Perfect Graphs and an Application to Optimizing Municipal ServicesAuthor(s): Alan TuckerSource: SIAM Review, Vol. 15, No. 3 (Jul., 1973), pp. 585-590Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2028578 .

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SIAM REVIEW

Vol. 15, No. 3, July 1973

PERFECT GRAPHS AND AN APPLICATION TO OPTIMIZING MUNICIPAL SERVICES*

ALAN TUCKERt

Abstract. The theory of perfect graphs deals with the fundamental graph-theoretic concepts of a clique and independent set. This article discusses the properties of perfect graphs and mentions some recent results about these graphs. A novel application of perfect graphs is presented. The application relates to an urban science problem involving optimal routing of garbage trucks.

The theory of perfect graphs deals with the fundamental graph-theoretic concepts of a clique (a set of mutually adjacent vertices) and an independent set (a set of vertices no pair of which is adjacent). While perfect graphs are quite interesting to graph theorists in their own right, these graphs also have great importance because of their varied applications. Perfect graphs arise in graph coloring problems and in statistical completion of block designs. Perfect graphs are closely related to perfect channels in communication theory. In a current project in the Urban Science Department at Stony Brook, a theorem about perfect graphs is playing a central role in the solution of a very difficult refuse collection problem.

A graph G = (V, A) is a finite set V (vertices) with a binary relation A (ad- jacency) defined on V which is symmetric and irreflexive. We call G1 = (V1, Al) a subgraph of G if V1 c V and A 1 is the restriction of A to V1 (some authors call G 1 a vertex-generated subgraph). The complement of G is a graph Gc = (V, AC) with the same vertex set and adjacency defined: for x, y E Vand x =A y, then xACy - xAy (any pair of distinct vertices are adjacent in exactly one of G and GC). Note that (GC)c = G. Observe that the concepts of clique and independent set are complemen- tary in that a clique in G is an independent set in Gc and an independent set in G is a clique in GC. A primitive circuit in G is a sequence of vertices (x1, x2, ... , xn) such that xiAxi+1, 1 < i < n-1, and x1Axn, but -xiAxi for i + 1 < j except x1Axn. For example, in Fig. 1, the circuit (a, b, c, d, e) is primitive but the circuit (e,f, c, d) is not.

e b

d c

FIG. 1

* Received by the editors January 18, 1972, and in revised form June 19, 1972. t Division of Mathematical Sciences, State University of New York, Stony Brook, Long Island,

New York 11790. 585

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586 ALAN TUCKER

We associate the following four numbers with a graph G: A(G) is the size of the largest clique in G; a(G) is the size of the largest independent set in G; 0(G) is the minimum number of cliques that cover all vertices of G; and y(G) is the min- imum number of independent sets that cover all vertices of G. In a vertex coloring problem, one seeks a minimum number of colors (called the chromatic number of G) such that adjacent vertices have different colors. Since the set of all vertices having a given color forms an independent set, the chromatic number is just y(G). For any given clique C and any independent set covering {IiJ, we observe that no two vertices in C can be contained in the same Ii. It follows that A(G) < y(G). By a complementary argument, we obtain a(G) < 0(G). For the graph G in Fig. 1, A(G) = 3 = y(G) and a(G) = 2 < 3 = 0(G), but in the subgraph obtained by removing vertex a (or any other vertex except f), equality holds in both cases. If the vertex f is removed, strict inequality holds in both cases. In the graph G1 obtained by adding an adjacency between a and f in G, strict inequality also holds in both cases. By complementarity, A(G) =-x(GC) and y(G) = 0(GC). Thus A(G) = y(G) if and only if cx(GC) = 0(GC). Then, for example, the inequalities are strict for the complement of G1. We shall be concerned with graphs where A(G) = y(G) and ax(G) = 0(G). A related question is whether there exists an upper bound on y(G) of the form A(G) + k (and similarly for 0(G)). The answer is no: for any integer k > 0, one can construct a graph G without triangles (i.e., A(G) = 2) yet having y(G) = k (see [10]).

The problem of determining A(G), y(G), ax(G) and 0(G) is quite difficult and generally requires finding a minimal cover or a set that forms a maximal clique (or independent set). A common technique for finding y(G) in a general graph is to pose the problem as an integer program. Let the vertices of G be indexed x1, x2 , -, xn and form a list I1, I2, * **, Ik of all maximal independent sets of G. Let the n x k matrix A be defined with aij = 1 if Xi E Ij and aij = 0 otherwise. Now, y(G) is the value of the integer program

minu ulk

Au _ l" u > 0, u integer,

where u is a k-component column vector and l1 is a j-component column vector with each entry equal to 1. Note that the value of the dual integer program

max v 1",

vA < lk v > 0, v integer

is just A(G) (i.e., the dual program seeks a maximal set of vertices, no two (or more) of which are in the same Ij; there is a more direct way to obtain A(G)). By duality theory, we know that max v 1 < min u 1, and thus we have rederived A(G) < y(G).

Now we turn to graphs in which A(G) = y(G) and a(G) = 0(G). One common example is a bipartite graph G = (V, A), where V partitions into two disjoint subsets V1 and V2 such that all adjacencies are between a vertex in V1 and a vertex in V2. Assuming there is at least one adjacency, then the largest clique is of size 2, and V1 and V2 are both independent sets. Thus A(G) = 2 = y(G). The fact that aX(G) = 0(G) in bipartite graphs is known as Konig's theorem (see [6]). Graphs in which ax(G) = 0(G) correspond to perfect channels in communication theory (see

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OPTIMIZING MUNICIPAL SERVICES 587

Shannon [13]). Berge [3], [4], [5] has shown that A(G) = y(G) and a(G) = 0(G) for several familiar classes of graphs. Actually Berge was concerned with stronger properties. A graph is called y-perfect if for all subgraphs H of G (including G itself) A(H) = y(H). A graph is called x-perfect if for all subgraphs H of G, a(H) = 0(H). A graph is called perfect if it is both y-perfect and aX-perfect. Berge [4] showed that bipartite graphs, comparability graphs, triangulated graphs, and unimodular graphs are all perfect. He made the following conjecture [5].

CONJECTURE 1 (Weak Perfect Graph Theorem). Thefollowing are equivalent: (a) G is perfect, (b) G is o-perfect, (c) G is y-perfect. Observe that by complementarity, the last condition is superfluous. Note

that the graph G in Fig. 1 is not y-perfect (and obviously not ax-perfect) since in the subgraph G*, obtained by deleting f, A(G*) = 2 < 3 = y(G*). Fulkerson [9] has generalized the ideas of ax-perfection and y-perfection to antiblocking pairs of polyhedra and has proved what he calls the "pluperfect graph theorem" (these antiblocking pairs are generalizations of the integer program above). Recently Lovasz [12] extended Fulkerson's work to obtain a proof of Conjecture 1 from the pluperfect graph theorem.

Observe that the "reason" for nonperfection in the graph in Fig. 1 is the exis- tence of a primitive circuit of odd length k ? 5. (Let OPC denote such a circuit; the graph G* mentioned above is an OPC.) If a graph G consists of an OPC of length 2k + 1 (k > 2), then A(G) = 2 < 3 = y(G) and oa(G) = k < k + 1 = 0(G). Clearly one must also watch out for graphs which are the complement of an OPC. Bipartite graphs, known to be perfect, are characterized as graphs with no odd primitive circuits (including length 3). Thus for a graph G with no triangles, G is perfect if and only if G has no OPC's. Gilmore conjectured that this result generalized as follows.

CONJECTURE 2 (Strong Perfect Graph Conjecture). Thefollowing are equiva- lent:

(a) G is oi-perfect, (b) G is y-perfect, (c) neither G nor GC contains any OPC's. Actually, one of the first two conditions is superfluous: for instance, if any

graph G is ai-perfect whenever (c) holds, then Gc is also ai-perfect when (c) holds; but this is equivalent to G being y-perfect. This author [14] recently proved that Conjecture 2 is valid for planar graphs (graphs that can be drawn without edges crossing). For this special case, OPC's in Gc are not involved, since an OPC of length k _ 7 in Gc would correspond to a nonplanar subgraph in G (and an OPC of length 5 in Gc corresponds to an OPC of length 5 in G). Conjecture 2 has the following useful corollary for coloring problems.

COROLLARY 1 (Conjecture). If G and GC contain no OPC's and G has no clique of size k + 1, then G can be k-colored.

The graph in Fig. 1 shows that the converse of Corollary 1 is not true. Other conditions are known for insuring k-colorability, but most are either unnecessarily strong, e.g., the degree of every vertex should not exceed k - 1, or else apply to just a special class of graphs. Unfortunately, the conditions of Corollary 1 are

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588 ALAN TUCKER

quite difficult to test. It is hoped (for reasons mentioned later) that other weak conditions for k-colorability will be found. However there has been no strong impetus for efficient, yet reasonably weak, coloring tests because there is no men- tion in the literature, to this author's knowledge, of operations research-type coloring problems which have needed to make use of graph-theoretic coloring tests. The following is a typical example of an applied coloring problem (see [7]). A state legislature wishes to minimize the number of hours to be set aside weekly for committee meetings. So a meetings schedule of minimal duration is needed (in which no two committees meet at the same hour if they have a common mem- ber). We set up a graph with one vertex for each committee and make two vertices adjacent if they correspond to two committees with a common member. Then an independent set in this graph corresponds to a set of committees whose member- ships do not overlap, that is, a set of committees that could all meet at the same hour. Thus, the solution to this problem is a minimal collection of independent sets which cover this graph a minimal coloring. The problem was converted [7] into the integer program for y(G) formulated above and solved by an algorithm due to Salkin [11].

Recently, this author was shown a routing problem whose solution required frequent testing for the k-colorability of a certain type of graph (in only one in- stance need an actual coloring be found): Since the associated integer program (which gives a k-coloring when it exists) takes a long time to solve, it was hoped that graph theory could provide a quicker sufficiency test which would not be too strong (would not eliminate too many graphs which are k-colorable). As implied above, this appears to be the first nontrivial application of coloring theory to operations research.

The k-colorable graphs arise in the following problem which was posed to the Urban Science Department at Stony Brook by the City of New York [1]. There is a set of sites Si which must be serviced ki times each week (1 < ki < 6; in this case, the visit was to pick up garbage). One wishes to derive a minimal, or near-minimal, set of (day-long) truck tours for a week such that each site is visited ki times and, in addition, such that these tours can be partitioned among the six days of the week (Sunday is excluded) in a manner so that no site is visited twice on one day. Even if each site is visited just once a week, this is, in general, an extremely difficult problem (which is complicated in this case by the fact that the garbage trucks must return periodically to their dumpsites). The method proposed for attacking this multiple-visit problem was a modification of an algorithm due to Clarke and Write [8]. This algorithm starts with an inefficient set of tours and successively tries to improve the set of tours. This method only gives a near-minimal set of tours. Further, it does not check to see that the derived tour set can be partitioned among the days of the week as required. This is where the coloring problem arises. Given a set of tours, we form an associated tour graph with one vertex for each tour and make two vertices adjacent if they correspond to two tours which visit a common site. As in the committee assignment problem, partitioning the tours among the 6 days is equivalent to 6-coloring the tour graph. Suppose we had a simplified situation where the week contained just 3 work days and we had generated the tours shown in Fig. 2a. In the tour graph in Fig. 2b, we desire a 3-coloring (which indeed exists). If the Clarke-Wright algorithm were to combine

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OPTIMIZING MUNICIPAL SERVICES 589

B tour B

toUr A

tourA ~~~~~~tour D A C

tourC

FD

tourF

-.-tour E

FIG. 2a. Tour routes; do tours A and B FIG. 2b. Tour graph on day 1, B and F on day 2, and C and

E on day 3

tours A and F in Fig. 2b (to get a smaller set of tours), such a move would have to be blocked (and other optimizing moves tried instead) because the resulting tour graph would have a clique of size 4. Thus in the 6-day case, to insure that the final set of tours has a tour graph which can be 6-colored, we would check each successive improvement tried by the Clarke-Wright algorithm to see that the resulting tour graph Gi does not have a clique of size 6 and that neither Gi nor Gi has an OPC. (This test assumes the validity of Conjecture 2, but were this test to fail in some routing problem, then at least operations research's loss would be mathematics' gain.)

Before they had knowledge of Corollary 1, the urban science researchers working on this problem had limited themselves to the case where each site is visited either 3 or 6 times. Then they solved the problem for the first two days of the week (a 3-time-a-week site would be visited once in the first two days) and here the tour graph only needed to be checked for bipartiteness (for this, it suffices to exclude all odd circuits, primitive or not). The resulting 2-day schedule was repeated three times to fill the week's demand in an even pattern. In the general case, we want to require that a site visited three (or two) times a week not only must have each visit on a different day but must also have the visits spaced evenly through the week. There are several ways to realize this constraint. One can manually alter the near-minimal tour set to insure even spacing, or one can solve the routing problem separately for each half (or third) of the week. In the extreme case, one assigns each site a specific set of days on which it is to be visited and then the simple routing problem is solved for each day separately. These and other approaches are currently being investigated.

In closing, we note that once the garbage gets to the dumpsites (situated along rivers) and is loaded on barges, then there is a whole new (but similar) routing problem for the tours of the tug boats that push groups of barges down to a landfill site on Staten Island. Because of special constraints in this case (one is that a tugboat pushing full barges must move with the tides), a different (randomized) variation of the Clarke-Wright algorithm was used [2].

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590 ALAN TUCKER

REFERENCES

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[2] E. BELTRAMI, N. BHAGAT AND L. BODIN, Refuse disposal in New York City: An analysis of barge dispatching, Transportation Sci., to appear.

[3] C. BERGE, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ., Halle-Wittenberg Math.-Natur. Reihe, (1961), pp. 114-115.

[4] , Some classes of perfect graphs, Graph Theory and Theoretical Physics, Academic Press, New York, 1967, pp. 155-165.

[5] , The rank of a family of sets and some applications to graph theory, Recent Progress in Combinatorics, W. Tutte, ed., Academic Press, New York, 1969, pp. 246-257.

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(1956), pp. 8-19. [14] A. TUCKER, The strong perfect graph theorem for planar graphs, Canad. J. Math., to appear.

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