On the Approximation Performance of Degree Heuristics for Matching
aa r X i v : . [ c s . D S ] D ec On the Approximation Performance of DegreeHeuristics for Matching
Bert Besser ⋆ and Bastian Werth Institut für Informatik, Goethe-Universität Frankfurt am Main, Germany
Abstract.
In the design of greedy algorithms for the maximum car-dinality matching problem the utilization of degree information whenselecting the next edge is a well established and successful approach.We define the class of “degree sensitive” greedy matching algorithms,which allows us to analyze many well-known heuristics, and provide tightapproximation guarantees under worst case tie breaking. We exhibit al-gorithms in this class with optimal approximation guarantee for bipartitegraphs. In particular the
KarpSipser algorithm, which picks an edge in-cident with a degree-1 node if possible and otherwise an arbitrary edge,turns out to be optimal with approximation guarantee ∆ ∆ − , where ∆ is the maximum degree. Keywords: matching, greedy, approximation, priority algorithms
Matching problems occur in many applications such as online advertising [20],image feature matching [9], or protein structure comparison [3].In Maximum Cardinality Matching a set of node-disjoint edges of maximumsize is to be determined. This problem can be solved in time O ( m √ n ) for bipar-tite as well as general graphs [5,13,15,27]. The O ( n . ) barrier was finally brokenin [22] with a runtime of O ( n ω ) , where ω < . holds.In scenarios where obtaining exact solutions is of less importance than easeof implementation and fast runtime, an approximate greedy algorithm is anadequate choice. Moreover, greedy matchings can be used as input for exactalgorithms to obtain considerable speed-ups [18].The following randomized greedy algorithms can be implemented in lineartime O ( n + m ) [12,16,19,25]. The Greedy algorithm [26] picks an edge which isnode disjoint from all previously picked edges, the
KarpSipser algorithm workslike
Greedy but picks an edge incident with a node of degree one, if such a nodeexists [16]. The
MRG algorithm (“modified random greedy”) [26] first selectsa node and then matches it with a neighbor, its variation
MinGreedy [26]first selects a node of minimum degree. The
Shuffle algorithm [14] computesa permutation π , processes nodes according to π and each time picks the π -lexicographically first edge. ( Ranking [17] works similar to
Shuffle but istailored for an on-line setting in bipartite graphs.) ⋆ Partially supported by DFG SCHN 503/6-1. revious Work.
Experiments show that large matchings are produced bythe above algorithms if ties are broken uniformly at random [12,18,19,26].All mentioned algorithms compute maximal matchings, i.e. matchings towhich no further edge can be added. A maximal matching is at least half aslarge as a maximum matching, hence the above algorithms trivially achieve ap-proximation ratio at least . An expected approximation ratio larger than ,namely + . , was shown first for MRG in [1]. However, the best knowninapproximability bound on the expected approximation ratio of
MRG is ,using methods in [11]. For Shuffle , only recently an expected approximationratio of at least ≈ . was shown in [8], whereas it is only known from [14]that this ratio cannot be larger than . The expected approximation ratio of Greedy and
MinGreedy is at most + ε , for any ε > [11,24].The expected performance on degree bounded graphs remains open for allmentioned algorithms. On graphs with degrees at most three, no algorithm dis-cussed so far achieves an expected approximation ratio better than [24]. Anexpected approximation ratio of at least ( p ( ∆ − + 1 − ∆ + 2) is achievedby Greedy on graphs with degrees at most ∆ [19].Furthermore MinGreedy leaves o ( n ) nodes unmatched in large random3-regular graphs [12]. In large sparse random graphs KarpSipser computesmatchings within o ( n ) of optimum size [2].Assuming worst case instead of random uniform tie breaking, in [4] it is shownthat MinGreedy is guaranteed to compute a matching of size at least ∆ − / ∆ − times optimal, if degrees are at most ∆ , but cannot guarantee a factor betterthan ∆ − ∆ − . For ∆ = 3 the factor is exactly , as is also shown in [4]. Our Contributions.
What is the benefit of using degree information whenpicking the next edge? We show tight approximation guarantees for
KarpSipser and
MinGreedy on bipartite graphs, assuming worst case instead of randomizedtie breaking.We introduce the class of deterministic degree sensitive greedy algorithmsand show that
KarpSipser , MinGreedy , Greedy , MRG , Shuffle , and allalgorithms for the query commit problem [21] belong to this class. (We alsoconsider a class of ‘two-sided’ algorithms like e.g.
MDS , which repeatedly picksan edge with minimum degree sum .) Our main result is that
MinGreedy and
KarpSipser are optimal degree sensitive algorithms.
Theorem 1.
The
KarpSipser algorithm always computes a matching of sizeat least ∆ ∆ − times optimal for any bipartite graph with degrees at most ∆ . Observe that the guarantee ∆ ∆ − for the KarpSipser algorithm implies atleast the same guarantee for
MinGreedy . If a degree-1 nodes exist, then bothalgorithms proceed identically, otherwise the
KarpSipser algorithm picks anarbitrary edge whereas
MinGreedy employs a finer edge selection routine.On general graphs,
KarpSipser and
MinGreedy do not perform equallywell. For ∆ =3 , MinGreedy achieves guarantee , see [4], whereas KarpSipser can only guarantee (the chord of a length-four cycle might be picked).2t is optimal to pick an edge with a degree-1 node, since such an edge belongsto some maximum matching. This observation in a sense explains KarpSipser .To prove Theorem 1 we devise a charging scheme which implicitly builds uponthis fact. Consider the connected components of the graph H on edge set M ∪ M ∗ ,where M is the matching computed by KarpSipser and M ∗ is an arbitrary max-imum matching. Connected components of H with small “local” approximationratios are amortized by “neighboring” components with large local approxima-tion ratios, where two components are neighbors if they are connected by anedge of the input graph. When a node gets matched, a charge depending on itscurrent degree is applied. A node which gets matched when it has degree one isnot charged, and has the potential to increase the local approximation ratio ofits own or of a neighboring component.To study limitations of greedy matching algorithms we utilize the frameworkof adaptive priority algorithms introduced by Borodin, Nielsen and Rackoff [7].It was successfully applied to e.g. Scheduling [7], Max-Sat [23], Sum-Coloring [6],graph problems like Steiner-Tree or Independent-Set [10], or matching in generalgraphs [4,24]. Inapproximability results are obtained similar to the adversarialarguments found in the analysis of competitive ratios of online algorithms.An adaptive priority algorithm A is defined relative to the notion of a dataitem , in which only part of the input is revealed. At the beginning of eachround A computes, incorporating all information gathered in previous rounds,a total priority order of all possible data items and receives the data item d of highest priority contained in the input. Then A has to make an irrevocabledecision based on d , thereby constructing part of the solution once and forever.The notion of “greedy” is captured by the submitted orders and the irrevoca-ble decisions. Adaptive priority algorithms have no resource constraints, henceinapproximability results apply to correspondingly large classes of algorithms.We define degree sensitive algorithms which utilize data items of the form h u, d u , v i , where u, v are nodes and d u ≥ is an integer. In any data item h u, d u , v i receivedby algorithm A nodes u and v are neighbors and u has degree d u . Here we referto the reduced graph , which contains exactly the edges incident with nodes notmatched in earlier rounds. If h u, d u , v i is received, then u and v must be matched.Additionally, before the first round an algorithm may access a priori knowl-edge on the input. We allow access to the number of nodes in the input graph. Theorem 2.
