Finding Optimal Solutions With Neighborly Help
Elisabet Burjons, Fabian Frei, Edith Hemaspaandra, Dennis Komm, David Wehner
aa r X i v : . [ c s . CC ] J un Finding Optimal Solutions With Neighborly Help
Elisabet Burjons
Department of Computer Science, ETH Zürich [email protected]
Fabian Frei
Department of Computer Science, ETH Zürich [email protected]
Edith Hemaspaandra
Department of Computer Science, Rochester Institute of Technology [email protected]
Dennis Komm
Department of Computer Science, ETH Zürich [email protected]
David Wehner
Department of Computer Science, ETH Zürich [email protected]
Abstract.
Can we efficiently compute optimal solutions to instances of a hard problem from optimalsolutions to neighboring (i.e., locally modified) instances? For example, can we efficiently computean optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studyingsuch questions not only gives detailed insight into the structure of the problem itself, but also into thecomplexity of related problems; most notably graph theory’s core notion of critical graphs (e.g., graphswhose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoreticnotion of minimality problems (also called criticality problems, e.g., recognizing graphs that become3-colorable when an arbitrary edge is deleted).We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show thatit is NP-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deletedsubgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior,demonstrating the power of our model to delineate problems from each other more precisely on astructural level.Moreover, we provide a number of new complexity results for minimality and criticality problems. Forexample, we prove that
Minimal- -UnColorability is complete for DP (differences of NP sets),which was previously known only for the more amenable case of deleting vertices rather than edges. ForVertex Cover, we show that recognizing β -vertex-critical graphs is complete for Θ p2 (parallel access toNP), obtaining the first completeness result for a criticality problem for this class. Keywords.
Critical Graphs, Computational Complexity, Structural Self-Reducibility, Minimality Prob-lems, Colorability, Vertex Cover, Satisfiability, Reoptimization, Advice
Funding.
Edith Hemaspaandra:
Research done in part while on sabbatical at ETH Zürich. Introduction and Related Work
In Subsection 1.1, we introduce and motivate our new model, which we then compare and contrastto related notions in Subsection 1.2. Finally, we present in Subsection 1.3 an overview of our mostinteresting results and place them into the context of the wider literature.
In view of the almost complete absence of progress in the question of P versus NP, it is naturalto wonder just how far and in what way these sets may differ. For example, how much additionalinformation enables us to design an algorithm that solves an otherwise NP-hard problem in polynomialtime? We are specifically interested in the case where this additional information takes the form ofoptimal solutions to neighboring (i.e., locally modified) instances. This models situations such as thatof a newcomer who may ask experienced peers for advice on how to solve a difficult problem, forinstance finding an optimal work route. Similar circumstances arise when new servers join a computernetwork. Formally, we consider the following oracle model: An algorithm may, on any given input,repeatedly select an arbitrary instance neighboring the given one and query the oracle for an optimalsolution to it. Occasionally, it will be interesting to limit the number of queries that we grant thealgorithm. In general, we do not impose such a restriction, however.What precisely constitutes a local modification and thus a neighbor depends on the specific problem,of course. We examine the prototypical graph problems Colorability and Vertex Cover, consideringthe following four local modifications, which are arguably the most natural choices: deleting an edge,adding an edge, deleting a vertex (including adjacent edges), and adding a vertex (including anarbitrary, possibly empty, set of edges from the added vertex to the existing ones). For example,we ask whether there is a polynomial-time algorithm that computes a minimum vertex cover for aninput graph G if it has access to minimum vertex covers for all one-edge-deleted subgraphs of G . Wewill show that questions of this sort are closely connected to and yet clearly distinct from researchin other areas, in particular the study of critical graphs, minimality problems, self-reducibility, andreoptimization. Criticality.
The notion of criticality was introduced into the field of graph theory by Dirac [8] in 1952in the context of Colorability with respect to vertex deletion. Thirty years later, Wessel [21] generalizedthe concept to arbitrary graph properties and modification operations. Nevertheless, Colorability hasremained a central focus of the extensive research on critical graphs. Indeed, a graph G is called critical without any further specification if it is χ -critical under edge deletion, that is, if its chromaticnumber χ ( G ) (the number of colors used in an optimal coloring of G ) changes when an arbitrary edgeis deleted. Besides Colorability, one other problem has received a comparable amount of attentionand thorough analysis in three different manifestations: Independent Set, Vertex Cover, and Clique.The corresponding notions are α -criticality, β -criticality, and ω -criticality, where α is the independencenumber (size of a maximum independent set), β is the vertex cover number (size of a minimum vertexcover), and ω is the clique number (size of a maximum clique). Note that these graph numbers areall monotone—either nondecreasing or nonincreasing—with respect to each of the local modificationsexamined in this paper. Minimality.
Another strongly related notion is that of minimality problems. An instance is called minimal with respect to a property if only the instance itself but none of its neighbors has this property;that is, it inevitably loses the property under the considered local modification. The correspondingminimality problem is to decide whether an instance is minimal in the described sense. For example,a graph G is minimally 3-uncolorable (with respect to edge deletion) if it is not 3-colorable, yet all itsone-edge-deleted neighbors are. The minimality problem Minimal- -UnColorability is the set of2ll minimally 3-uncolorable graphs. Note that a graph is critical exactly if it is minimally k -uncolorablefor some k .While minimality problems tend to be in DP (i.e., differences of two NP sets, the second levelof the Boolean hierarchy), DP-hardness is so difficult to prove for them that only a few have beenshown to be DP-complete so far; see for instance Papadimitriou and Wolfe [15]. Note that the notionof minimality is not restricted to graph problems. Indeed, minimally unsatisfiable formulas figureprominently in many of our proofs. Auto-Reducibility.
Our model provides a refinement of the notion of functional auto-reducibility; seeFaliszewski and Ogihara [9]. An algorithm solves a function problem R ⊆ Σ ∗ × Σ ∗ if on input x ∈ Σ ∗ it outputs some y ∈ Σ ∗ with ( x, y ) ∈ R . The problem R is auto-reducible if there is a polynomial-time algorithm with unrestricted access to an oracle that provides solutions to all instances except x itself. The task of finding an optimal solution to a given instance is a special kind of function problem.Defining all instances to be neighbors (local modifications) of each other lets the two concepts coincide. Self-Reducibility.
Self-reducibility is auto-reducibility with the additional restriction that the algo-rithm may query the oracle only on instances that are smaller in a certain way. There are a multitudeof definitions of self-reducibility that differ in what exactly is considered to be “smaller,” the twoseminal ones stemming from Schnorr [17] and from Meyer and Paterson [14]. For Schnorr, an instanceis smaller than another one if its encoding input string is strictly shorter. While his definition doesallow for functional problems (i.e., more than mere decision problems, in particular the problem offinding an optimal solution), it is too restrictive for self-reducibility to encompass our model since notall neighboring graphs have shorter strings under natural encodings. Meyer and Paterson are less rigidand allow instead any partial order having short downward chains to determine which instances areconsidered smaller than the given one. The partial orders induced by deleting vertices, by deletingedges, and by adding edges all have short downward chains. The definition by Meyer and Patersonis thus sufficient for our model to become part of functional self-reducibility for all local modifica-tions considered in this paper but one, namely, the case of adding a vertex, which is too generous amodification to display any particularly interesting behavior.As an example, consider the graph decision problem
Colorability = { ( G, k ) | χ ( G ) ≤ k } , whichis self-reducible by the following observation. Any graph G with at least two vertices that is not aclique is k -colorable exactly if at least one of the polynomially many graphs that result from mergingtwo non-adjacent vertices in G is k -colorable. This works for the optimization variant of the problemas well. Any optimal coloring of G assigns at least two vertices the same color, except in the trivialcase of G being a clique. An optimal coloring for the graph that has two such vertices merged thenyields an optimal coloring for G . This contrasts well with the findings for Colorability’s behaviorunder our new model discussed below. Reoptimization.
Reoptimization examines optimization problems under a model that is tightly con-nected to ours. The notion of reoptimization was coined by Schäffter [18] and first applied by Archettiet al. [1]. The reoptimization model sets the following task for an optimization problem:
Given an instance, an optimal solution to it, and a local modification of this instance, computean optimal solution to the modified instance.
The proximity to our model becomes clearer after a change of perspective. We reformulate the reop-timization task by reversing the roles of the given and the modified instance.
Given an instance, a local modification of it, and an optimal solution to the modified instance,compute an optimal solution to the original instance. Formally, a partial order is said to have short downward chains if the following condition is satisfied: There is apolynomial p such that every chain decreasing with respect to the considered partial order and starting with some string x is shorter than p ( | x | ) and such that all strings preceding x in that order are bounded in length by p ( | x | ). able 1. An overview of our results regarding the hardness of Colorability and Vertex Cover in our model for themost common definitions of a local modification. The v stands for a vertex and the e stands for an edge. The questionmark indicates an interesting open problem. The results in the vertex-addition columns are trivial; see Theorem 16 inAppendix A. The NP-hardness results for the 1-query case all follow from rather simple Turing reductions; see Theorem 17in Appendix B.No. ofQueries Colorability Vertex CoverAdd v Delete v Add e Delete e Add v Delete v Add e Delete e Note that this perspective switch flips the definition of local modification; for example, edge deletionturns into edge addition. Aside from this, the task now reads almost identical to that demanded inour model. The sole but crucial difference is that in reoptimization, the neighboring instance andthe optimal solution to it are given as part of the input, whereas in our model, the algorithm mayselect any number of neighboring instances and query the oracle for optimal solutions to them. Evenif we limit the number of queries to just one, our model is still more generous since the algorithm ischoosing (instead of being given) the neighboring instance to which the oracle will supply an optimalsolution. Thus, hardness in our model always implies hardness for reoptimization, but not vice versa.In fact, all problems examined under the reoptimization model so far remain NP-hard. Only for someof them could an improvement of the approximation ratio be achieved after extensive studies, the firstdiscovered examples being tsp under edge-weight changes [3] and addition or deletion of vertices [2].This stands in stark contrast to the results for our model, as outlined in the next section.
We shed a new light on two of the most prominent and well-examined graph problems, Colorabilityand Vertex Cover.Our results come in two different types.The first type concerns the hardness of the two problems in our model for the most common localmodifications; Table 1 summarizes the main results of this type. In addition, Corollaries 2 and 14 showthat Satisfiability and Vertex Cover remain NP-hard for any number of queries if the local modificationis the deletion of a clause or a triangle, respectively. The results for the vertex-addition columns aretrivial since we can just query an optimal solution for the graph with an added isolated vertex; seeTheorem 16. The hardness results for the one-query case all follow from the same simple Theorem 17,variations of which appear in the study of self-reducibility and many other fields; see Appendix B. Thefindings of Theorems 10, 12 and 19 clearly delineate our model from that in reoptimization, wherethe NP-hard problems examined in the literature remain NP-hard despite the significant amount ofadvice in form of the provided optimal solution; see Böckenhauer et al. [4].The results of the second type locate criticality problems in relation to the complexity classesDP and Θ p2 . The class Θ p2 was introduced by Wagner [19] and represents the languages that can bedecided in polynomial time by an algorithm that has access to an NP oracle under the restrictionthat all queries are submitted at the same time. The definitions of the classes immediately yield theinclusions NP ∪ coNP ⊆ DP ⊆ Θ p2 .Papadimitriou and Wolfe [15] have shown that Minimal-UnSat (the set of unsatisfiable formulasthat become satisfiable when an arbitrary clause is deleted) is DP-complete. Cai and Meyer [7] builtupon this to prove DP-completeness of
VertexMinimal- k -UnColorability (the set of graphs thatare not k -colorable but become k -colorable when an arbitrary vertex is deleted), for all k ≥
3. WithTheorems 7 and 8, we were able to extend this result to classes that are analogously defined for themuch smaller local modification of edge deletion, which is considered the default setting; namely, we4rove DP-completeness of
Minimal- k -UnColorability , for all k ≥ p2 and DP-hard.As Joret [13] points out, a construction by Papadimitriou and Wolfe [15] proves the DP-hardnessof recognizing β -critical graphs. This problem also lies in Θ p2 , but no finer classification has beenachieved so far. In Theorem 15, we show that this problem is in fact Θ p2 -hard, yielding the first knownΘ p2 -completeness result for a criticality problem. AT Our main technique for proving the nontrivial hardness results in our model is the following: Webuild in polynomial-time computable solutions for each locally modified problem instance. That way,the solutions to the locally modified problem instances do not give away any information about theinstance to be solved. A similar approach is taken in some proofs of DP-completeness for minimalityproblems. Indeed, we can occasionally combine the proof of DP-hardness with that of the NP-hardnessof computing an optimal solution from optimal solutions to locally modified instances. Denote by the set of nonempty
CNF -formulas with exactly three distinct literals per clause. We beginby showing in Theorem 1 that there is a reduction from (the set of satisfiable -formulas)to that builds in polynomial-time computable solutions for all one-clause-deleted subformulasof the resulting -formula. At first glance, this very surprising result may seem dangerouslyclose to proving P = NP; Corollary 2 will make explicit where the hardness remains. We will then usethe reduction of Theorem 1 as a preprocessing step in reductions from to other problems.
Theorem 1.
There is a polynomial-time many-one reduction f from to and a polynomial-time computable function s such that, for every -formula Φ and for every clause C in f (Φ) , s ( f (Φ) − C ) is a satisfying assignment for f (Φ) − C .Proof. Papadimitriou and Wolfe [15, Lemma 1] give a reduction from 3 -UnSat to Minimal-UnSat (the set of
CNF -formulas that are unsatisfiable but that become satisfiable with the removal of anarbitrary clause). In Appendix C, we show how to enhance this reduction such that it has all propertiesof our theorem. First, we carefully prove that there is a function s that together with the originalreduction satisfies all properties of our theorem, except that we may output a formula that is not in . In order to rectify this, we show that the standard reduction from Sat to Sat that decreasesthe number of literals per clause to at most three maintains all the required properties. The sameis then shown for the standard reduction that transforms
CNF -formulas with at most three literalsper clause into -formulas that have exactly three distinct literals per clause. Combining thesethree reductions, we can therefore satisfy all requirements of our theorem.
Corollary 2.
Computing a satisfying assignment for a -formula whose one-clause-deleted sub-formulas all have a satisfying assignment from these assignments is NP -hard.Proof. Given a -formula Φ, compute f (Φ), where f is the reduction from Theorem 1. Nowcompute s ( f (Φ) − C ) for every clause C in f (Φ) and compute a satisfying assignment for f (Φ) fromthese solutions. Use this assignment to determine whether Φ is satisfiable. As mentioned in the previous section, the constructions of some DP-completeness results for minimal-ity problems can be adapted to obtain NP-hardness for computing optimal solutions from optimalsolutions to locally modified instances. There are remarkably few complexity results for minimal-ity problems; fortunately, however,
VertexMinimal- -UnColorability (the graphs that are not This set is often denoted
E3-CNF in the literature. is DP-complete by reduction from Minimal- -UnSat [7]. We will show how to extract from said reduction a proof of the fact thatcomputing an optimal coloring for a graph from optimal colorings for its one-vertex-deleted subgraphsis NP-hard (Theorem 4). However, the standard notion of criticality is χ -criticality under edge dele-tion, and the construction by Cai and Meyer [7] does unfortunately not yield the analogous result fordeleting edges. This was to be expected, since working with edge deletion is much harder. Surprisingly,however, a targeted modification of the constructed graph allows us to establish, through a far moreelaborate case distinction, that computing an optimal coloring for a graph from optimal colorings forits one-edge-deleted subgraphs is NP-hard (Theorem 6) as well as that the related minimality problem Minimal- -UnColorability is DP-complete (Theorem 7). Lemma 3.
There is a polynomial-time many-one reduction g from to and apolynomial-time computable function opt such that, for every -formula Φ and for every vertex v in g (Φ) , opt( g (Φ) − v ) is an optimal coloring for g (Φ) − v .Proof. Given a -formula Φ, let g (Φ) = h ( f (Φ)), where f is the reduction from Theorem 1and h is the reduction from Minimal- -UnSat to VertexMinimal- -UnColorability by Cai andMeyer [7]. Since h also reduces to [7, Lemma 2.2], so does g . A carefulinspection of the reduction g reveals that there is a polynomial-time computable function opt suchthat, for every vertex v in g (Φ), opt( g (Φ) − v ) is a 3-coloring of g (Φ) − v . We can also verify that g (Φ) − v does not have a 2-coloring, hence opt( g (Φ) − v ) is an optimal coloring. We do not dive intothe details as this lemma immediately follows from the proof of the analogous result for edge deletion(Lemma 5), as explained in Appendix D. Theorem 4.
Computing an optimal coloring for a graph from optimal colorings for its one-vertex-deleted subgraphs is NP -hard.Proof. Given a -formula Φ, compute g (Φ), where g is the reduction from Lemma 3, computeopt( g (Φ) − v ) for every vertex v in g (Φ), and from these optimal solutions compute one for g (Φ). Thisdetermines whether g (Φ) is 3-colorable and thus whether Φ is satisfiable. Lemma 5.
There is a polynomial-time many-one reduction g from to and apolynomial-time computable function opt such that, for every -formula Φ and for every edge e in g (Φ) , opt( g (Φ) − e ) is an optimal coloring of g (Φ) − e .Proof. Given a -formula Φ, let g (Φ) = h ( f (Φ)) − e , where f is the reduction from Theorem 1, h is the reduction to VertexMinimal- -UnColorability by Cai and Meyer [7], and e is the edge { v c , v s } , with v c being the unique vertex adjacent to all variable-setting vertices and v s being the onlyremaining neighbor vertex of v c . We prove in detail that g has all the desired properties in Appendix D.See Figure 1 for an example of the construction. Theorem 6.
Computing an optimal coloring for a graph from optimal colorings for its one-edge-deleted subgraphs is NP -hard.Proof. The same argument as for Theorem 4 can be applied here.
Theorem 7.
Minimal- -UnColorability is DP -complete.Proof. Membership in DP is immediate, since given a graph G = ( V, E ), determining whether G − e is 3-colorable for every e ∈ E is in NP and so is determining whether G is 3-colorable. As forDP-hardness, the argument from the proof of Lemma 5 shows that mapping Φ to h (Φ) − { v c , v s } ,where h is the reduction from Minimal- -UnSat to VertexMinimal- -UnColorability by Caiand Meyer [7], gives a reduction from Minimal- -UnSat to Minimal- -UnColorability (and to VertexMinimal- -UnColorability as well). Recall that Minimal- -UnSat is DP-hard [15]. It should be noted that
VertexMinimal- -UnColorability is denoted by Minimal-3-UnColorability by Caiand Meyer [7] despite the fact that minimality problems usually refer to the case of edge deletion.
VertexMinimal- k -UnColorability , for all k ≥ Theorem 8.
Minimal- k -UnColorability is DP -complete, for every k ≥ .Proof. Membership in DP is again immediate. To show hardness for k ≥
4, we reduce
Minimal- -Un - Colorability to Minimal- k -UnColorability . We use the construction for deleting vertices [7,Theorem 3.1] and map graph G to G + K k − . Note that χ ( K k − ) = k − χ ( H + H ′ ) = χ ( H ) + χ ( H ′ ) for any two graphs H and H ′ . First suppose G + K k − is in Minimal- k -UnColor - ability . Then G + K k − is not k -colorable, and so G is not 3-colorable. Let e be an edge in G . Then( G − e ) + K k − = ( G + K k − ) − e is k -colorable, and thus G − e is 3-colorable. It follows that G is in Minimal- -UnColorability .Now suppose G is in Minimal- -UnColorability . Then G + K k − is not k -colorable. Let e be anedge in G + K k − . If e is an edge in G , then G − e is 3-colorable and so ( G + K k − ) − e = ( G − e )+ K k − is k -colorable. If e is an edge in K k − , then K k − − e is ( k − G is 4-colorable (let ˆ e be any edge in G , take a 3-coloring of G − ˆ e , and change the color of one of the vertices incident toˆ e to the remaining color), so ( G + K k − ) − e = G + ( K k − − e ) is k -colorable. Finally, if e = { v, w } for a vertex v in G and a vertex w in K k − , let ˆ e be an edge in G incident to v , take a 3-coloring of G − ˆ e , take a disjoint ( k − K k − , and change the color of v to the color of w . As a result,for all edges e in G + K k − , ( G + K k − ) − e is k -colorable. It follows that G + K k − is in Minimal- k -UnColorability .The construction above does not prove the analogues of Lemmas 3 and 5: Note that G is 3-colorableif and only if ( G + K k − ) − v and ( G + K k − ) − e are both ( k − v in K k − and for every edge e in K k − , and so we can certainly determine whether a graph is 3-colorable fromthe optimal solutions to the one-vertex-deleted subgraphs and one-edge-deleted subgraphs of G + K k − in polynomial time. Turning to criticality and vertex-criticality, we can bound their complexity asfollows. Theorem 9.
The two problems of determining whether a graph is critical and whether it is vertex-critical are both in Θ p2 and DP -hard.Proof. For the Θ p2 -membership of the two problems, we observe that the relevant chromatic numbersof a graph G = ( V, E ) and its neighbors can be computed by querying the NP oracle
Colorability = { ( G, k ) | χ ( G ) ≤ k } for every ( G, k ), ( G − e, k ), and ( G − v, k ) for every e ∈ E , v ∈ V , and k ≤ k V ( G ) k in parallel.For the DP-hardness of the two problems, we prove that h (Φ) − { v c , v s } is a reduction from Minimal- -UnSat to both of them. We have already seen that it reduces Minimal- -UnSat to Minimal- -UnColorability . Hence, for every Φ ∈ Minimal- -UnSat , the graph h (Φ) − { v c , v s } is in Minimal- -UnColorability ⊆ VertexMinimal- -UnColorability and thus both criticaland vertex-critical. For the converse it suffices to note that, for every Φ ∈ CNF with clauses of size atmost 3, h (Φ) − { v c , v s } is 4-colorable and thus in Minimal- -UnColorability (in VertexMinimal- -UnColorability , respectively) if and only if it is critical (vertex-critical, respectively).The exact complexity of these problems remains open, however. In particular, it is unknownwhether they are Θ p2 -hard. This contrasts with the case of Vertex Cover, for which we prove inTheorem 15 that recognizing β -vertex-criticality is indeed Θ p2 -complete.Before that, however, we return to our model and consider Colorability under the local modificationof adding an edge. If we allow only one query, the problem stays NP-hard via a simple Turing reduction:Iteratively adding edges to the given instance eventually leads to a clique as a trivial instance, seeTheorem 17 in Appendix B. Note that the restriction to one query is crucial for this reduction to work; For two graphs G and G , the graph join G + G is the disjoint union G ∪ G plus a join edge added from everyvertex of G to every vertex of G ; see, e.g., Harary’s textbook on graph theory [11, p. 21]. Theorem 10.
There is a polynomial-time algorithm that computes an optimal coloring for a graphfrom optimal colorings of all its one-edge-added supergraphs; in fact, two optimal colorings, one foreach of two specific one-edge-added supergraphs, suffice.
For the proof of this theorem, we naturally extend the notion of universal vertices as follows.
Definition 11.
An edge { u, v } ∈ E of a graph G = ( V, E ) is called universal if, for every vertex x ∈ V − { u, v } , we have { x, u } ∈ E or { x, v } ∈ E . A graph is called universal-edged if all its edgesare universal. Additionally, we denote, for any given graph G = ( V, E ) and any vertex x ∈ V , the open neighbor-hood of x in G by N ( x ) := { y | { x, y } ∈ E } and the closed neighborhood of x in G by N [ x ] := N ( x ) ∪{ x } .We are now ready to give the proof of Theorem 10. Proof of Theorem 10.
We show that
Colorer (Algorithm 1), which uses the oracle of our model and
Subcol (Algorithm 2) as subroutines, has the desired properties.
Algorithm 1
Colorer
Input:
An undirected graph G = ( V, E ). Output:
An optimal coloring for G . Description:
Optimizes universal-edged graphs with two queries to
Oracle , which provides optimalsolutions to one-edge-added supergraphs; other graphs are optimized via
Subcol . for every edge { u, v } ∈ E do for every vertex x ∈ V − { u, v } do if { u, x } / ∈ E ∧ { v, x } / ∈ E then f ← Oracle ( G ∪ { u, x } ) f ← Oracle ( G ∪ { v, x } ) if f uses fewer colors on G than f then return f else return f k ← while Subcol ( G, k ) = NO do k ← k + 1 return Subcol ( G, k )We begin by proving that
Colorer is correct. Assume first that the input graph G = ( V, E ) is notuniversal-edged. Then
Colorer can find an edge { u, v } ∈ E with a non-neighboring vertex x ∈ V and query the oracle on G ∪{ u, x } and G ∪{ v, x } for optimal colorings f and f . We argue that at leastone of them is also optimal for G . Let f be any optimal coloring of G . Since u and v are connectedby an edge, we have f ( u ) = f ( v ) and hence f ( x ) = f ( u ) or f ( x ) = f ( v ); see Figure 2 in Appendix E.Thus, f is also an optimal coloring of G ∪ { x, u } or G ∪ { x, v } , and so we have χ ( G ) = χ ( G ∪ { x, u } ) or χ ( G ) = χ ( G ∪ { x, v } ). Therefore, f or f is an optimal coloring for G as well and returned on line 7or 9, respectively.The while loop can be entered only if the graph G is universal-edged. This allows us to computean optimal solution to G with no queries at all by using Subcol (Algorithm 2). We will show that
Subcol is a polynomial-time algorithm that computes, for any universal-edged graph G and anypositive integer k , a k -coloring of G if there is one, and outputs NO otherwise. The while loop of8 lgorithm 2 Subcol
Input:
An undirected, universal-edged graph G = ( V, E ) and a positive integer k . Output: A k -coloring f for G if there is one; NO if there is none. Description:
Works by recursion on k , with k = 1 and k = 2 serving as the base cases. if G has no edge then return the constant 1-coloring with f ( x ) = 1 for all x ∈ V . else if k = 1 then return NO. if G has bipartition { A, B } then return the 2-coloring f ( x ) = ( x ∈ A and2 for x ∈ B. else if k=2 then return NO. Choose an arbitrary edge { ℓ, r } ∈ E . L ← N ( ℓ ) − N [ r ]; R ← N ( r ) − N [ ℓ ]; M ← N ( ℓ ) ∩ N ( r ) g ← Subcol ( G [ M ] , k − if g = NO then return NO return the k -coloring f ( x ) = g ( x ) for x ∈ M , k − x ∈ L ∪ { r } , and k for x ∈ R ∪ { ℓ } . Colorer thus searches the smallest integer k such that G has a k -coloring, that is, k = χ ( G ). Hence,an optimal coloring of G is returned on line 13. Due to k = χ ( G ) ≤ k V k , Colorer has polynomialtime complexity.It remains to prove the correctness and polynomial time complexity of
Subcol . This can be doneby bounding its recursion depth and verifying the correctness for each of the six return statements;this is hardest for the last two. The proof relies on the properties of the partition M ∪ L ∪ R ∪ { ℓ } ∪ { r } as illustrated in Figure 3; see Appendix E for all details.In this section, we have proven that Minimal- k -UnColorability is complete for DP for every k ≥ This section will show that the behavior of Vertex Cover in our model is distinctly different from theone that we demonstrated for Colorability in the previous section. In particular, Theorem 12 provesthat computing an optimal vertex cover from optimal solutions of one-vertex-deleted subgraphs canbe done in polynomial time, which is impossible for optimal colorings according to Theorem 4 unlessP = NP.First, we note that the NP-hardness proof for our most restricted case with only one query stillworks (i.e., Theorem 17 in Appendix B is applicable): Deleting vertices, adding edges, or deleting edgesrepeatedly will always lead to the null graph, an edgeless graph, or a clique through polynomially manyinstances. As we have seen for Colorability in the previous section, hardness proofs of this type mayfail due to exponential branching as soon as multiple queries are allowed. We can show that, unlessP = NP, this is necessarily the case for edge addition and vertex deletion since two granted queriessuffice to obtain a polynomial-time algorithm.
Theorem 12.
There is a polynomial-time algorithm that computes an optimal vertex cover for a graphfrom two optimal vertex covers for some one-vertex-deleted subgraphs. roof. Observe what can happen when a vertex v is removed from a graph G with an optimal vertexcover of size k . If v is part of any optimal vertex cover of G , then the size of an optimal vertex coverfor G − v is k −
1. Given any graph G , pick any two adjacent vertices v and v . Since there is an edgebetween them, one of them is always part of an optimal vertex cover, thus either G − v or G − v orboth will have an optimal vertex cover of size k −
1. Two queries to the oracle return optimal vertexcovers for G − v and G − v . The algorithm chooses the smaller of these two covers (or any, if theyare the same size) and adds the corresponding v i . The resulting vertex cover has size k and is thusoptimal for G .Theorem 19 in Appendix F proves that the analogous result for adding an edge holds as well.At this point, we would like to prove either an analogue to Theorem 6, showing that computingan optimal vertex cover is NP-hard even if we get access to a solution for every one-edge-deletedsubgraph, or an analogue to Theorem 19, showing that the problem is in P if we have access to asolution for more than one one-edge-deleted subgraph. We were unable to prove either, however. Thelatter is easy to do for many restricted graph classes (e.g., graphs with bridges), yet we suspect thatthe problem is NP-hard in general. We will detail a few reasons for the apparent difficulty of provingthis statement after the following theorem and corollary, which look at deleting a triangle as the localmodification. Theorem 13.
There is a reduction g from to VertexCover such that, for every -formula Φ and for every triangle T in g (Φ) , there is a polynomial-time computable optimal vertexcover of g (Φ) − T . The proof of Theorem 13 relies on the standard reduction from to VertexCover ; see [10],where clauses correspond to triangles; see Appendix G for the details. Applying the same argumentas in the proof of Theorem 4 yields the following corollary.
Corollary 14.
Computing an optimal vertex cover for a graph from optimal vertex covers for theone-triangle-deleted subgraphs is NP -hard. What can we say about optimal vertex covers for one-edge-deleted graphs? Papadimitriou andWolfe show [15, Theorem 4] that there is a reduction g from Minimal- -UnSat to Minimal- k -No - VertexCover (called
Critical-VertexCover in [15]; asking, given a graph G and an integer k , whether G does not have a vertex cover of size k but all one-edge-deleted subgraphs do). Thereduction builds in a polynomial-time computable vertex cover of size k for every one-edge-deletedsubgraph. And so g is a reduction from to VertexCover such that there exists a polynomial-time computable function opt such that for every -formula Φ and g (Φ) = ( G, k ), it holds, forevery edge e in G , that opt( G − e ) is a vertex cover of size k . Unfortunately, it may happen that anoptimal vertex cover of G − e has size k −
1; namely, if e is an edge connecting two triangles, an edgebetween two variable-setting vertices, or any edge of the clause triangles. The function opt does thusnot give us an optimal vertex cover, thwarting the proof attempt. This shows that we cannot alwaysget results for our model from the constructions for criticality problems.The following would be one approach to design a polynomial-time algorithm that computes anoptimal vertex cover from optimal vertex covers for all one-edge-deleted subgraphs: It is clear thatdeleting an edge does not increase the size of an optimal vertex cover and decreases it by at most one.If, for any two neighbor graphs, the provided vertex covers differ in size, then we can take the smallerone, restore the deleted edge, and add any one of the two incident vertices to the vertex cover; thisgives us the desired optimal vertex cover. If the optimal vertex cover size decreases for all deletions ofa single edge, we can do the same with any of them. Thus, it is sufficient to design a polynomial-timealgorithm that solves the problem on graphs whose one-edge-deleted subgraphs all have optimal vertexcovers of the same size as an optimal vertex cover of the original graph. One might suspect that onlyvery few and simple graphs can be of this kind. However, we obtain infinitely many such graphs bythe removal of any edge from different cliques, as already mentioned in the introduction. In fact, there10s a far larger class of graphs with this property and no apparent communality to be exploited for theefficient construction of an optimal vertex cover.We now turn to our complexity results of β -(vertex-)criticality. The reduction from Minimal- -UnSat to Minimal- k -NoVertexCover by Papadimitriou and Wolfe [15] establishes the DP-hardness of deciding whether a graph is β -critical. However, it seems unlikely that β -criticality is inDP. The obvious upper bound is Θ p2 , since a polynomial number of queries to a VertexCover oracle,namely (
G, k ) and ( G − e, k ) for all edges e in G and all k ≤ k V ( G ) k , in parallel allows us to determine β ( G ) and β ( G − e ) for all edges e in polynomial time, and thus allows us to determine whether G is β -critical. While we have not succeeded in proving a matching lower bound, or even any lower boundbeyond DP-hardness, we do get this lower bound for β -vertex-criticality, thereby obtaining the firstΘ p2 -completeness result for a criticality problem. Theorem 15.
Determining whether a graph is β -vertex-critical is Θ p2 -complete.Proof. Membership follows with the same argument as above, this time querying the oracle
Vertex - Cover in parallel for all (
G, k ) and ( G − v, k ) for all vertices v in G and all k ≤ k V ( G ) k . To show thatthis problem is Θ p2 -hard, we use a similar reduction as the one by Hemaspaandra et al. [12, Lemma4.12] to prove that it is Θ p2 -hard to determine whether a given vertex is a member of a minimumvertex cover. We reduce from the Θ p2 -complete problem VC = = { ( G, H ) | β ( G ) = β ( H ) } [20].Let n = max( k V ( G ) k , k V ( H ) k ), let G ′ consist of n + 1 − k V ( G ) k isolated vertices, let H ′ consistof n + 1 − k V ( H ) k isolated vertices, and let F = ( G ∪ G ′ ) + ( H ∪ H ′ ). Note that β ( F ) = ( n +1) + min( β ( G ) , β ( H )). If β ( G ) = β ( H ), then β ( F ) = ( n + 1) + β ( G ) = ( n + 1) + β ( H ) and forevery vertex v in F , β ( F − v ) = n + β ( G ). Thus, F is critical. If β ( G ) = β ( H ), assume withoutloss of generality that β ( G ) < β ( H ). Then β ( F ) = n + 1 + β ( G ). Let v be a vertex in G ′ . Then β ( F − v ) = min( n + 1 + β ( G ) , n + β ( H )) = n + 1 + β ( G ), and therefore F is not critical. We defined a natural model that provides new insights into the structural properties of NP-hardproblems. Specifically, we revealed interesting differences in the behavior of Colorability and VertexCover under different types of local modifications. While Colorability remains NP-hard when the localmodification is the deletion of either a vertex or an edge, there is an algorithm that finds an optimalcoloring by querying the oracle on at most two edge-added supergraphs. Vertex Cover, in contrast,becomes easy in our model for both deleting vertices and adding edges, as soon as two queries aregranted. The question of what happens for the local modification of deleting an edge remains asan intriguing open problem that defies any simple approach, as briefly outlined above. Moreover,examples of problems where one can prove a jump from membership in P to NP-hardness at a givennumber of queries greater than 2 might be especially instructive.With its close connections to many distinct research areas, most notably the study of self-reducibilityand critical graphs, our model can serve as a tool for new discoveries. In particular, we were ableto exploit the tight relations to criticality in the proof that recognizing β -vertex-critical graphs isΘ p2 -hard, yielding the first completeness result for Θ p2 in the field. Acknowledgments
We thank the anonymous referees and Hans-Joachim Böckenhauer, Rodrigo R. Gumucio Escobar,Lane Hemaspaandra, Juraj Hromkovič, Rastislav Kralovič, Richard Kralovič, Xavier Muñoz, MartinRaszyk, Peter Rossmanith, Walter Unger, and Koichi Wada for helpful comments and discussions.11 eferences [1] Claudia Archetti, Luca Bertazzi, and Maria Grazia Speranza. Reoptimizing the traveling salesmanproblem.
Networks , 42(3):154–159, 2003.[2] Giorgio Ausiello, Bruno Escoffier, Jérôme Monnot, and Vangelis Paschos. Reoptimization ofminimum and maximum traveling salesman’s tours. In
Proceedings of the 10th ScandinavianWorkshop on Algorithm Theory (SWAT 2006) , volume 4059 of
Lecture Notes in Computer Science ,pages 196–207. Springer-Verlag, 2006.[3] Hans-Joachim Böckenhauer, Luca Forlizzi, Juraj Hromkovič, Joachim Kneis, Joachim Kupke,Guido Proietti, and Peter Widmayer. Reusing optimal TSP solutions for locally modified inputinstances. In
Proceedings of the 4th IFIP International Conference on Theoretical ComputerScience (IFIP TCS 2006) , pages 251–270. Springer-Verlag, 2006.[4] Hans-Joachim Böckenhauer, Juraj Hromkovič, and Dennis Komm. Reoptimization of hard opti-mization problems. In Teofilo F. Gonzalez, editor,
AAM Handbook of Approximation Algorithmsand Metaheuristics , volume 1, chapter 25, pages 427–454. CRC Press 2018, 2nd edition, 2018.[5] Hans-Joachim Böckenhauer, Juraj Hromkovič, Tobias Mömke, and Peter Widmayer. On thehardness of reoptimization. In
Proceedings of the 34th Conference on Current Trends in Theoryand Practice of Computer Science (SOFSEM 2008) , volume 4910 of
Lecture Notes in ComputerScience , pages 50–65. Springer-Verlag, 2008.[6] Hans K. Büning and Oliver Kullmann. Minimal unsatisfiability and autarkies. In Armin Biere,Marijn Heule, Hans van Maaren, and Toby Walsh, editors,
Handbook of Satisfiability 2009 , pages339–401. IOS Press, 2009.[7] Jin-Yi Cai and Gabriele E. Meyer. Graph minimal uncolorability is D P -complete. SIAM Journalon Computing , 16(2):259–277, 1987.[8] Gabriel A. Dirac. Some theorems on abstract graphs.
Proceedings of the London MathematicalSociety , s3-2(1):69–81, 1952.[9] Piotr Faliszewski and Mitsunori Ogihara. On the autoreducibility of functions.
Theory of Com-puting Systems , 46(2):222–245, 2010.[10] Michael Garey and David S. Johnson.
Computers and Intractability: A Guide to the Theory ofNP-Completeness . W. H. Freeman and Company, 1979.[11] Frank Harary.
Graph Theory . Addison-Wesley, 1991.[12] Edith Hemaspaandra, Holger Spakowski, and Jörg Vogel. The complexity of Kemeny elections.
Theoretical Computer Science , 349(3):382–391, 2005.[13] Gwenaël Joret.
Entropy and Stability in Graphs . PhD thesis, Université Libre de Bruxelles,Faculté des Sciences, 2008.[14] Albert R. Meyer and Mike Paterson. With what frequency are apparently intractable problemsdifficult? Technical Report MIT/LCS/TM-126, Laboratory for Computer Science, MIT, Cam-bridge, MA, 1979.[15] Christos H. Papadimitriou and David Wolfe. The complexity of facets resolved.
Journal ofComputer and System Sciences , 37(1):2–13, 1988.[16] Jörg Rothe and Tobias Riege. Completeness in the Boolean Hierarchy.
Journal of UniversalComputer Science , 12(5):551–578, 2006. 1217] Claus-Peter Schnorr. Optimal algorithms for self-reducible problems. In
Proceedings of the 3rdInternational Colloquium on Automata, Languages, and Programming , pages 322–337. EdinburghUniversity Press, 1976.[18] Markus W. Schäffter. Scheduling with forbidden sets.
Discrete Applied Mathematics , 72(1–2):155–166, 1997.[19] Karl Wagner. Bounded query classes.
SIAM Journal on Computing , 19(5):833–846, 1990.[20] Klaus W. Wagner. More complicated questions about maxima and minima, and some closures ofNP.
Theoretical Computer Science , 51(1–2):53–80, 1987.[21] Walter Wessel. Criticity with respect to properties and operations in graph theory. In Lás-zló Lovász András Hajnal and Vera T. Sós, editors,
Finite and Infinite Sets. (6th HungarianCombinatorial Colloquium, Eger, 1981) , volume 2 of
Colloquia Mathematica Societatis JanosBolyai , pages 829–837. North-Holland, 1984. 13
Tractability for Vertex Addition
We show that Vertex Cover and Colorability are trivially tractable if we are allowed to query an oraclefor an optimal solution to the input graph with one vertex added.
Theorem 16.
Under our model with the local modification of adding a vertex, the two problems offinding an optimal vertex cover and finding an optimal coloring are in P .Proof. Let G be the given graph. Add an isolated vertex v and query the oracle for an optimal vertexcover or an optimal coloring of G + v , respectively. Clearly, the restriction of this solution to G isoptimal as well since an isolated vertex is never part of an optimal vertex cover and can be coloredarbitrarily. (Note that we could also avoid isolated vertices by adding a universal vertex instead sincea universal vertex needs to be colored in a unique color and is, without loss of generality, part of anyoptimal vertex cover.) B Hardness Results for Restriction to a Single Query
In the most restricted case of our model, where we grant the algorithm but one single query for anoptimal solution of a neighboring (i.e., locally modified) instance, all considered problems preservetheir NP-hardness. The proof for this is simple enough and based on a technique that is commonlyapplied in reoptimization, self-reducibility, and many other fields. We formulate the following theoremas a generalization of Lemma 1 by Böckenhauer et al. [5].
Theorem 17.
Let
OptProb be an optimization problem. Let T be a set of efficiently solvable in-stances of OptProb . (Or, to be more precise: Let T ⊆ Σ be a subset of instances for the problem OptProb ⊆ Σ × Σ such that there is a polynomial-time algorithm that computes an optimal solu-tion on every instance of T .) Let the considered local modification be such that applying arbitrary localmodifications repeatedly will inevitably transform any instance of OptProb into an instance in T ina polynomial number of steps. Then OptProb is NP -hard in our model with restriction to one query.Proof. We give a reduction from
OptProb in the classical setting to
OptProb in our model. Let A be a polynomial-time algorithm that computes on instance I i a locally modified instance I i +1 anduses an optimal solution to I i +1 to compute an optimal solution for I i . For any given instance I of OptProb , we thus get a chain I , I , . . . , I n of polynomial length such that I n is in T . We canefficiently compute an optimal solution to I n ∈ T and then use A to successively compute optimalsolutions to I n − , . . . , I , I in polynomial time. C Details of the Proof of Theorem 1
This section goes through the reduction from 3 -UnSat to Minimal-UnSat by Papdimitriou andWolfe [15] (mostly following the notation of Büning and Kullmann [6]) plus three standard reductionsfor additional form constraints, showing that the composition has all the properties that we need forour results.
C.1 The main reduction: -U N S AT to M INIMAL -U N S AT Let Φ ∈ be a Boolean formula over the variable set { x , . . . , x n } , n >
1; that is, Φ = C ∧ . . . ∧ C m with C i = ℓ i, ∨ ℓ i, ∨ ℓ i, and ℓ i,j ∈ { x , . . . , x n , x , . . . , x n } , where an overline denotesnegation. In the following, we construct in polynomial time an equivalent CNF -formula Ψ with theadditional property that each one-clause-deleted subformula of Ψ has an easy-to-compute satisfyingassignment. First, delete without replacement any clause that contains a variable and its negation.(Such a clause is satisfied for every assignment.) Assume thus without loss of generality that no14lause C i contains a variable and its negation. Now introduce m new variables { y , . . . , y m } and let π i := y ∨ . . . ∨ y i − ∨ y i +1 ∨ . . . ∨ y m . LetΨ = m ^ i =1 ( C i ∨ π i ) ∧ m ^ i =1 3 ^ j =1 ( ℓ i,j ∨ π i ∨ y i ) ∧ ^ ≤ i C.1.1 Φ satisfiable ⇒ Ψ satisfiable Let α be a satisfying assignment for Φ. Then β : ( x i α ( x i ) , for all i, and y i , for y ∈ Y, is a satisfying assignment for Ψ. Indeed, we have β ( C i ∨ π i ) = 1 since α ( C i ) = 1; and we have β ( ℓ i,j ∨ π i ∨ y i ) = 1 and β ( y i ∨ y j ) = 1 since β ( y i ) = 1. C.1.2 Ψ satisfiable ⇒ Φ satisfiable Let β be a satisfying assignment for Ψ. We prove that β (or, to be precise, the restriction β | X ) alsosatisfies Φ. The satisfied clauses of the form y i ∨ y j in Ψ guarantee that β ( y ˆ ı ) = 0 for at most oneˆ ı ∈ { , . . . , m } . Case 1: Assume that β ( y i ) = 1 for all i ∈ { , . . . , m } . All clauses that contain a literal y i areclearly satisfied. The only remaining clauses have the form C i ∨ π i . Moreover, β ( y i ) = 0 for all i ∈ { , . . . , m } implies β ( π i ) = 0 for all i ∈ { , . . . , m } . Thus, we have β ( C i ∨ π i ) = 1 if and onlyif β ( C i ) = 1. By assumption β satisfies all clauses of Ψ, in particular those of the form C i ∨ π i ;therefore β also satisfies Φ. Case 2: Assume that β ( y ˆ ı ) = 0, that is, β ( y ˆ ı ) = 1 for exactly one ˆ ı ∈ { , . . . , m } . Then β ( π ˆ ı ) = 0and β ( π i ) = 1 for the remaining i = ˆ ı . Thus, all clauses that contain y i or π i with i = ˆ ı are triviallysatisfied. The only four remaining clauses are( C ˆ ı ∨ π ˆ ı ) ∧ ^ j =1 ( ℓ ˆ ı,j ∨ π ˆ ı ∨ y ˆ ı ) , which, due to β ( π ˆ ı ) = β ( y ˆ ı ) = 0, simplify to C ˆ ı ∧ ℓ ˆ ı, ∧ ℓ ˆ ı, ∧ ℓ ˆ ı, . This is unsatisfiable since C ˆ ı = ℓ ˆ ı, ∧ ℓ ˆ ı, ∧ ℓ ˆ ı, . Thus, case 2 cannot occur. C.1.3 Ψ is satisfiable after deletion of an arbitrary clause There are three cases, which we handle separately. Case 1: The deleted clause is y ˆ ı ∨ y ˆ . We show that in this case, the following assignment issatisfying: β : y ˆ ı ,y ˆ ,y i , for ˆ ı = i = ˆ , and x i arbitrary . We have β ( π i ) = 1 for all i ∈ { , . . . , m } since any π i contains either y ˆ ı or y ˆ . The remainingclauses y i ∧ y j with ( i, j ) = (ˆ ı, ˆ ) are trivially satisfied.15 ase 2: The deleted clause is C ˆ ı ∨ π ˆ ı . In this case, the assignment β : y ˆ ı ,y i , for i = ˆ ı,x i , for x i ∈ { ℓ ˆ ı, , ℓ ˆ ı, , ℓ ˆ ı, } ,x i , for x i ∈ { ℓ ˆ ı, , ℓ ˆ ı, , ℓ ˆ ı, } , and x i arbitrary , otherwise,is satisfying. All clauses of the form y i ∨ y j are satisfied since only y ˆ ı is assigned 1 and i = j .We also have β ( π i ) = 1 for all i = ˆ ı , so all clauses containing π i for i = ˆ ı are satisfied. Since C ˆ ı ∨ π ˆ ı is deleted, the only three remaining clauses are ℓ ˆ ı,j ∨ π ˆ ı ∨ y ˆ ı for j ∈ { , , } . These aresatisfied because β ( ℓ ˆ ı,j ) = 1 for j ∈ { , , } . (Such an assignment is valid since no clause C i contains a variable and its negation, as mentioned in the first paragraph; in particular C ˆ ı , that is, { ℓ ˆ ı, , ℓ ˆ ı, , ℓ ˆ ı, } ∩ { ℓ ˆ ı, , ℓ ˆ ı, , ℓ ˆ ı, } = ∅ .) Case 3: The deleted clause is ℓ ˆ ı, ˆ ∨ π ˆ ı ∨ y ˆ ı . Also in this case, the assignment β : y ˆ ı ,y i , for i = ˆ ı,x i , for x i ∈ { ℓ ˆ ı, ˆ } ∪ { ℓ ˆ ı,j | j = ˆ } ,x i , for x i ∈ { ℓ ˆ ı, ˆ } ∪ { ℓ ˆ ı,j | j = ˆ } ,x i arbitrary , otherwise,is satisfying. The same argument as in Case 2 shows that the assignment to y ˆ ı satisfies all clausesbut the three clauses C ˆ ı and ℓ ˆ ı,j ∨ π ˆ ı ∨ y ˆ ı for j ∈ { , , } − { ˆ } . The clause C ˆ ı is satisfied because β ( ℓ ˆ ı, ˆ ) = 1; the other two are satisfied due to β ( ℓ ˆ ı,j ) = 1 for j = ˆ . C.2 Additional Form ConstraintsC.2.1 CNF to OR is the set of all CNF -formulas with exactly two or three literals in every clause. Construction. Ψ can be replaced by an equivalent -formula Ψ ′ while retaining the property that each one-clause-deleted subformula has an easy-to-compute satisfying assignment. We can use the standardreduction which replaces a clause C i = ℓ i, ∨ . . . ∨ ℓ i, k C i k by( ℓ i, ∨ z i, ) | {z } C i, ∧ ( z i, ∨ ℓ i, ∨ z i, ) | {z } C i, ∧ . . . ∧ ( z | C i |− ∨ ℓ i, k C i k− ∨ z i, k C i k ) | {z } C i, k Ci k− ∧ ( z i, k C i k ∨ ℓ k C i k ) | {z } C i, k Ci k , where z i, , . . . , z i, k C i k are k C i k new variables. Equivalence. Ψ and Ψ ′ are equivalent because we can use the assignment of truth values to z i, , . . . , z i, k C i k to satisfyall but an arbitrary one of the substituted clauses above. Easy-to-compute satisfying assignments for one-clause-deleted subformulas. Deleting a clause C ˆ ı, ˆ from Ψ ′ corresponds to the deletion of the clause C ˆ ı from Ψ because the clauses C ˆ ı,j with j = ˆ can always be satisfied by assigning 1 to the variables z ˆ ı, , . . . , z ˆ ı, ˆ − and 0 to thevariables z ˆ ı, ˆ +1 , . . . , z ˆ ı, k C ˆ ı k . 16 .2.2 to CCURRENCES -2 OR is the set of -formulas where each variable occurs at most oncein each clause and at most three times in the entire formula. Construction. Let Φ be a -formula over { x , . . . , x n } . Assume that x occurs in Φ a total of a times in theaffirmative and b times negated. We replace the a affirmative occurrences by x , , . . . , x ,a and the b negated occurrences by x ,a +1 , . . . , x ,a + b . Moreover, we add the following new clauses:( x , ∨ x , ) ∧ . . . ∧ ( x ,a + b − ∨ x ,a + b ) . Repeating this for x , . . . , x n results in a formula Ψ. Observe that the added clauses are equivalent tothe implication chain x , ⇒ x , ⇒ . . . ⇒ x ,a + b − ⇒ x ,a + b for all i ∈ { , . . . , m } . We repeat this construction for x , . . . , x n and obtain our formula Ψ. Now, we show that Ψ is equivalentto Φ. Correctness. Given a satisfying assignment α for Φ, the assignment β : x i,j α ( x i ) trivially satisfies the constructedformula Ψ. For the converse, assume there is a satisfying assignment β for Ψ. We prove that themodified assignment β ′ ( x i,j ) = β ( x i,a ) for all j ∈ { , . . . , a + b } also satisfies Ψ. Obviously, β ′ satisfiesthe implication chains since there is no dependence on j . To see that the other clauses are satisfiedas well, consider the two possible assignments for x i,a . Case 1. If β ( x i,a ) = 1, then β ( x i,j ) = 1 for all j ≥ a by the implication chain. These variables arealso assigned 1 by β ′ , which has β ′ ( x i,j ) = 1 for all j . Thus β ′ can only differ from β on thevariables x i,j with j < a . These are the positively occurring variables and β ′ assigns 1 to all ofthem. Therefore, the changes to the assignment keep all the satisfied clauses satisfied. Case 2. If β ( x i,a ) = 0, then β ( x i,j ) = 0 for all j ≤ a by the contrapositive of the implication chain.These variables are also assigned 0 by β ′ , which has β ′ ( x i,j ) = 0 for all j . Thus β ′ can only differfrom β on the variables x i,j with j > a . These are the negatively occurring variables and β ′ assigns0 to all of them. As before, we conlcude that none of the changes to the assignment renders anysatisfied clause unsatisfied.Now, we trivially obtain from β ′ a satisfying assignment for Φ. Easy-to-compute satisfying assignments for one-clause-deleted subformulas. Assume that a clause x ˆ ı, ˆ ∨ x ˆ ı, ˆ +1 is deleted. (For all other clauses, the correspondence between Φ andΨ is immediate.) Then, the ˆ ı th implication chain breaks in two and we are left with x , ⇒ . . . ⇒ x ˆ ı, ˆ and x ˆ ı, ˆ +1 ⇒ . . . ⇒ x ,a + b . Consider the four (partial) assignments β : x ˆ ı,j j,β ′ : ( x ˆ ı,j j = ˆ ,x ˆ ı, ˆ , : x ˆ ı,j j,β ′ : ( x ˆ ı,j j = ˆ + 1 ,x ˆ ı, ˆ +1 . As already seen, the two assignments β and β correspond to the possible assignments for x i in Φ. The option of β ′ and β ′ , however, allows us to freely switch the assignment to one variable,either x ˆ ı, ˆ or x ˆ ı, ˆ +1 . This means that the clause where this variable occurs can always be satisfied;which is tantamount to deleting this clause. For the remaining clauses, we use the assignment fromAppendix C.2.1. C.2.3 CCURRENCES -2 OR to With the following construction, we gain the property that every clause contains exactly 3 literals(instead of either 2 or 3), but lose the property that every variable occurs at most three times andevery literal at most twice. Construction. Let Φ be a given formula in . We construct Ψ in the following way:Clauses with exactly three literals remain unchanged. A clause C i = ( ℓ i, ∨ ℓ i, ) with two literals isreplaced by e C i = ( ℓ i, ∨ ℓ i, ∨ y i ) ∧ ( ℓ i, ∨ ℓ i, ∨ y i ), with a new variable y i . A clause C i = ( ℓ i, ) withonly one literal is replaced by e C i = ( ℓ i, ∨ y i ∨ z i ) ∧ ( ℓ i, ∨ y i ∨ z i ) ∧ ( ℓ i, ∨ y i ∨ z i ) ∧ ( ℓ i, ∨ y i ∨ z i ) , with new variables y i and z i . Correctness. By assigning the right values to y i and z i , respectively, we satisfy any of the two clauses (any three ofthe four clauses, respectively) of e C i , leaving one that simplifies to the original C i . Easy-to-compute satisfying assignments for one-clause-deleted subformulas. If a clause of e C i is deleted, we can again use the assignment to y i and z i to satisfy the remaining ones;thus virtually deleting the whole of e C i . D Full Proof of Lemma 5 For convenience, we restate Lemma 5 before giving its proof. Lemma 5. There is a polynomial-time many-one reduction g from to and apolynomial-time computable function opt such that, for every -formula Φ and for every edge e in g (Φ) , opt( g (Φ) − e ) is an optimal coloring of g (Φ) − e .Proof. Given a -formula Φ, let g (Φ) = h ( f (Φ)) − { v c , v s } , where f is the reduction fromTheorem 1 and h is the reduction from Minimal- -UnSat to VertexMinimal- -UnColorability by Cai and Meyer [7] described below. We will show that g reduces to andthat there is a polynomial-time computable function opt such that, for every -formula Φ andfor every edge e in g (Φ), opt( g (Φ) − e ) is an optimal coloring of g (Φ) − e .For completeness, we briefly describe the reduction h from Minimal- -UnSat to VertexMinimal- -UnColorability [7] (also excellently explained by Rothe and Riege [16]). Let Φ be a -formula with variables { x , . . . , x n } and clauses { c , . . . , c m } . The graph h (Φ) is defined as follows;18 c v s x x · · · x i x i · · · x j x j · · · x n x n a b a b a b a m b m a m b m a m b m t t t t m t m t m · · · C C m Figure 1. The graph h (Φ) − { v c , v s } for a -formula with C = x ∨ x i ∨ x j and C m = x j ∨ x n ∨ x n . see Figure 1. First, we create two vertices v c and v s connected by an edge. Then, for each variable x i , we create two vertices x i and x i and connect them to each other and each of them to v c . For eachclause C k = ℓ k ∨ ℓ k ∨ ℓ k of Φ, we create nine new vertices, namely a triangle t k , t k , t k and a pair a ki , b ki for each literal ℓ ki , where t ki is connected to b ki , a ki is connected to b ki , and both a ki and b ki are connected to v s ; if and only if the literal ℓ ∈ { x j , x j } appears as the i th literal in C k , there is anedge from ℓ to a ki .We first show that g is a reduction from to . Cai and Meyer [7, Lemma 2.2]show that h is a reduction from to . This implies that for every -formulaΦ, Φ is satisfiable if and only if h ( f (Φ)) is 3-colorable, so it suffices to show that if h ( f (Φ)) − { v c , v s } is 3-colorable, then so is h ( f (Φ)). Consider a 3-coloring of h ( f (Φ)) − { v c , v s } such that v c and v s getthe same color. Following the original proof [7], we call the colors T, F, and C. Assume that v c and v s are colored T. Now change the color of v c to C and change the color of every literal vertex originallycolored C to T. It is easy to check that this new coloring is a 3-coloring of h ( f (Φ)). Let e be an edge in g (Φ). We need to show that there is a polynomial-time computable optimalcoloring of g (Φ) − e . We show that there is a polynomial-time computable 3-coloring. (This is optimalbecause g (Φ) − e is not 2-colorable since it contains triangles.)Let C be a clause in f (Φ). Let α be a polynomial-time computable assignment for f (Φ) − C . Fromthis assignment, we can compute in polynomial time a 3-coloring of g (Φ) − C , i.e., g (Φ) minus the nineclause-vertices representing C , in such a way that the literal-vertices are colored T or F according to α , v c is colored C, and v s is colored T.1. If e = { x i , x i } , let C be a clause in f (Φ) such that x i occurs positively in C (note that it followsfrom the definition of f that every literal appears positively in at least one clause of f (Φ)). Color g (Φ) − C as explained above. If x i is colored F, change its color to T. This is still a 3-coloring of g (Φ) − C , and since x i occurs positively in C , we can extend this coloring to a 3-coloring of g (Φ).2. If e = { v c , ℓ i } , where ℓ i ∈ { x i , x i } , let C be a clause in f (Φ) such that ℓ i occurs positively in C . Color g (Φ) − C as explained above. If ℓ i is colored T, then we can extend the coloring toa 3-coloring of g (Φ) in polynomial time. If ℓ i is colored F, change the color of ℓ i to C, and forevery a -vertex connected to ℓ i , change its color from C to F, and for every b -vertex connected toa changed a -vertex, change its color from F to C. It is possible that because of this, the b -verticesin a clause are all colored C, and the attached triangle cannot be colored. If that is the case, thereis a b -vertex in the clause that is connected to an a -vertex that is connected to a literal that is Note that this also shows that deleting the edge { v c , v s } is crucial for the lemma and that the original construction [7]does not work for deleting edges. vx (a) An arbitrary k -coloring of G . { u, x } u vx (b) For G ∪ { u, x } ,this is no longer a k -coloring. { v, x } u vx (c) However, it re-mains a k -coloring for G ∪ { v, x } . Figure 2. Any k -coloring of G is also a k -coloring for G ∪ { u, x } or G ∪ { v, x } or both. If a coloring is optimal for G ,then it is also optimal for G ∪ { u, x } or G ∪ { v, x } . The figure depicts only the induced subgraphs of { u, v, x } . colored T. Change the color of the a -vertex to C and that of the b -vertex to F. Now we can colorthe triangle. This results in a 3-coloring of g (Φ) − C , and it is easy to check that we can extendthis coloring to a 3-coloring of g (Φ).3. Let C be a clause such that e is connected to a clause vertex of C . Again, color g (Φ) − C asabove. If α satisfies f (Φ), we can in polynomial time compute a 3-coloring of g (Φ). So supposethat α does not satisfy f (Φ). Then all literal-vertices connected to a clause-vertex of C are coloredF. When we try to extend this coloring, all a -vertices in the clause must be colored C and all b -vertices must be colored F, which means that we cannot color the triangle with 3 colors. If e isone of the triangle edges, we can color the triangle-vertices T, C, and T. If e connects a b -vertexto a t -vertex, we can color that t -vertex F, and the other t -vertices T and C. For the remainingcases, we show that we can change the color of one of the b -vertices, which again allows us to colorthe triangle. If e connects a literal to an a -vertex, we can color the a -vertex F and the connecting b -vertex C. If e connects an a -vertex to a b -vertex, we can color the b -vertex C. If e connects v s to an a -vertex, we can color the a -vertex T and the connecting b -vertex C. If e connects v s to a b -vertex, we can color the b -vertex T.This completes the proof of Lemma 5. We now explain why g also fulfills the requirements ofLemma 3. Let v be a vertex in g (Φ), and let e be an edge incident with v (such an edge always existssince g (Φ) does not contain isolated vertices). Then opt( g (Φ) − e ) is a 3-coloring. This gives us a3-coloring of g (Φ) − v , which is optimal since g (Φ) − v does not have a 2-coloring. E Additional Explanations for the Proof of Theorem 10 This appendix provides supplementing material for the proof of Theorem 10. On the one hand, we giveFigure 2, which illustrates how Algorithm 1 can optimally color graphs that are not universal-edgedusing only two queries. On the other hand, we prove the correctness and polynomial-time complexityof Subcol , which is used to optimally color universal-edged graphs without any queries, in Lemma 18and exemplify the used construction in Figure 3. Lemma 18. The subroutine Subcol (Algorithm 2) is correct and runs in polynomial time. For convenience in referencing the lines of the algorithm Subcol , we reprint it here.20 ℓ, r } dℓ rL RM Figure 3. An example of the construction that we use in Subcol (Algorithm 2) for a k -colorable graph G , exploitingthe fact that G is known to be universal-edged. In the example, we have k = 4. In general, the graph G is k -colorableif and only if the induced subgraph G [ M ] is ( k − G [ L ] and G [ R ] are independent sets. Thefollowing relations hold in general as well: L = N ( ℓ ) − N [ r ] = V − N [ r ] , M = N ( ℓ ) ∩ N ( r ) ,R = N ( r ) − N [ ℓ ] = V − N [ ℓ ] , and V = N [ ℓ ] ∪ N [ r ] = L ∪ M ∪ R ∪ { ℓ, r } .. In the example, only edge d prevents G [ M ] from being 1-colorable and thus G from being 3-colorable. Algorithm 2 Subcol Input: An undirected, universal-edged graph G = ( V, E ) and a positive integer k . Output: A k -coloring f for G if there is one; NO if there is none. Description: Works by recursion over k , with k = 1 and k = 2 serving as the base cases. if G has no edge then return the constant 1-coloring with f ( x ) = 1 for all x ∈ V . else if k = 1 then return NO. if G has bipartition { A, B } then return the 2-coloring f ( x ) = ( x ∈ A, and2 for x ∈ B. else if k=2 then return NO. Choose an arbitrary edge { ℓ, r } ∈ E . L ← N ( ℓ ) − N [ r ]; R ← N ( r ) − N [ ℓ ]; M ← N ( ℓ ) ∩ N ( r ) g ← Subcol ( G [ M ] , k − if g = NO then return NO return the k -coloring f ( x ) = g ( x ) for x ∈ M , k − x ∈ L ∪ { r } , and k for x ∈ R ∪ { ℓ } . Proof of Lemma 18. Note first that the input for Subcol is a pair ( G, k ), where G is a universal-edgedgraph and k a positive integer. It is thus clear that Subcol runs in polynomial time: The recursiondepth is ⌊ ( k − / ⌋ and for the two base cases it is easy to check whether G has edges and whether G has a bipartition and, if there is any, find one in polynomial time.We now show that Subcol is correct by going through all six return statements. The first one21n line 2 is correct since the constant coloring is a k -coloring for any k ∈ N − { } . With the secondone in line 4, the case k = 1 is completely and correctly covered. Analogously, the third one in line 6is correct since a 2-coloring is a k -coloring for any k ∈ N − { , } , and the case k = 2 is correctlycovered together with the fourth return statement in line 8. If none of the first four return statementsof Subcol are executed, the graph G has an edge and the choice of an edge { ℓ, r } ∈ E is possible.For the last two return statements in lines 13 and 14, we will prove the correctness by inductionover k , with the above two cases k = 1 and k = 2 serving as the induction basis. We will rely onthe properties of a partition of G that we describe in what follows; see Figure 3 for an illustratingexample with a graph that is k -colorable for k = 4 but not for k = 3. Let { ℓ, r } be the edge of G as chosen by the algorithm. The remaining vertices V − { ℓ, r } are partitioned, depending on the waythey are connected to { ℓ, r } , into the three sets L , R , and M : L contains the vertices adjacent to l but not to r , R contains the vertices connected to r but not to l , and M contains the vertices thatare adjacent to both l and r . Note that the sets L , R , and M are disjoint. They cover V − { ