Let k = 2 ⌈ log k ′ ⌉ . The construction OrGate ( y ) consists two node sets L ( OrGate ( y )) and R ( OrGate ( y )). Consider a binary tree on the nodes { u , . . . , u k } where u k ′ + i = u k ′ for i ∈ [ k − k ′ ]. Then L ( OrGate ( y )) consists of nodes corresponding to theinternal nodes of the tree, i.e. L ( OrGate ( y )) = { v u ∨ u , . . . , v u k − ∨ u k , v u ∨ u ∨ u ∨ u , . . . , v u k − ∨ u k − ∨ u k − ∨ u k , . . . , v u ∨ u ∨ ... ∨ u k } For each internal node v with children u and w , we add four new check nodes C ( v ) := { c ( v ) , c ( v ) , c ( v ) , c ( v ) } :all are connected v , the first and third are connected to u and the first and second are connected to w . If v a) OrGate ( y ) (b) ZigZag ( x ) (c) Ballast
Figure 5: Reduction from
MinCodeword to MinTrap
AWGN − maj .is the root of the tree, we also add one more new check node which is connected only to v . We call this node z . Let R ( OrGate ( y )) be the set of such nodes, and let E ( OrGate ( y )) be the set of such edges. Finally, let f ( S, y ) = { v u i ∨ ... ∨ u j ∈ L ( OrGate ( y )) : | S ∩ { u i , . . . , u j }| ≥ } . Lemma 2.
For all y ∈ G S ∩ R , Disc S ( OrGate ( y )) ≤ − with equality if S ∩ L ( OrGate ( y )) = f ( S, y ) .Proof. Consider the four check nodes C ( v ) for some internal node v of the tree used in the construction of OrGate ( y ). Let u, w be the children of v in the original binary tree tree. Then, if either u, v, w ∈ S thenDisc S ( C ( v )) ≤ v ∈ S and at least one of u, w ∈ S . Consequently, if Γ( y ) ∩ L ′ ∩ S = ∅ ,Disc S ( ∪ v C ( v )) ≤ S ∩ L ( OrGate ( y )) = f ( S, y ). In particular, the root of the binary tree isin S and therefore the final check node z has odd degree. Therefore, if Γ( y ) ∩ L ′ ∩ S = ∅ , Disc S ( ∪ v C ( v )) ≤ − S ∩ L ( OrGate ( y )) = f ( S, y ). Note that the graph G that has been constructed has | L | ≤ mn + 2 n ( n − k ) and | R | ≤ ( n − m ) + n . Lemma 3.
Disc S ( G ) ≥ iff S ∩ L ′ is a codeword and for each x ∈ S ∩ L ′ , ZigZag ( x ) ∩ L ⊂ S .Proof. According to Lemma 1 and 2,Disc S ( G ) = Disc S ( G ′ ) + X x ∈ L ′ Disc S ( ZigZag ( x )) + X y ∈ R ′ Disc S ( OrGate ( y )) ≤ Disc S ( G ′ ) − X x ∈ L ′ I ZigZag ( x ) ∩ L S − | Γ( S ) ∩ R ′ | . Note that Disc S ( G ′ ) ≤ | Γ( S ) ∩ R ′ | and therefore Disc S ( G ) ≥ d G S ( y ) is even for all y ∈ R ′ and ZigZag ( x ) ⊂ G S for all x ∈ S ∩ L ′ . Again, according to Lemma 1 and 2 if ∀ y ∈ R ′ , d G S ( y ) = 0 mod 2, and ∀ x ∈ S ∩ L ′ , ZigZag ( x ) ⊂ G S , then Disc S ( G ) ≥ Theorem 6.
For any constant α , there is no polynomial-time, α -approximation algorithm for MinGood ,unless RP = N P . (xZigZag )(xZigZag )(xZigZag )(xZigZag x x x x y y y )(yOrGate )(yOrGate )(yOrGate BallastBallastBallast
Figure 6: Combining the
ZigZag , OrGate , and Ballast constructions.