For each degree sensitive algorithm A and for any ε > , there isa bipartite graph of degree at most ∆ (and with a perfect matching) such that A computes a matching of size at most ∆ ∆ − + ε times optimal. Consequently,
KarpSipser is an optimal degree sensitive algorithm. Why?To implement
KarpSipser as a degree sensitive algorithm, in each round thepriority order begins with all possible data items h u, , v i , in arbitrary order,and continues with all remaining data items, also in arbitrary order. Similarly,3 reedy , MRG , MinGreedy , and
Shuffle can be implemented as degreesensitive algorithms. All algorithms for the query commit problem are degreesensitive as well: such an algorithm has access to the set of nodes of a graph buthas no knowledge of its edges, repeatedly tests whether two unmatched nodesare connected by an edge, and adds each found edge to the matching.Consider so called two-sided algorithms like e.g.
MDS , which repeatedlypicks an edge such that the degree sum of both incident nodes is minimum.Such algorithms are contained in the natural generalization of degree sensitivealgorithms to data items of the form h u, d u , v, d v i , where u, d u , and v are defined as before and d v is the current degree of node v . Weshow that such algorithms cannot achieve approximation ratio larger than ∆ +12 ∆ − .Note that this bound is only marginally weaker than our ∆ ∆ − bound fordegree sensitive algorithms, and we conjecture that it can be strengthened tothe same factor. To support our conjecture, we prove it for ∆ = 3 and show thatthe approximation ratio of MDS is bounded by ∆ ∆ − . Structure of the Paper.
We prove Theorems 1 and 2 in Sections 2 and 3,respectively. In Section 3.1 we discuss two-sided algorithms. Results on graphswith bounded average degree are discussed in Section 4. Conclusions and openproblems are presented in Section 5.
KarpSipser
This section proves Theorem 1. Let G =( L ∪ R, E ) be the bipartite input graph.We fix the matching M ⊆ E computed by KarpSipser and a maximum match-ing M ∗ ⊆ E . Nodes in the graph H =( L ∪ R, M ∪ M ∗ ) have degree at most two:The connected components of H are paths, cycles, and isolated nodes. We ig-nore isolated nodes. W.l.o.g. we choose M ∗ such that each H -component iseither an augmenting path or a singleton . A path X alternates between m X ≥ edges of M and m X +1 edges of M ∗ . The two path endpoints of X are notcovered by M ( ... where M ∗ -edges and M -edgesare drawn double resp. crossed). A singleton is an edge contained in both M and M ∗ ( ). Each other component, i.e. each even-length path or cycle, isturned into singletons by replacing its maximum matching edges with its M -edges ( ). Since we ignore isolated H -nodes any nodeis M -covered or a path endpoint, which never gets matched. Local Approximation Ratios.
We lower bound local approximation ra-tios of paths and singletons. A path X has local approximation ratio m X m X +1 , asingleton has local approximation ratio =1 . Small local approximation ratiosof short paths will be amortized by those of long paths and singletons.We transfer coins between H -components, each coin is worth κ of ‘ M -funds’.If component X receives c X coins and pays d X coins, then c X − d X is the balance X . The local approximation ratio of X becomes ( m X + κ · ( c X − d X ) ) / m ∗ X , where m X , m ∗ X are the numbers of M -edges respectively M ∗ -edges of X . Weestablish balances of at least c X − d X ≥ − ∆ − for each singleton X and (1) c X − d X ≥ − m X · ∆ −
2) + 2 ∆ for each path X . (2)The local approximation ratio of a singleton X is at least l := 1 − κ · ∆ − ,since we have m X = m ∗ X =1 . Choosing κ := ∆ − we obtain a lower boundof l = ∆ ∆ − . The local approximation ratio of a path X attains the same lowerbound, since it is at least m X − m X · κ · ∆ −
2) + κ · ∆m X + 1 = m X · l + κ · ∆m X + 1 = l + κ · ∆ − − m X + 1 = l . Since the minimum local approximation ratio over all components in H is l = ∆ ∆ − , KarpSipser achieves (global) approximation ratio at least ∆ ∆ − : | M || M ∗ | = P X m X P X m ∗ X = P X m X + κ · ( c X − d X ) P X m ∗ X ≥ P X ∆ ∆ − · m ∗ X P X m ∗ X = ∆ ∆ − . To establish Theorem 1 it remains to verify the balance bounds (1) and (2).Here is our plan. We claim that each M -covered node of a path X pays atmost ∆ − coins. Hence the balance of X is at least c X − d X ≥ − m X · ( ∆ − .To verify (2) we prove a balance increase for X of at least ∆ . Increase for X comes from X -nodes which pay less than ∆ − coins or receive coins.The first X -node u L ∈ L in the left partition is matched in the creation step of X . The left end step of X matches the X -node x L ∈ L in the edge { x L , w R }∈ M ∗ with the X -endpoint w R ∈ R . Note that u L = x L might hold. The node matchedwith x L is called x ′ R . Nodes in the opposite partitions are defined analogously(double drawn edges belong to M ∗ and crossed edges belong to M ): w L x R x ′ L . . . u R u L . . . x ′ R x L w R Our plan is to show that a balance increase for X of at least ∆ can be achievedby some of nodes u L , x L , x ′ R , w R and a certain G -neighbor v R of u L . The actualselection of increase nodes is determined later. We say that increase ∆ is achieved for partition L of X . W.l.o.g. in our analysis we discuss partition L . A balanceincrease of ∆ for partition R of X is obtained from the analogous set of nodes. Transfers.
We move coins over edges in F = E \ ( M ∪ M ∗ ) , where F -edgesconnect “neighboring” components of H . An F -edge which moves coins is called a transfer , and moves coins in exactly one direction. Therefore we denote a transferas a directed edge ( u, w ) and call it a debit from u and a credit to w . We define common transfers and donation transfers.5 efinition 1. Let edge { u, w } ∈ F connect an M -covered node u with a pathendpoint w . Then ( u, w ) is a common transfer and moves one coin, iff after thestep which matches u and removes { u, w } from G the degree of w is at most one. If u L has a common debit ( u L , w ) , then after creation of X node w hasbecome a degree-1 node, i.e. after creation of X node w has degree exactly one.Why? Before u L , u R are matched, both are incident with an M -edge and an M ∗ -edge. So when X is created, all degrees are at least two, since KarpSipser picksan edge with a degree-1 node if possible. Furthermore, observe that degrees aredecreased by at most one in each step since G is bipartite. In particular, in thecreation step of X the degree of w is decreased from exactly two to exactly one.If after creation of X there is a (is no) degree-1 path endpoint among the G -neighbors of u L , then we say that a (no) right degree-1 endpoint exists aftercreation of X . In the (no)-case, some of nodes u L , x L , x ′ R , w R achieve a balanceincrease of at least ∆ for partition L of X (Lemma 1). To discuss the rest ofour plan assume the other case, i.e. that a right degree-1 endpoint exists aftercreation of X , call it w . A certain G -neighbor of u L in the right partition of w , callit v R , pushes the balance increase for partition L of X to at least ∆ (Lemma 2). How to Choose v R ? Recall that the right path endpoint w never getsmatched. After creation of X , node w has degree one, thus KarpSipser matchesa degree-1 node next. In particular, by Proposition 1 (shown later) the right par-tition of w also contains degree-1 nodes v , . . . , v s , s ≥ which will get matched. Proposition 1.
If there is a right degree-1 path endpoint w , then in the rightpartition there is a degree-1 node which is not a path endpoint. We choose v R as the first of v , . . . , v s which gets matched. Note that v R is notnecessarily matched in the step after creation of X , since after creation of X partition L might contain a degree-1 node as well.No F -edges are incident with v R when it gets matched with degree one. So, byDefinition 1, zero common debits leave v R . Thus v R can increase the balance of itscomponent. If v R belongs to X , then we will see that some of u L , x L , x ′ R , w R , v R achieve increase at least ∆ . If v R belongs to a component Y = X then we donatethe increase for Y back to X using a donation transfer ( v R , u L ) . Definition 2. If v R belongs to another component than u L , then edge ( v R , u L ) is a donation transfer . Transfer ( v R , u L ) moves ∆ − coins unless the follow-ing holds, in which case it moves ∆ − coins: Before v R gets matched the rightpartition contains exactly ∆ − degree-1 nodes besides v R which are all endpoints. Our claim that a path node pays at most ∆ − coins holds, as we show now.(Whenever the component for which a node is defined is not clear from contextwe use superscripts to indicate the component.) We first argue that v XR = v YR holds for paths X = Y . Node v XR has degree one after creation of X , hence v XR is matched before KarpSipser picks an edge without a degree-1 node. In par-ticular, node v XR is matched before the next path is created, call it Y . But v YR ismatched after Y is created, hence we get v XR = v YR . Consequently, at most one6onation debit leaves v XR . Now recall that v XR has no common debits, since v XR is matched with degree one. Our argument applies in particular if v XR is a pathnode. Thus each path node either pays at most ∆ − coins in one donation debit,or one coin in each of at most ∆ − common debits. We have to verify that the increase of a node of a path X is counted either for L or for R , but not for both partitions. We define node sets which increase thebalance for partitions L resp. R of X , and argue that they do not intersect. – If u L = x L holds, then we obtain increase from nodes in I = L = { u L , w R } . – If u L = x L holds, then increase comes from nodes in I = L = { u L , x L , x ′ R , w R } .If a right degree-1 endpoint exists after creation, then I = L , I = L additionally con-tain v R . Sets I = R , I = R are defined analogously, depending on u R = x R resp. u R = x R .Observe that we have v R / ∈ { x R , u R } and v L / ∈ { x L , u L } since a donationtransfer source node v gets matched when it has degree one whereas an x -nodeor u -node gets matched when it is incident with an M -edge and an M ∗ -edge.One of the following holds: – u L = x L ∧ u R = x R : In this case observe that I = L ∩ I = R = ∅ holds. – u L = x L ∧ u R = x R (analogous to u L = x L ∧ u R = x R ): For L we obtain increasefrom nodes in I = L . From u R = x R we get u L = x ′ L , thus I = L ∩ I = R = ∅ holds. – u L = x L ∧ u R = x R : Here we have u R = x ′ R ∧ u L = x ′ L , therefore I = L ∩ I = R = ∅ holds. Isolated Nodes in H . Recall that our analysis ignores isolated H -nodes.Why is our guarantee valid? Isolated H -nodes are never matched by the Karp-Sipser algorithm. We assume that each node which is never matched is a pathendpoint. Hence an isolated H -node might receive but does not pay transfers.Thus it only decreases but does not increase local approximation ratios. Recall that we use a donation transfer ( v R , u L ) only if a right degree-1 pathendpoint w exists after creation of a path X , where w is a G -neighbor of u L .If w belongs to a component other than X , then a common transfer ( u L , w ) goesfrom u L to w . If w belongs to X , then we have w = w R and u L = x L , i.e. path X is created in an end step. In this case w R receives only one common credit: Proposition 2.
Node w R receives exactly one common credit iff w R has degreeat most one after x L gets matched. Else w R receives exactly two common credits. Nodes of an end step increase their path’s balance by 2. In particular, in-crease 2 is achieved no matter if one of the nodes has a donation debit.
Proposition 3. If u L = x L holds, then x ′ R , x L , w R achieve increase at least 2. Propositions 2 and 3 are shown later. We are ready to verify the balancebound (2) for a path X : increase ∆ is achieved for each of partitions L, R of X .7 emma 1. Let X be a path. If no right degree-1 endpoint exists after creationof X , then nodes in I = L resp. I = L increase the balance of X by ∆ .Proof. Recall that no nodes but u L , u R get isolated at creation. Since thereafteralso no right degree-1 endpoint exists, no common debit leaves u L . Moreover,recall that no donation debit leaves u L . Hence u L increases the balance by ∆ − .If we have u L = x L , then after creation of X node w R remains with at leasttwo incident F -edges. Both are common credits to w R and further increase thebalance of X by 2. So nodes in { u L , w R } ⊆ I = L increase the balance of X by ∆ .Otherwise we have u L = x L . Using Proposition 3, we obtain additional increaseat least 2 from x ′ R , x L , w R . Here we have { u L , x ′ R , x L , w R }⊆ I = L . Lemma 2.
Let X be a path. If a right degree-1 endpoint exists after creationof X , then nodes in I = L resp. I = L increase the balance of X by ∆ .Proof. Recall that I = L , I = L also contain v R , since a right degree-1 endpoint existsafter creation of X . We distinguish four cases, which are restated below beforetheir respective analysis. Assume that u L = x L holds. If v R is a node in X , then wehave v R = x ′ R or v R = x ′ R , which are the first two cases. In the third case v R is nota node in X . If u L = x L holds, then v R is not a node in X . Why? After creationof X all M - and M ∗ -edges of X but those incident with u L = x L and u R = x ′ R arestill in the graph. So the only M -covered X -node which could have degree onenow is the M ∗ -neighbor of x ′ R , call it l . But v R = l , since l is in the left partition. u L = x L , v R in X , v R = x ′ R : No common or donation transfer leaves v R ,since v R has degree one when it gets matched and belongs to the same path as u L .Thus v R achieves increase ∆ − for partition L of X . Since we have v R = x ′ R ,the balance increase of 2 for nodes x ′ R , x L , w R by Proposition 3 pushes the totalincrease to at least ∆ . Observe that we have { v R , x ′ R , x L , w R } ⊆ I = L . u L = x L , v R in X , v R = x ′ R : Note that { x ′ R , u L } is an M ∗ -edge of X . As inthe first case, zero debits leave v R and v R achieves increase ∆ − . So we are doneif w R receives 2 common credits, since then we have { v R , w R } ⊆ I = L . From here onassume that w R receives less than two common credits. By Proposition 2 node w R receives at least one common credit. A further increase of 1 is obtained if u L or x L has less than ∆ − common debits. Here we have { v R , w R , u L , x L } ⊆ I = L .If both u L and x L have ∆ − common debits, then we show a contradiction toProposition 1: we argue that, after x ′ R , x L are matched, there is a right degree-1node and all right degree-1 nodes are endpoints. After x ′ R , x L are matched, thedestination endpoints of common debits from u L , x L have degree at most one.Node w R has degree at most one as well, by Proposition 2, since we have as-sumed that w R receives only one common credit. So the number of endpointsneighboring u L (in G ) is ∆ − , while x L has ∆ − neighbors (in G ) which areendpoints. Therefore after x ′ R , x L are matched an endpoint neighbor of x L (in G )has degree one. Also, all degree-1 nodes in the right partition are endpoints. u L = x L , v R not in X : At most ∆ − common transfers leave u L , since nocommon transfer goes from u L to v R . Therefore u L achieves an increase of 1.Observe that after creation at most ∆ − degree-1 endpoints exist in the right8artition. Hence by Definition 2, a donation transfer ( v R , u L ) moves ∆ − coinsto X . Using the increase of 2 for nodes x ′ R , x L , w R due to Proposition 3, the totalincrease is ∆ . The increase is obtained from nodes { u L , v R , x ′ R , x L , w R } ⊆ I = L . u L = x L ( v R not in X ): Again, at most ∆ − common debits leave u L .Recall that each destination node of a common debit from u L has degree exactlyone after creation of X . Also, node w R has degree one after creation if and onlyif w R receives exactly one common credit, as Proposition 2 shows.Assume that w R receives two common credits or u L has at most ∆ − common debits, in which case the increases of u L and w R sum up to at least three,since w R receives at least one common credit by Proposition 2. After creationthe right partition contains at most ∆ − degree-1 endpoints. By Definition 2,a donation transfer ( v R , u L ) moves additional ∆ − coins to X . We are donewith an increase of at least ∆ for partition L of X by nodes { u L , w R , v R } ⊆ I = L .Lastly, assume that w R receives one common credit and ∆ − common debitsleave u L , i.e. the increases of u L and w R sum up to at least two. Observe thatafter creation the right partition contains ∆ − many degree-1 endpoints andthat v R is the only right degree-1 node which is not an endpoint. Therefore, byDefinition 2, a donation transfer ( v R , u L ) moves additional ∆ − coins to X .We get an increase of at least ∆ for L of X by nodes { u L , w R , v R } ⊆ I = L .Next, we prove that the balance of singletons is large enough. Lemma 3.
A singleton pays at most ∆ − coins and therefore satisfies (1) .Proof. Recall that a node has either common or donation debits, but not both,and at most one donation debit leaves each node. We distinguish three cases fornodes z L , z R of a singleton: both have a donation debit, or both have commondebits, or w.l.o.g. a donation debit ( z L , u R ) leaves z L and z R has common debits. A Donation Debit Leaves Each of z L , z R : Exactly two donation debitsleave the singleton. By definition, each moves at most ∆ − coins. Both z L , z R Have Common Debits:
We show that each of z L , z R hasat most ∆ − common debits. Assume that z L has ∆ − common debits.When z L , z R are matched, both are incident with an F -edge and by definitionof KarpSipser all nodes have degree at least two. Thereafter the destinationnodes of common debits from z L have degree one, and these endpoints are theonly degree-1 nodes in their partition since the only other G -neighbor of z L is z R .A contradiction to Proposition 1. An analogous argument applies to z R . A Donation Debit Leaves z L and z R Has Common Debits:
We aredone if ( z L , u R ) moves at most ∆ − coins, since at most ∆ − common debitsleave z R . If ( z L , u R ) moves ∆ − coins, then z R has at most ∆ − common debits:assuming that z R has ∆ − common debits, say to nodes w L , . . . , w ∆ − L , we showa contradiction. By definition of ( z L , u R ) , before z L gets matched the partitionof z L contains ∆ − degree-1 path endpoints and no other degree-1 nodes but z L .But then after z L is matched, at least one of the w iL has degree one, since thedegree of at most ∆ − endpoints was decreased to zero. Furthermore, since z L is now matched, all degree-1 nodes in the left partition are path endpoints. Thiscontradicts Proposition 1. 9o complete the proof of Theorem 1 we have to show Propositions 1 to 3.We start with the result that solely depends on the definition of path endpointsand the bipartiteness of G . Proposition 1.
If there is a right degree-1 path endpoint w , then in the rightpartition there is a degree-1 node which is not a path endpoint.Proof. Assume that all degree-1 nodes in the partition of w are path endpoints.Since these are never matched, an edge with a degree-1 node u in the other parti-tion is picked next, say u gets matched with v . Observe that v is in the partitionof w and that all degrees in this partition, but that of v , are not changed. So theset of degree-1 nodes in the partition of w remains unchanged. By repeating theargument the degree of w is never decreased to zero. A contradiction.Next, we prove the result on the number of common credits to an endpoint. Proposition 2.
Node w R receives exactly one common credit iff w R has degreeat most one after x L gets matched. Else w R receives exactly two common credits.Proof. First, recall that no degree-1 node is matched in the creation step of thepath of w R . At creation, node w R is not yet isolated and consequently has degreeat least two as well. Since G is bipartite, edges incident with w R are removed inpairwise different steps. Hence there is a step when w R has degree two.An edge is not a common credit to w R if it is removed before w R has degreetwo. Thereafter, each F -edge removed from w R is a common credit to w R . Henceif w R has degree two when x L is already matched, then both remaining F -edgesare common credits. If w R has degree two when x L is not yet matched, then w R has only one incident F -edge and receives one common credit, and after x L ismatched w R has degree at most one. Proposition 3. If u L = x L holds, then x ′ R , x L , w R achieve increase at least 2.Proof. Observe that no donation debit leaves x L , since x L has degree at leasttwo when it is matched. We distinguish if a donation debit leaves x ′ R or not. No Donation Debit Leaves x ′ R : If w R receives two common credits, thenwe are done. Otherwise w R receives exactly one common credit, by Proposition 2.Therefore it suffices to find an additional increase of one. If one of x ′ R , x L has lessthan ∆ − common debits, then we are done. So let each of x ′ R , x L have ∆ − common debits. Consequently each of x ′ R , x L is incident with ∆ − many F -edgesjust before being matched, i.e. both their degrees—and hence all degrees—areat least two. After x ′ R , x L are matched, the destination nodes of common debitsfrom x L have degree exactly one, since their degrees are decreased by exactlyone. Since w R receives one common credit, node w R also has degree one as aconsequence of Proposition 2. Hence all degree-1 nodes in the right partition arepath endpoints. A contradiction to Proposition 1. A Donation Debit ( x ′ R , u L ) Leaves x ′ R : Recall that no common debitleaves x ′ R , since x ′ R is matched when it has degree one. If ( x ′ R , u L ) moves ∆ − coins, then x ′ R increases the balance by 1. Using a common credit to w R , whichexists by Proposition 2, we get a total increase of at least 2.10ow assume that ( x ′ R , u L ) moves ∆ − coins. If w R receives two commoncredits, or w R receives one common credit and at most ∆ − common debitsleave x L , then we are done. So assume that w R receives one common creditand ∆ − common debits leave x L , say to nodes w R , . . . , w ∆ − R . We show acontradiction to Proposition 1. After x ′ R , x L are matched, the w iR have degree atmost one by definition, and w R has degree at most one due to Proposition 2. Weclaim that at least one of w R and the w iR has degree exactly one after x ′ R , x L arematched. Why? Since ( x ′ R , u L ) moves ∆ − coins, before x ′ R , x L are matched theright partition contains exactly ∆ − degree-1 endpoints. Hence thereafter atmost ∆ − of w R and the w iR are isolated, as claimed. Furthermore, before x ′ R , x L are matched node x ′ R is the only degree-1 node in its partition which is not anendpoint, and thereafter x ′ R is matched. So after x ′ R , x L are matched all degree-1nodes in the right partition are endpoints. This contradicts Proposition 1. In this section we prove Theorem 2. We describe the adaptive priority game between algorithm A and an adversary B , who processes the priority orderssubmitted by A in order to construct a hard input instance. In each round,adversary B presents the highest priority data item h u, d u , v i in the current orderwhich should be in the graph: Each presented data item must be consistent withthe previous construction, i.e. giving the final construction as input to A mustresult in the same sequence of submitted priority orders and received data items.We first prove our ∆ ∆ − bound for bipartite graphs with degrees at most ∆ ≥ and without a perfect matching. Thereafter we modify B such that the construc-tion also works for ∆ = 3 , and such that the graph has a perfect matching.Adversary B constructs a graph which contains k traps T , T , . . . , T k . Foreach trap T i algorithm A will insert ∆ edges into its matching (crossed edgesin Figure 1), whereas T i contains ∆ − edges of a maximum matching (doubleedges). Besides traps the graph contains a constant number of additional nodesand edges. Hence A achieves approximation ratio at most ∆ ∆ − + ε for large k .Trap T i contains a left cycle on nodes c i , c i , c i , c i which is connected via anedge { c i , p i } to a left path on nodes p i , p i . Trap T i also contains a right cycle onnodes d i , d i , d i , d i connected via { d i , q i } to a right path on nodes q i , q i . The leftpath is connected to the right cycle via edges { p i , d i } , { p i , d i } , and analogouslythe right path of T i is connected to the left cycle of the next trap T i +1 viaedges { q i , c i +11 } , { q i , c i +13 } ; the right path of the last trap T k is connected to anextra cycle on nodes e , e , e , e via edges { q k , e } , { q k , e } ; an extra node e connects to the left cycle nodes c , c of the first trap. The left and right cyclesin T i are connected by Λ = ∆ − many length-three paths on nodes w ij , x ij , y ij , z ij via edges { c i , w ij } , { c i , w ij } and { z ij , d i } , { z ij , d i } for ≤ j ≤ Λ . During thegame B will add more edges to this graph, depending on the actions taken by A .To start the game, adversary B announces the number k · (12 + 4 Λ )+5 ofnodes. The construction of B proceeds such that after the first ∆ rounds all11 c c c c p p d d d d q q e e e e T , . . . , T k w x y z w Λ x Λ y Λ z Λ ... T Fig. 1.
The construction of adversary B . Algorithm A receives data items for boldnodes. The partitions of the graph are marked with white and gray nodes. Gray edgesform connected components in rounds Λ +2 resp. Λ +4 of the adaptive priority game. nodes in T but q are isolated. The graph to be constructed thereafter is onetrap ‘shorter’ with q instead of e connected to the leftmost trap. Adversary B repeats its strategy for T , T , . . . , T k . After B finishes the construction of T k ,algorithm A scores at most two edges for nodes q k , e , e , e , e .Observe that in the first round the minimum degree is two. In each ofrounds ≤ j ≤ Λ , adversary B presents the highest priority data item h u, d u , v i with ≤ d u ≤ ∆ in the respective priority order submitted by A . Adversary B then relabels nodes in the graph such that u = x j and v = y j holds, i.e. algo-rithm A picks the crossed edges in the length-three paths.In each round B has committed to u = x j having current degree d u . Since d u may be larger than two, adversary B inserts additional edges incident with x j into the graph in Figure 1. The d u − additional edges connect x j with arbitrarynodes in the set { w , . . . , w j − , w j +1 , . . . , w Λ , c , c , q } . This set has cardinal-ity ∆ − ≥ d u − and only contains nodes outside the partition of x j .The additional edges are consistent: In previous rounds A could not gatherknowledge about the neighborhood of u = x j , or any other still unmatched node,therefore the additional edges do not have effect on previous actions taken by A .Edges incident with u, v —including additional edges—are removed from thegraph in the next round, hence in round j + 1 the minimum degree is two, again.The G -degrees of the nodes receiving additional edges are increased to atmost ∆ − during rounds ≤ j ≤ Λ : The w j have degree at most Λ − ∆ − ,both c , c have degree at most Λ = ∆ − , and q has degree at most Λ = ∆ − .In round Λ + 1 adversary B again presents the highest priority item h u, d u , v i with ≤ d u ≤ ∆ in the submitted order. This time B relabels nodes suchthat u = p and v = c (hence A picks the crossed edge connecting the left cycleand path), and inserts d u − ≤ ∆ − additional edges connecting u witharbitrary nodes in the set { z , . . . , z Λ , d , d } . The G -degrees of nodes receivingan additional edge are increased by only one, i.e. they do not exceed ≤ ∆ .In round Λ +2 a star centered at c is disconnected from the rest of the graph.Since A computes a maximal matching, these star nodes get isolates when A matches c . W.l.o.g. we assume that A isolates these nodes in round Λ +2 .Similarly, adversary B constructs the right cycle and path. In round Λ +3 ,algorithm A matches u = q with v = d , where additional d u − edges connect q { w , . . . , w Λ , c , c } of left path and cycle nodesin trap T . In round Λ +4= ∆ a star centered at d is disconnected from the restof the graph. W.l.o.g. again, algorithm A scores this edge in this round.Adversary B repeats its strategy for the construction of trap T . As before, G -degrees of nodes which receive additional edges are not increased above ∆ . How-ever, we have to pay attention to nodes w , . . . , w Λ , c , c . For these nodes ad-versary B might already have constructed one additional edge from q . So ad-ditional edges in T increase the degrees of these nodes to at most ∆ —and notto at most ∆ − as discussed for T . This applies analogously to T , T , . . . , T k . ∆ = 3 : Paths on nodes w ij , x ij , y ij , z ij do not exist. Left and right paths havefour nodes p i , . . . , p i resp. q i , . . . , q i instead of two, and are still connected tocycles via { p i , c i } resp. { q i , d i } . Edges { p i , d i } , { p i , d i } and { q i , c i +11 } , { q i , c i +13 } connecting paths with nodes of the ‘next’ cycle are replaced by { p i , q i } , { p i , d i } , resp. { q i , p i +12 } , { q i , c i +13 } . During the game adversary B does not insert anyadditional edges. All nodes have degrees two or three. In particular, nodes p , p have degree three resp. two: In the first round B presents the highest prioritydata item h u, d u , v i and relabels nodes such that A picks edge { p , p } , no matterif d u =2 or d u =3 holds. The remainder of the left cycle and path in T is nowseparated from the rest of the graph. Therein A can pick at most two edges:Algorithm A scores three out of four. Adversary B repeats this constructionanalogously for edges { q , q } , { p , p } , { q , q } , . . . , { q k , q k } and their pathsand cycles. Hence we obtain the claimed convergence to = ∆ ∆ − . Note.
Since in case ∆ = 3 the set of edges is fixed, we can strengthen ourbound by giving the algorithm additional a priori knowledge on the input, namelythe number of edges. Moreover, observe that the algorithm cannot counter theadversary’s strategy even if the degree sequence of the input graph is given asa priori knowledge. Perfect Matching:
Adversary B replaces the extra node e with a length-four cycle C and connects c , c (resp. p , c for ∆ = 3 ) to different nodes of C such that degrees in C are two and three and the graph is bipartite (similar to thecycle on e , . . . , e ). The construction starts as discussed. However, when the starcentered at node c is disconnect from the rest of the traps, it is still connectedto C . W.l.o.g. we assume that A isolates all nodes still connected to c in the nextrounds. (To compensate for the two additional edges scored by A , adversary B increases k .) Thereafter the construction proceeds as discussed above. The class of degree sensitive algorithms is defined based on data items h u, d u , v i ,which state the degree of one node in an edge. The minimum degree sum al-gorithm and the algorithm which selects a minimum degree node and then aminimum degree neighbor use degree information of both nodes, hence they can-not be analyzed with the help of our class.In this section we discuss a generalization of degree sensitive algorithmsto ‘two-sided’ algorithms. Therefore we extend the definition of a data item13o h u, d u , v, d v i , i.e. we allow an algorithm to specify the degrees d u and d v ofboth nodes u and v of an edge. Otherwise, two-sided algorithms are definedexactly like degree sensitive algorithms.We conjecture that no two-sided algorithm can perform better than Karp-Sipser . In this section we support this conjecture by showing three relatedinapproximability bounds.First, we prove that for ∆ = 3 two-sided algorithms cannot beat the perfor-mance of degree sensitive algorithms, i.e. they are bounded by the same approx-imation ratio. Theorem 3.
Consider a two-sided algorithm A . For any ε > there is a bi-partite graph with degrees at most ∆ = 3 (and a perfect matching) such that A computes a matching of size at most + ε = ∆ ∆ − + ε times optimal.Proof. We slightly change the adversary B from the proof of Theorem 2 to obtainan adversary B ′ for A . Adversary B ′ removes right paths and cycles and theirincident edges from all traps, and connects the path node p i to nodes c i +13 , p i +12 ofthe next cycle and path. Cycle C and the cycle on nodes e , e , e , e along withall their incident edges are replaced by two edges in the first and the k -th cycleand path, namely edges { c , p } , { c , p } resp. { c k , p k } , { p k , p k } , see Figure 2. p p p p c c c c p p p p c c c c p k p k p k p k c k c k c k c k ... Fig. 2.
The construction of adversary B ′ . No additionaledges are inserted during the game. Fig. 3.
An additionalcomponent.
Before the first round, adversary B ′ announces that the number of nodes is n for some large integer n . The parameter k will be determined by B ′ based onthe actions taken by A . In particular, the graph has n − k additional connectedcomponents, each with two length-four cycles connected by two edges like inFigure 3. Observe that any edge in any connected component is incident eitherwith a degree-2 node and a degree-3 node or with two degree-3 nodes. No edgeis incident with two degree-2 nodes.The following Invariant holds throughout the game: At the beginning ofround i + 1 there is an integer ≤ k ∗ ≤ i such that algorithm A has matched orisolated all nodes in i − k ∗ additional components as well as all nodes but p k ∗ in14he first k ∗ paths and cycles. No other nodes are matched or isolated. For i = 0 ,before the first round no nodes are isolated and the Invariant holds.Consider round i + 1 . Observe that the minimum degree is two, and thatevery edge is incident with at least one node of degree three. Adversary B ′ presents the highest priority data item h u, d u , v, d v i in the order submitted by A with d u , d v ∈ { , } and at least one of d u , d v equals three.If d u = d v =3 , then B ′ constructs the next additional component and relabelsnodes such that u, v are the two leftmost nodes in Figure 3. Observe that A scores at most three out of four edges in this component, since therein only grayedges are left. W.l.o.g. we assume that A scores the additional two edges in thenext two rounds. Since k ∗ is not increased, the Invariant continues to hold.If d u = d v , then w.l.o.g. let d u = 3 and d v = 2 . By the Invariant ,algorithm A has already matched nodes p , p , p , p , . . . , p k ∗ , p k ∗ in previousrounds. Adversary B ′ relabels nodes such that u = p k ∗ +12 and v = p k ∗ +13 hold. Af-ter nodes p k ∗ +12 , p k ∗ +13 are matched, the remainder of the k ∗ +1 -th path and cycleis disconnected from the rest of the graph, see gray edges in Figure 2. In thisremainder A scores at most two more edges. W.l.o.g. we assume that A doesso in the next two rounds. Hence k ∗ is incremented by one and the Invariant holds before round i + 1) + 1 .We assume that the last path and cycle resp. the last additional componentis solved optimally, i.e. algorithm A scores four out of four edges. In each otherpath and cycle and in each other additional component, algorithm A scoresthree out of four edges. Hence B ′ can choose sufficiently large n such that theapproximation ratio of A is at most ( n − · ≤ + ε .Next, we show that two-sided algorithms can perform at most marginally bet-ter than degree sensitive algorithms, i.e. they cannot beat the inapproximabilitybound ∆ ∆ − considerably. Theorem 4.
Let A be a two-sided algorithm. There is a bipartite input graphof degree at most ∆ ≥ for which A computes a matching of size at most ∆ +12 ∆ − times optimal.Proof. The adaptive priority game between A and an adversary B lasts for ∆ +1 rounds. Let δ = ∆ − . The final construction G contains the graph G ′ depictedin Figure 4 as a subgraph. In particular, graph G contains additional edges whichare not depicted in G ′ , but G does not have any additional nodes.The construction of B proceeds such that in rounds , . . . , δ algorithm A picksedges { u , v } , . . . , { u δ , v δ } and after round δ the reduced graph consists onlyof gray nodes and of edges connecting gray nodes. Observe that all remainingedges touch exactly four gray nodes, namely the unlabeled ones in the figure.We assume that in this reduced graph algorithm A scores four edges in fourrounds, which is optimal. Since G ′ contains a perfect matching of size ∆ − and A scores one edge in each of δ + 4 = ∆ + 1 rounds, the approximation ratiois ∆ +12 ∆ − , as claimed. In the rest of the proof it remains to discuss the first δ rounds. 15 a ′ bb ′ a u v b a i u i v i b i a δ u δ v δ b δ ...... Fig. 4.
The subgraph G ′ of the final construction (the algorithm receives data itemsfor bold nodes) Recall that A does not receive identifiers of neighbors of the nodes in adata item. As a consequence, in each round adversary B is free—without beinginconsistent—to relabel nodes in G ′ according to the data item presented to A .We proceed inductively. Assume that at the beginning of round i algorithm A has picked edges { u , v } , . . . , { u i − , v i − } . The minimum degree in the reducedgraph G i is . Adversary B uses the set D = { , . . . , ∆ } of allowed degrees: Fromthe order submitted by A in round i adversary B presents the highest prioritydata item h u, d u , v, d v i with d u ∈ D and d v ∈ D . Adversary B relabels nodessuch that u = u i and v = v i hold: algorithm A picks edge { u i , v i } , as desired.Now B delivers on its promise that both nodes have degree d u resp. d v .Therefore B inserts additional edges into the graph. In particular, since u i al-ready has two incident edges in G ′ , adversary B adds d u − edges, each incidentwith u i and one of nodes a, a ′ , a , . . . , a δ . Analogously, adversary B adds d v − edges, each incident with v i and one of nodes b, b ′ , b , . . . , b δ .It remains to show that B does not violate degree constraints when insertingnew edges. Since we have d u ≤ ∆ and thus d u − ≤ ∆ − , for all u -nodesat most δ ( ∆ −
2) = ( ∆ − ∆ −
2) = ( ∆ − + ( ∆ − edges are inserted. Sinceall a -nodes can receive up to δ ( ∆ −
3) + 2( ∆ −
2) = ( ∆ − + 2( ∆ − edges,their degrees are increased to at most ∆ if new edges are distributed evenly.Analogously, degrees of b -nodes are at most ∆ .Finally, we show that the two-sided MDS algorithm does not achieve betterapproximation ratio than any degree sensitive algorithm, for all ∆ . Theorem 5.
For each ∆ ≥ there is a bipartite graph of degree at most ∆ forwhich MDS computes a matching of size at most ∆ ∆ − times optimal.Proof. Choose k = ∆ − . The hard instance is depicted in Figure 5. Observe thatthe degree sum of any edge is at least 4, since nodes c L and c R have degree atleast 3. Hence we may assume that in the first step MDS picks edge { u L , u R } (thetop crossed edge). Assume that edges { u L , u R } , . . . , { u iL , u iR } with i < k have16 L u L u R u kL u kR c R ... Fig. 5.
A hard instance for
MDS already been picked. The minimum degree sum is still four, and edge { u i +1 L , u i +1 R } is picked next. In the end, for each of nodes c L and c R an incident edge is picked.Hence the computed matching has k + 2 = ∆ edges, whereas a maximummatching consists of the k + 2 = ∆ − double drawn edges. In Section 3 we have shown that
MinGreedy and
KarpSipser achieve the op-timal approximation guarantee ∆ ∆ − on bipartite graphs with degrees boundedby ∆ . Our inapproximability results carry over to graphs of bounded averagedegree. However, both MinGreedy and the
KarpSipser algorithm achieveapproximation guarantee only + ε even if the average degree is constant. Theorem 6.
The approximation guarantee of
MinGreedy and the
Karp-Sipser algorithm is bounded by at most + ε for bipartite graphs with averagedegree at most , for any ε > . We note that our construction also applies to
Greedy , MRG , Shuffle , the minimum degree sum algorithm, the algorithm which first selects a minimumdegree node and then a minimum degree neighbor, and to all algorithms for thequery commit problem.
Proof.
Nodes of the graph are partitioned into sets
L, U, V, W, X, R , where wehave | L | = | R | =2 and Y = { v Y , . . . , v nY } for Y ∈{ U, V, W, X } and n ∈ N . For A, B ∈{ U, V, W, X } we denote the set {{ v iA , v iB } : 1 ≤ i ≤ n } as A ∗ B . The edge set is ( L × U ) ∪ ( U ∗ V ) ∪ ( V ∗ W ) ∪ ( W ∗ X ) ∪ ( X × R ) . Nodes in
L, R have degree n , nodes in U, X have degree , and nodes in V, W have degree two. We argue that any of the given algorithms proceeds as follows,considering worst case tie breaking: in each of the first n rounds an edge in V ∗ W is picked. Why? Assuming that only edges in V ∗ W have already been picked,the minimum degree over all non-isolated nodes is two; furthermore, both nodesof each remaining edge in V ∗ W have minimum degree degree two.After round n all remaining edges are incident with nodes in L, R and the al-gorithm scores at most four more edges, i.e. a matching of size n +4 is computed.However, observe that ( U ∗ V ) ∪ ( W ∗ X ) is a matching of size n . Therefore an al-gorithm computes a matching of size at most n +42 n times optimal, which convergesto as n → ∞ . The average degree in the graph is · · n +2 · n · · n · n +4 = n n +4 ,which converges to from below. 17 Conclusion and Open Problems
MinGreedy and
KarpSipser achieve optimal approximation guarantee ∆ ∆ − among degree sensitive algorithms, on bipartite graphs with degrees at most ∆ .If degree sensitive algorithms are allowed to use data items with degreesof both neighbors (‘two-sided’ algorithms), then we conjecture that the sameinapproximability factor ∆ ∆ − applies. However, we can only provide partialproofs, namely for ∆ =3 and for the MDS algorithm.The
KarpSipser algorithm is a refinement of
Greedy , since it picks arandom edge unless there is a degree-1 node. What is the expected approximationratio of the
KarpSipser algorithm and the analogous refinement of
MRG ? References
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