Optimal Index Codes with Near-Extreme Rates

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1 0 IEEE International Symposium on Information Theory Proceedings Optimal Index Codes with Near-Extreme Rates Son Hoang Dau Division of Mathematical Sciences School of Phys. and Math. Sciences Nanyang Technological University Nanyang Link, Singapore Vitaly Skachek Department of ECE McGill University 480 University Street Montréal, QC HA A7, Canada Yeow Meng Chee Division of Mathematical Sciences School of Phys. and Math. Sciences Nanyang Technological University Nanyang Link, Singapore Abstract The min-rank of a digraph was shown by Bar- Yossef et al. 006 to represent the length of an optimal scalar linear solution of the corresponding instance of the Index Coding with Side Information ICSI problem. In this work, the graphs and digraphs of near-extreme min-ranks are characterized. Those graphs and digraphs correspond to the ICSI instances having near-extreme transmission rates when using optimal scalar linear index codes. It is also shown that the decision problem of whether a digraph has min-rank two is NP-complete. By contrast, the same question for graphs can be answered in polynomial time. I. INTRODUCTION Index Coding with Side Information ICSI [] is a communication scheme dealing with broadcast channels in which receivers have prior side information about the messages to be broadcast. By using coding and by exploiting the knowledge about the side information, the sender may significantly reduce the number of required transmissions compared with the straightforward approach. Apart from being a special case of the well-known non-multicast Network Coding problem [], [], the ICSI problem has also found various potential applications on its owns, such as audio- and video-on-demand, daily newspaper delivery, data pushing, and oppoturnistic wireless networks [], [4], [5], [6]. In the work of Bar-Yossef et al. [4], the optimal transmission rate of scalar linear index codes for an ICSI instance was neatly characterized by the so-called min-rank of the side information digraph corresponding to that instance. The concept of min-rank of a graph goes back to Haemers [7]. Min-rank serves as an upper bound for the celebrated Shannon capacity of a graph [8]. It was shown by Peeters [9] that computing the min-rank of a general graph that is, the Min-Rank problem is a hard task. More specifically, Peeters showed that deciding whether the min-rank of a graph is smaller than or equal to three is an NP-complete problem. The interest in the Min-Rank problem has grown after the work of Bar-Yossef et al. [4]. Exact and heuristic algorithms for finding min-rank over the binary field of a digraph were developed in [0]. The minrank of a random digraph was investigated in []. A dynamic programming approach was proposed in [] to compute minranks of outerplanar graphs in polynomial time. The work of this author was done in part while he was with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 680, USA. In this paper, we study graphs and digraphs that have nearextreme min-ranks. In other words, we study ICSI instances with n receivers for which optimal scalar linear index codes have transmission rates, n, n, or n. Consequently, we show that the problem of deciding whether a digraph has min-rank two over F is NP-complete. By contrast, the same decision problem for graphs can be solved in polynomial time. The characterizations of graphs and digraphs with nearextreme min-ranks are summarized in the table below. The star mark indicates that the result is established in this paper. Min-Rank Graph G Digraph D G is complete trivial D is complete trivial D is not complete G is not complete and G is - and D is fairly - colorable [9] colorable n G connected has a maximum matching of size two and does not unknown contain F Fig. 4 as a subgraph n G connected is a star graph unknown n G has no edges trivial D has no circuits The paper is organized as follows. Basic notations and definitions are presented in Section II. In Section III, we present characterization of graphs and digraphs of near-extreme minranks. In Section IV, we prove the hardness of the decision problem of whether a digraph has min-rank two. The full version of this paper is available at web.spms.ntu.edu.sg/ daus000/extremeic.pdf. II. NOTATIONS AND DEFINITIONS Let [n] = {,,...,n}. Let F q denote the finite field of q elements. The support of a vector u F n q is defined to be the set suppu = {i [n] : u i 0}. For an n k matrix M, let M i denote the ith row of M. For a set E [n], let M E denote the E k sub-matrix of M formed by rows of M which are indexed by the elements of E. For any matrix M over F q, we denote by rank q M the rank of M over F q. We use e i to denote the unit vector, which has a one at the ith position, and zeros elsewhere. A graph is a pair G = VG,EG where VG is the set of vertices of G, and EG, the edge set, is a set of unordered pairs of distinct vertices of G. A typical edge of G is of the form {i,j} where i,j VG, and i j. For a graph with n vertices, we often take [n] as the vertex set. A digraph //$.00 0 IEEE 5

2 D = VD,ED is defined similarly, except that ED is called the arc set, which is a set of ordered pairs of distinct vertices of D. A typical arc of D is of the form e = i,j where i,j VD, and i j. We note that any graph can be viewed as a digraph with the same set of vertices, where each edge in EG is replaced with two antiparallel arcs in ED. Ifi,j ED thenj is called an out-neighbor ofi. The set of out-neighbors of a vertex i in the digraph D is denoted by N D O i. We simply use N Oi whenever there is no potential confusion. For a graph G, we denote by N G i the set of neighbors of i, namely, the set of vertices adjacent to i. The order of a digraph is its number of vertices. The complement of a digraph D, denoted D, is a digraph with vertex set VD and arc set {i,j : i,j VD, i j, i,j / ED}. The complement of a graph is defined analogously. A circuit in a digraph D = VD,ED is a sequence of pairwise distinct vertices C = i,i,...,i r, where i s,i s+ ED for all s [r ] and i r,i ED as well. A digraph is called acyclic if it contains no circuits. The ICSI problem is formulated as follows. Suppose a sender S wants to send a vector x = x,x,...,x n, where x i F t q for alli [n], tonreceiversr,r,...,r n. EachR i possesses some prior side information, consisting of the blocks x j, j X i [n], and is interested in receiving a single block x i. The sender S broadcasts a codeword Ex F κ q, κ N, that enables each receiver R i to recover x i based on its side information. Such a mapping E is called an index code. We refer to t as the block length and κ as the length of the index code. The ratio κ/t is called the transmission rate of the index code. The objective of S is to find an optimal index code, that is, an index code which has the minimum transmission rate. The index code is called linear if E is an F q -linear mapping, and nonlinear otherwise. The index code is called scalar if t = and vector if t >. The length and the transmission rate of a scalar index code t = are identical. Example II.. Consider the following ICSI instance in which n = 5 and X = {}, X = {}, X = {,4}, X 4 = {5}, X 5 = {,4}. Assume that x i F i [5]. S R R R x +x x +x x 4 +x 5 R 5 requests x R 4 requests x 5 owns x owns x,x 4 requests x requests x requests x 4 owns x owns x owns x,x 5 4 Fig. : Example of an ICSI instance and an index code The sender broadcasts three packets x +x, x +x, and x 4 + x 5. This index code is of length three. The decoding process goes as follows. As R knows x and receives x +x, it obtains x = x + x + x ; Similarly, R obtains x = x +x +x ; R obtains x = x +x +x +x +x ; R 4 obtains x 4 = x 5 +x 4 +x 5 ; R 5 obtains x 5 = x 4 +x 4 +x 5. This index code saves two transmissions compared with the naïve approach when S broadcasts all five messages. Each instance of the ICSI problem can be described by the so-called side information digraph [4]. Given n and X i, i [n], the side information digraph D is defined as follows. The vertex set VD = [n]. The arc set ED = i [n] {i,j : j X i }. The side information digraph corresponding to the ICSI instance in Example II. is depicted in Fig. a. Definition II. [7], [4]. Let D = VD,ED be a digraph of order n. A matrix M = m i,j Fq n n is said to fit D if m i,i 0 and m i,j = 0 whenever i j and i,j / ED. The min-rank of D over F q, denoted minrk q D, is defined to be the minimum rank of a matrix in Fq n n that fits D. Since a graph can be viewed as a special case of a digraph, the above definitions also apply to a graph. 5 4 a Side information digraph D b A rank-three matrix that fits D Fig. : Corresponding side information digraph Example II. Bar-Yossef et al. [4] showed that the length transmission rate of an optimal scalar linear index code for the ICSI instance described by D is precisely the min-rank of D. Let β q t,d denotes the length of an optimal vector index code of block length t over F q for an ICSI instance described by a digraphd. Lubetzky et al. [] defined the broadcast rateβ q D of the corresponding ICSI instance to be lim t β q t,d/t see also [4]. Then minrk q D is an upper bound on β q D. Theorem II. [7], [4], [4]. For any digraph D we have αd β q D minrk q D, where αd denotes the size of a maximum acyclic induced subgraph of D. The same inequalities hold for a graph G, where αg denotes the size of a maximum independent set of G. III. DIGRAPHS OF NEAR-EXTREME MIN-RANKS A. Digraphs of Min-Rank One Proposition III.. A digraph has min-rank one over F q if and only if it is a complete digraph. Due to Theorem II., it is also trivial to see that β q D = if and only if D is a complete digraph. B. Digraphs of Min-Rank Two It was observed by Peeters [9] that a graph has min-rank two if and only if it is not a complete graph and its complement is a bipartite graph. We now focus only on digraphs of min-rank two. In this section, only the binary alphabet is considered. We first introduce the following concept of a fair coloring of a digraph. Recall that a k-coloring of a graph G = VG,EG is a mapping φ : VG [k] which satisfies 5

3 the condition that φi φj whenever {i,j} EG. We often refer to φi as the color of i. If there exists a k-coloring of G, then we say that G is k-colorable. Definition III.. Let D = VD,ED be a digraph. A fair k-coloring of D is a mapping φ : VD [k] that satisfies the following conditions C If i,j ED then φi φj; C For each vertex i of D, it holds that φj = φh for all out-neighbors j and h of i. If there exists a fair k-coloring of D, we say that we can color D fairly by k colors, or, D is fairly k-colorable. Lemma III.. A digraph D = VD, ED is fairly - colorable if and only if there exists a partition of VD into three subsets A, B, and C that satisfy the following conditions For every i A: either N O i B or N O i C; For every i B: either N O i A or N O i C; For every i C: either N O i A or N O i B. Theorem III.4. Let D = VD,ED be a digraph. Then minrk D if and only if D, the complement of D, is fairly -colorable. Proof: Suppose VD = [n]. The ONLY IF direction: By the definition of min-rank, minrk D implies the existence of an n n binary matrix M of rank at most two that fits D. There must be some two rows of M that span its entire row space. Without loss of generality, suppose that they are the first two rows of M, namely, M and M these two rows might be linearly dependent if minrk D <. Let B = suppm suppm, A = suppm \ B, and C = suppm \B. Since the binary alphabet is considered and the matrix M has no zero rows, for every i [n], one of the following must hold: M i = M ; M i = M ; M i = M + M. Hence i suppm i A B C for every i [n]. This implies that A B C = [n]. Suppose that i A. Then either M i = M or M i = M + M. The former condition holds if and only if suppm i = A B, which in turns implies that i,j ED for all j A B \{i}. In other words, i,j / ED for all j A B. Here D = VD,ED is the complement of D. The latter condition holds if and only if suppm i = A C, which implies that i, j / ED for all j A C. In summary, for every i A we have i,j / ED, for all j A; Either i,j / ED, for all j B, or i,j / ED, for all j C; In other words, for every i A, either NO D i B or NO D i C. Analogous conditions hold for every i B and for every i C as well. Therefore, by Lemma III., D is fairly -colorable. The IF direction: Suppose now that D is fairly -colorable. It suffices to find an n n binary matrix M of rank at most two which fits D. By Lemma III., there exists a partition of VD into three subsets A, B, and C that satisfy the following three conditions For every i A: either NO Di B or ND O i C; For every i B: either NO Di A or ND O i C; For every i C: either NO Di A or ND O i B. We construct an n n matrix M = m i,j as follows. For each i A, if NO Di B then let m i,j = for j A C, and m i,j = 0 for j B. Otherwise, if NO D i C then let m i,j = for j A B, and m i,j = 0 for j C. For i B and i C, M i can be constructed analogously. It is obvious that M fits D. Moreover, each row of M can be written as a linear combinations of the two binary vectors whose supports are A B and B C, respectively. Therefore, rank q M. We complete the proof. The following corollary characterizes the digraphs of minrank two. Corollary III.5. A digraph D has min-rank two over F if and only if D is fairly -colorable and D is not a complete digraph. For a graph G, it was proved by Blasiak et al. [5] that β G = if and only if G is bipartite and G is not a complete graph. A characterization by forbidden subgraphs of digraphs D with β D = was also obtained therein. C. Digraphs of Min-Ranks Equal to Their Orders Proposition III.6. Let G be a graph of order n. Then minrk q G = n if and only if G has no edges. Proposition III.7. Let D be a digraph of order n. Then minrk q D = n if and only if D is acyclic. Proof: Equivalently, we show that minrk q D n if and only if D has a circuit. Let VD = [n]. Suppose D has a circuit C = i,i,...,i r. We construct a matrix M fitting D as follows. Let VC = {i,...,i r }. For j / VC, letm j = e j. Fors [r ], letm is = e is e is+. Finally, let M ir = e i e ir. Clearly, M fits D. Moreover, as M ir = r s= M i s, we have rank q M VC r. Hence rank q M = rank q MVC +rankq M[n]\VC r +n r = n. Therefore, minrk q D n. Conversely, suppose that minrk q D n. By Theorem II., αd n. Therefore, D contains a circuit. By Theorem II., it is not hard to see that Proposition III.6 and III.7 also hold if we replace minrk q by β q. D. Graphs of Min-Ranks One Less Than Their Orders Definition III.8. A graph G = VG,EG is called a star graph if VG and there exists a vertex i VG such that EG = { {i,j} : j VG\{i} }. Fig. : A star graph Theorem III.9. Let G be a connected graph of order n. Then minrk q G = n if and only if G is a star graph. 5

4 Proof: We first suppose that minrk q G = n. If n = then G must be a complete graph, which is a star graph. We assume thatn. AsG is connected, there exists a vertex i of degree at least two. Let i and i be any two distinct vertices adjacent to i. Our goal is to show that for every vertex j i, we have {i,j} EG, and those are all possible edges. Firstly, suppose for a contradiction that {i, j} / EG for some j i. Since G is connected, there exists k such that {j,k} EG. Then either k i or k i. Suppose that k i. We create a matrix M as follows. Let M j = M k = e j + e k, M i = M i = e i + e i, and M h = e h for h / {i,j,i,k}. Then M fits G and rank q M n. Therefore, minrk q G < n. We obtain a contradiction. Thus, all other vertices are adjacent to i. Secondly, suppose for a contradiction that there exist two adjacent vertices, namely j and k, both are different from i. As we just proved, both j and k must be adjacent to i. We create a matrix M as follows. We take M i = M j = M k = e i + e j + e k, and M h = e h for h / {i,j,k}. Clearly M fits G and moreover, rank q M n, which implies that minrk q G < n. We obtain a contradiction. Conversely, assume that G is a star graph, where EG = { {i,js } : s [n ] }, j s VG\{i} for all s [n ]. We create a matrix M fitting G by taking M js = e i +e js for s [n ], and M i = e i +e j. Since M i M j, we deduce that rank q M n. Hence minrk q G n. On the other hand, since{j s : s [n ]} is a maximum independent set in G, we obtain that αg = n. By Theorem II., minrk q G αg = n. Thus, minrk q G = n. It is not hard to see that Theorem III.9 still holds with minrk q being replaced by β q. E. Graphs of Min-Ranks Two Less Than Their Orders A matching in a graph is a set of edges without common vertices. A maximum matching is a matching that contains the largest possible number of edges. The number of edges in a maximum matching in G is denoted by mmg. Theorem III.0. Suppose G is a connected graph of order n 6. Then minrk q G = n if and only if mmg = and G does not contain a subgraph isomorphic to the graph F depicted in Fig. 4. minrk q G n mmg = n. As αg minrk q G, it suffices to show that αg = n. Let {a,b} and {c,d} be the two edges in a maximum matching in G. Let U = {a,b,c,d} and V = VG\U. The main idea is to show that we can always find two nonadjacent vertices in U which are not adjacent to any vertex in V. Such two vertices can be added to V to obtain an independent set of size n, which establishes the proof. It is also not hard to see that Theorem III.0 still holds with minrk q being replaced by β q. IV. THE HARDNESS OF THE MIN-RANK PROBLEM FOR DIGRAPHS In this section, we first prove that the decision problem of whether a given digraph is fairly k-colorable see Definition III. is NP-complete, for any given k. The hardness of this problem, by Proposition III. and Corollary III.5, leads to the hardness of the decision problem of whether a given digraph has min-rank two overf. The fairk-coloring problem is defined formally as follows. Problem: FAIR k-coloring Instance: A digraph D, an integer k Output: True if D is fairly k-colorable, False otherwise Theorem IV.. The fair k-coloring problem is NP-complete for k. Proof: This problem is obviously in NP. For NP-hardness, we reduce the k-coloring problem to the fair k-coloring problem. Suppose that G = VG,EG is an arbitrary graph. We aim to build a digraph D = VD,ED so that G is k- colorable if and only if D is fairly k-colorable. Suppose that VG = [n]. For each i [n], we build the following gadget, which is a digraph D i = V i,e i. The vertex set of D i is V i = {i} { ω i,ij : i j N G i }, where ω i,ij are newly introduced vertices. We refer to ω i,ij as a clone in D i of the vertex i j [n]. The arc set of D i is E i = { ω i,ij,i : i j N G i }. Let N G i = {i,i,...,i ni }. Then D i is depicted in Fig. 5. i Fig. 4: The forbidden subgraph F Sketch of the proof: For the ONLY IF direction, suppose that minrk q G = n. It is not hard to see that minrk q G n mmg. Hence mmg. As mmg {0,} and VG 6 imply that either G has no edges minrk q G = n > n or G is a star graph minrk q G = n > n, we deduce that mmg =. Moreover, as the graph F has min-rank three less than its order, G should not contain any subgraph isomorphic to F. We now turn to the IF direction. Suppose that mmg = and G does not contain any subgraph isomorphic to F. Then ω i,i ω i,i ω i,ini Fig. 5: Gadget D i for each vertex i of G Additionally, we also introduce n new vertices p,p,...,p n. The digraph D = VD,ED is built as follows. The vertex set of D is Let VD = n i= V i {p,p,...,p n }. Q i = { p i,i } { p i,ω i,i j : i [n], j [n i ], i j = i }, 54

5 be the set consisting of p i,i and the arcs that connect p i and all the clones of i. The arc set of D is then defined to be ED = n i= E i n i= Q i. For example, if G is the graph in Fig. 6, then D is the digraph in Fig. 7. Fig. 6: An example of the graph G ω, ω, ω, ω, p p p Fig. 7: The digraph D built from the graph G in Fig. 6 Our goal now is to show that G is k-colorable if and only if D is fairly k-colorable. Suppose that G is k-colorable and φ G : [n] [k] is a k- coloring of G. We define φ D : VD [k] as follows For every i [n], φ D i = φ G i; If i j = i, then φ Dω i,i j = φ D i = φ G i, in other words, clones of i have the same color as i; For every i [n], φ D p i is chosen arbitrarily, as long as it is different from φ D i. We claim that φ D is a fair k-coloring for D. We first verify the condition C see Definition III.. It is straightforward from the definition of φ D that the endpoints of each of the arcs of the forms p i,i for i [n], and p i,ω i,i j for i j = i, have different colors. It remains to verify that i and ω i,ij have different colors. On the one hand, ω i,ij is a clone of i j, and hence has the same color as i j. In other words, φ D ω i,ij = φ D i j = φ G i j. On the other hand, since i j N G i, we obtain that φ G i j φ G i = φ D i. Hence, φ D ω i,ij φ D i i [n]. Thus, C is satisfied. We now check if C see Definition III. is also satisfied. The out-neighbors of p i are i and its clones, namely, ω i,i j with i j = i. These vertices have the same color in D by the definition of φ D. Thus C is also satisfied. Therefore φ D is a fair k-coloring of D. Conversely, suppose that φ D : VD [k] is a fair k- coloring of D. Condition C guarantees that all clones of i have the same color asi, namely,φ D ω i,i j = φ Di ifi j = i. Therefore, by C, if {i,j} EG, that is, j N G i, then φ D i φ D ω i,j = φ D j. Hence, if we define φ G : [n] [k] by φ G i = φ D i for all i [n], then it is a k-coloring of G. Thus G is k-colorable. Finally, notice that the order of D is a polynomial with respect to the order of G. More specifically, VD = VG + EG and ED = VG + 4 EG. Moreover, building D from G, and also obtaining a coloring of G from a coloring of D, can be done in polynomial time in VG. Since the k-coloring problem k is NP-hard [6], we conclude that the fair k-coloring problem is also NP-hard. Corollary IV.. Given an arbitrary digraph D, the decision problem of whether minrk D = is NP-complete. Recall that for a graph G, it was observed by Peeters [9] that G has min-rank two if and only if G is a bipartite graph and G is not a complete graph, which can be verified in polynomial time. Another related result, presented by Blasiak et al. [5], stated that there is a polynomial time algorithm to verify whether β D = for a general digraph D. REFERENCES [] Y. Birk and T. Kol, Informed-source coding-on-demand ISCOD over broadcast channels, in Proc. IEEE Conf. on Comput. Commun. INFOCOM, San Francisco, CA, 998, pp [] R. Ahlswede, N. Cai, S. Y. R. Li, and R. W. Yeung, Network information flow, IEEE Trans. Inform. Theory, vol. 46, pp. 04 6, 000. [] R. Koetter and M. Médard, An algebraic approach to network coding, IEEE/ACM Trans. Netw., vol., pp , 00. [4] Z. Bar-Yossef, Z. Birk, T. S. Jayram, and T. Kol, Index coding with side information, in Proc. 47th Annu. IEEE Symp. on Found. of Comput. Sci. FOCS, 006, pp [5] S. El Rouayheb, A. Sprintson, and C. Georghiades, On the index coding problem and its relation to network coding and matroid theory, IEEE Trans. Inform. Theory, vol. 56, no. 7, pp , 00. [6] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Médard, and J. Crowcroft, Xors in the air: Practical wireless network coding, in Proc. ACM SIGCOMM, 006, pp [7] W. Haemers, An upper bound for the shannon capacity of a graph, Algebr. Methods Graph Theory, vol. 5, pp. 67 7, 978. [8] C. E. Shannon, The zero-error capacity of a noisy channel, IRE Trans. Inform. Theory, vol., pp. 5, 956. [9] R. Peeters, Orthogonal representations over finite fields and the chromatic number of graphs, Combinatorica, vol. 6, no., pp. 47 4, 996. [0] M. A. R. Chaudhry and A. Sprintson, Efficient algorithms for index coding, in Proc. IEEE Conf. on Comput. Commun. INFOCOM, 008, pp. 4. [] I. Haviv and M. Langberg, On linear index coding for random graphs, 0, preprint. [] Y. Berliner and M. Langberg, Index coding with outerplanar side information, in Proc. IEEE Symp. on Inform. Theory ISIT, Saint Petersburg, Russia, 0, pp [] E. Lubetzky and U. Stav, Nonlinear index coding outperforming the linear optimum, IEEE Trans. Inform. Theory, vol. 55, no. 8, pp , 009. [4] N. Alon, A. Hassidim, E. Lubetzky, U. Stav, and A. Weinstein, Broadcasting with side information, in Proc. 49th Annu. IEEE Symp. on Found. of Comput. Sci. FOCS, 008, pp [5] A. Blasiak, R. Kleinberg, and E. Lubetzky, Index coding via linear programming, 0, preprint. [6] R. M. Karp, Reducibility among combinatorial problems, Complexity Comput. Comput., vol. 40, no. 4, pp. 85 0,