Proof.
Assume that S is a good set such that S ≤ α MinGood for some constant α . By Lemma 3 andLemma 2, | S | = | S ∩ L ′ | m + X y ∈ Γ( S ∩ L ′ ) | f ( S, y ) | , and S ∩ L ′ corresponds to a codeword. But P y ∈ Γ( S ∩ L ′ ) | f ( S, y ) | ≤ n ( n − k ), and so by setting m sufficientlylarge we get a constant approximation for MinCodeword . But no such approximation exists unless RP = N P [15].
AWGN − maj To achieve the hardness result for
MinTrap
AWGN − maj we need to further augment our graph G with multiple“ Ballast ” constructions. We call the resulting graph G + . The intuition behind Ballast is that no nodes from
Ballast will be chosen in S while the multiple copies of Ballast will ensure that the complement of S is alsogood. A single Ballast consists of nodes L ( Ballast ) = { u , . . . , u l } , R ( Ballast ) = { v , . . . , v l , w , . . . , w l } , andedges, E ( Ballast ) = { ( u i , v i ) : i ∈ [ l ] } ∪ { ( v i , u i +1 ) : i ∈ [ l − } ∪ { ( v l , u ) } ∪ { ( u i , w i ) : 1 ≤ i ≤ l − } . We consider setting l = n | L | and adding | R | copies of Ballast to G . Lemma 4.
Disc S ( Ballast ) ≤ with equality iff L ( Ballast ) ⊂ S .Proof. Let A = S ∩{ u , . . . , u l } . Note that Γ( A ) contains at least | A |− L ( Ballast ) ⊂ S . Γ( A ) contains at most | A | nodes of even degree with equality iff L ( Ballast ) ⊂ S . Lemma 5.
Assuming there exists a non-zero codeword, there is a good set in G . Furthermore, any good setin G is a trapping set for G + .Proof. Let S ′ be the subset of L ′ corresponding to the minimum weight codeword. Let S = S ′ ∪ [ x ∈ S ′ L ( ZigZag ( x )) ! ∪ [ y ∈ Γ( S ′ ) f ( S, y ) . Then S is a good set in G . For the second part of the lemma note that by Lemma 4, for S ⊂ L , Disc ¯ S ( G + ) ≥| R | − | R | = 0. Theorem 7.
For any constant α , there is no polynomial-time, α -approximation algorithm for MinGood ,unless RP = N P .Proof.
Assume that S is a trapping set such that S ≤ α MinTrap
AWGN − maj for some constant α . ByLemma 5, we know that | S | ≤ α | L | and hence S does not include all left hand side nodes of any copy of Ballast because doing so would imply that | S | ≥ | L ( Ballast ) | = n | L | . But then by Lemma 4, we may assume11hat no nodes from Ballast are included in S because removing all such nodes from S increases Disc S ( G ).Consequently S must be a subset of L . Since any good subset of L is a trapping set, MinGood ( G ) = MinTrap
AWGN − maj ( G + ). But, by Theorem 6, there is no constant approximation of MinGood . The fact that a problem is NP-hard usually does not imply that a special instance of the problems is NP-hard.Since iterative decoding algorithms have both linear-time complexity and offer good decoding performanceonly for special classes of codes, it is important to establish the analogues of the results in Section 4 for suchcodes. We provide next a set of results establishing the hardness of approximating stopping and trappingsets for low-density parity-check (LDPC) codes.LDPC codes are linear block codes for which the parity-check matrix H is sparse -i.e., for which H hasa “small” number of non-zero entries. More formally, we define an LDPC code as follows. An LDPC code isa code with the property that each variable and check node in its Tanner graph G = ( L ∪ R, E ) has degreeat most δ v and δ c , respectively, for some constants δ v , δ c > n . Theorem 8.
There exists a constant α > such that it is NP-hard to α -approximate MinStop in theTanner graph of an LDPC code.
The proof follows along the same lines as the proof of NP-hardness using reduction from the problem
MinVertCov problem [27]: Let G = ( V, E ) be an undirected graph, which, without loss of generality, canbe assumed to be connected and of vertex degree bounded from above by three. Furthermore, also assumethat | V | = n , | E | = m , and that E = { e , . . . , e m } , V = { v , . . . , v n } . Without loss of generality, one can set e = ( v , v ) ∈ E . A bipartite graph G vc is constructed as follows: the left hand side vertices of the graphconsist of nodes L = L ∪ L , where L = V , and L = { e ′ , . . . , e ′ m } . The right hand side vertices of thegraph consist of nodes R = R ∪ R , with R = E , and R = { z , . . . , z m } . The set of edges of G vc is acollection of ordered pairs the following form: { ( e i ∈ R , u ∈ L ) , ( e i ∈ R , v ∈ L ) : e i = ( u, v ) ∈ E } ∪ { ( e i ∈ R , e ′ i ∈ L ) : 1 ≤ i ≤ m }∪{ ( z i ∈ R , e ′ i ∈ L ) , ( z i ∈ R , e ′ i +1 ∈ L ) : 1 ≤ i ≤ m − } ∪ { ( z m ∈ R , v ∈ L ) , ( z m ∈ R , e ′ ∈ L ) } . It is straightforward to show that if S is a stopping set in G , then S ∩ L is a vertex cover in G [27]. Asa consequence, there exists a constant ǫ > ǫ ) approximation algorithm for the MinStop problem, unless P=NP.Note that in the construction, each vertex in L has degree bounded from above by four (the auxiliaryvariable node e ′ , . . . , e | E | have, by construction, degree two, while all vertices in V other than v and v havedegree at most three; the vertices v and v can have degree at most four). Similarly, the check nodes havemaximum degree three, since by construction, the vertices z , . . . , z | E | have degree two, while the vertices in R have degree three.One can establish the even stronger result that the MinStop problem for LDPC codes remains NP hardeven for codes with Tanner graphs that avoid cycles of length four. This follows from the same arguments usedin the proof of the theorem above, with an additional reference to the hardness of the
MinSetCovInterOne problem, which also holds in the setting of sparse codes [23].
Theorem 9.
There exists a constant α > such that it is NP-hard to α -approximate MinTrap
AWGN − elem in the Tanner graph of an LDPC code.Proof. The proof follows along the same lines as the proof of Theorem 5, with the three-dimensional matchingproblem replaced by its constraint version involving a bounded number ℓ of appearances of each element in X . Theorem 10.
The problems
MaxLikeDecode and
MinCodeword are NP-hard for LDPC codes. roof. The proof is a direct consequence of the fact that the parity-check matrix used in the reduction fromthe
MaxThreeDimMatch to the
MaxLikeDecode problem is sparse (it has column weight three, andthe row weight can be made bounded as well by invoking the constraint that any element of X cannot appearmore than r ≥ MaxLikeDecode to the
MinCodeword problem [37, 38].As a consequence of the above finding, all trapping set problems described in Section 4, for which thehardness was established in terms of reductions from the
MinCodeword problem, remain NP-hard for theclass of LDPC codes.
The error floor is a phenomena inherent to iterative decoders that manifests itself as a sudden change in theslope of the BER performance of a code. Alternatively, it represents a phase transition in the dynamicalsystem of the decoder that prohibits it from attaining a sufficiently low BER. The error floor usually appearsat moderate to high signal-to-noise ratios, i.e. for small values of the erasure and error probability p of theBEC and BSC channel. For such values of p , the codeword error-rate R ( p ) has the formlog ( R ( p )) ≃ log( N κ ) + κ log( p ) , (1)where κ denotes the size of the smallest stopping/trapping sets, while N i represents the number of such sets.The dominating term in the expression is the linear term κ log( p ).As a consequence of the results in Section 4, we have the following result. Corollary 6.
Unless P = N P , there is no polynomial time algorithm for estimating the error-floor of codesused over the BEC and BSC within an O (1) term. For the AWGN channel with noise variance σ , a heuristic formula for the codeword error-rate was derivedin [31], where it was shown that R ( σ ) ≥ X T ∈T P ( T, σ ) , where T denotes the set of dominant (small) elementary trapping sets for the given code, and P ( T, σ ) isthe probability of decoder failure on a trapping set T . It was observed that simulation of decoding can beviewed as stochastic process for finding trapping sets [31]. This, and other methods that rely on combiningsimulation techniques with “aided flipping” methods and greedy search strategies, were all observed to beinefficient when estimating the error-floor of “good codes” - i.e. codes with large minimum stopping andtrapping set sizes. In the next section, we show that some problems discussed in the paper has complexitythat grows exponentially with the size of the smallest set being sought, but only polynomially with respectto the size of the input (i.e., code length). Consequently, one can easily find the smallest stopping sets offairly long codes, provided that the size of such stopping sets is not greater than 10 −
15 [14, 13, 34]. Thiswas observed in several papers, including [34].
Parameterized complexity represents a measure of the computational cost of problems that have several inputparameters. Problems for which one of the parameters, say π , is fixed are called parameterized problems.There exist problems that require exponential running time in the parameter π but that are computable ina time that is polynomial in the input size. Hence, if π is fixed at a small value, such problems can stillbe exactly solved in an efficient manner. A parameterized problem that allows for the existence of suchpolynomial time algorithms is termed a fixed-parameter tractable problem and it belongs to the class FPT,first studied by Downey and Fellows [13]. 13any NP-complete problems are fixed-parameter tractable. As an example, the MinVertCov is FPT,with complexity O ( κ n + (4 / κ κ ), where κ denotes the size of the smallest vertex cover, and n is the sizeof the input, i.e., the number of vertices in the graph. Despite the fact that MinVertCov is a specialinstant of
MinHitSet with set sizes equal to two, the latter is not known to have FPT algorithms whenparameterization is performed only with respect to the size of the smallest hitting set κ . Strong evidencesuggests that such an algorithm does not exist, since MinHitSet is W [2]-complete (for the non-trivialdefinition of the W [2] class, see [14]). It is only known that MinHitSet is FPT when the set sizes arebounded, and parameterization is performed with respect to, say, κ + δ max , where δ max denotes the size ofthe largest set in the MinHitSet formulation.In this section, we use the results of [8, 16, 33] to show that the
MinCStop problem is FPT. Furthermore,by invoking the recent results in [11], we show that the problem of enumerating all cover stopping sets isFPT as well.
Theorem 11.
The problem
MinCStop for LDPC codes of maximal constant check node degree δ c is inFTP, with best known complexity bound of the form O δ c − s δ c − !! κ + n ! . (2)The algorithm that achieves this bound is a tree search algorithm, see [16]. Theorem 12.
The problem of enumerating all minimal cover stopping sets in LDPC codes of maximalconstant check node degree δ c is in FTP, with best known complexity bound of the form O ⋆ (( δ c − o (1)) κ ) , where O ⋆ refers to an O ( · ) function for which all polynomial factors are suppressed, and where κ stands forthe size of the smallest cover stopping set. As a final remark, the problem
MinStop can be shown to be W[1]-hard, due to its connection to theExact Even Set problem [8].
We showed that a class of problems, pertaining to the size of the smallest stopping and trapping sets inTanner graphs is NP-hard to even approximate. Furthermore, we showed that similar results apply to theclass of LDPC codes. Our findings provide one of the few known families of codes for which the minimumdistance and stopping set problems are NP-hard. We also show that a simple instance of the stopping setproblem for LDPC codes, namely the complete stopping set problem, is fixed parameter tractable.
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