Shiftable Interval Graphs

A Shiftable Interval Graph (SIG) is defined by a set of intervals and a set of windows associated with the intervals. Each interval does not have a fixed position, but it is allowed to move, provided that it remains completely contained into its window. Once a position has been fixed for all the intervals, the graph becomes an usual interval graph. In this paper we address the problem of finding the position of the intervals, which minimizes or maximizes some classical measures of the graph, such as clique number, stability number, chromatic number, clique cover number. We mainly focus on complexity aspects, bounds and solution algorithms. Some problems are solvable in polynomial time, others are proved to be NP-hard. Moreover some subclasses of SIG's ,for which exist polynomial algorithms exist, are characterized. Many practical applications can be reduced to problems on SIG's, and SIG's seem to be an interesting modeling framework. 1.ÊIntroduction and general definitions In the present paper the class of Shiftable Interval Graphs (SIGÕs) is introduced as an extension of the class of interval graphs (IGÕs) (Golumbic 1980). For this class of graphs we will study the well known concepts of clique, independent set, cover by clique, coloring. In particular we will analyze the complexity of determining some characteristic measures on SIGÕs (such as min or max clique number, min or max stability number, etc.). When possible we will devise efficient algorithms. For the NP-complete problems, we shall propose lower and upper bounds and identify subclasses of easy instances. A SIG S is defined by a set of n triples tiÊ=Ê of non-negative integer numbers satisfying riÊÐÊliÊ3ÊliÊ>Ê0, i.e. SÊoÊ{tiÊ=ÊÊÎÊZ+: riÊÐÊliÊ3ÊliÊ>Ê0, for iÊ=Ê1,É,n}. The pair [li,ri] will be called window wi and the value li will be called the length of the interval associated with window wi. It is easy to think of a SIG as a set of intervals each of which is free to move within the corresponding window, i.e. such that the left endpoint of the ith interval does not lay on the left of li, and the right endpoint of the same interval does not lay on the right of ri. The exact position of each interval within its window is easily described by means of the placement jÊ=Ê[j1,Êj2,É,Êjn], a vector the jth component of which represents the distance between the left endpoint of the jth interval and the left endpoint, lj, of the corresponding window wj. A placement jÊis feasible if 0Ê£ÊjjÊ£ÊrjÊÐÊljÊÐÊlj for all j. Thus, once jj has been fixed to Ð jj, the coordinate of the left and right endpoints of the jth interval are given by ljÊ+Ê Ð jj and ljÊ+Ê Ð jjÊ+Êlj, respectively. In what follows, the pair (ti, Ð ji) will represent the fact that the ith interval has been placed according to Ð ji, that is the pair (ti, Ð ji) represents the interval [liÊ+Ê Ð ji,ÊliÊ+Ê Ð jiÊ+Êli]. (*) Dipartimento di Elettronica e Informazione Politecnico di Milano, Via Ponzio 34/5 20133 Milano, Italy. e-mail: [email protected] (**) IASI CNR, Viale Manzoni 30, 00185 Roma, e-mail [email protected] 2 By interval model M( Ê Ðj) we shall indicate the set M ( Ð j)Ê=Ê {(t1, Ð j1), (t2, Ð j2),É, (tn, Ð jn)}, which is, in fact, a set of intervals of the real line. The intersection graph of the intervals in M(ÊÐ j) will be denoted by G(ÊÐ j). We say that any two intervals [a,b] and [a',b'] intersect when a'Ê<ÊbÊ£Êb'. The intersection graph G of the intervals is, clearly, an interval graph. Interval graphs are deeply studied in the literature; often we will exploit some properties of interval graphs to approach the problems defined on SIGÕs. The set of all interval graphs G(j) obtained by varying j in all possible ways is called the family FS associated with the given SIG S. Notice that different values of the placement vectors j, hence different interval models, may give rise to the same interval graph G(j). A minimization (maximization, respectively) problem on a SIG S is defined as follows: Given: a SIG SÊoÊ{tiÊ=ÊÊÎÊZ3 +: riÊÐÊliÊ3ÊliÊ>Ê0, for iÊ=Ê1,É,n} and a function f:ÊFSÊ®ÊZ+, Find: a graph GÊÎÊFS, Such That: f(G) is minimum (maximum, resp.) over all graphs in FS. In other words, an optimization problem on a SIG S consists in identifying an interval graph GÊÎÊFS on which f(G) attains its optimum value. If f is defined as a maxÐtype function itself, a minimization problem on a SIG S turns out to be a minÐmax problem. This happens, for example, when f is defined as the clique number of G. In fact, in this case the optimization problem consists in finding a graph GÊÎÊFS whose MAXimum complete subgraph has MINimum size. By similar reasoning we obtain minÐmin, maxÐmin, and maxÐmax problems. Given an interval graph G we will denote by w(G), c(G), a(G), k(G), d(G) the clique number (i.e. the size of a complete subgraph of maximum size), the chromatic number (i.e. the size of a coloring of minimum size), the stability number (i.e. the size of an independent set of maximum size), the clique cover number (i.e. the size of a minimum sized covering by complete subgraphs), and the size of the minimum dominating set. The problems of determining a GÊÎÊFS which minimizes (maximizes) w(G), c(G), a(G), k(G) and d(G) will be denoted by min (max) w(S), c(S), a(S), k(S) and d(S). Note that given an interval graph defined on n nodes and m edges, the problems of determining w(G), c(G), a(G), k(G) takes O(n logn) time (Gupta et al. 1982), while determining d(G) takes O(n+m) time (Bertossi 1986, Farber 1984). For some of these problems we shall devise polynomial algorithms. Other problems will be proved to be NP-hard, and we shall prove some lower and upper bound for them. Finally, we shall try to characterize subclasses of SIGÕs for which the problems that are difficult in the general case, can be solved in polynomial time. Many practical applications can be reduced to these kinds of problems on SIGÕs. Take as an example some scheduling problems where jobs with ready and due dates are to be scheduled on a set of identical machines: the ready and due dates of a job can be seen as 3 the left and right end point of a window, respectively, and its processing time as the interval length associated to the same window. We will show how minimizing or maximizing w(S), c(S), a(S), k(S) and d(S) can be interpreted in this environment. In a companion paper (Bonfiglio et al. 1997) the problems related to the size of the dominating set are explored from both theoretical and computational viewpoints. The paper introduces first some definitions of particular classes of SIG's and some basic properties (Section 2). Then it considers the problems related to the clique and the chromatic number (Section 3), the problems related to the clique cover and the stability number (Section 4). A final section contains some concluding remarks and some directions for future work. 2. Definitions and basic properties This section is devoted to discussing the relationship between a given SIG S, the graphs of the family FS, the interval models which can arise, and the same entities of the so-called derived SIG which will be introduced in a while. It is convenient to define the intersection graph HÊ=Ê(V,EH) of the set of windows {[li,ri]: i=1,É, n} where the nodes of V are in oneÐtoÐone correspondence with the windows of the given SIG, and an edge connects two nodes u, v if and only if the corresponding windows intersect. We now observe the following: Observation 2.1: Any interval graph GÊ=Ê(V,EG)ÊÎÊFS is a partial subgraph of HÊ=Ê(V,EH), in the sense that EGÊÍÊEH. Clearly, it is not true that all the partial subgraphs of H are interval graphs (Fig. 1(c)), nor that every partial subgraph of H belongs to FS, even though it may be an interval graph (Fig. 1 (d)), as the following Figure shows, where SÊoÊ{t1=<1,7,4>, t2=<3,13,3>, t3=<2,7,1>, t4=<4,6,1>, t5=<8,12,2>}.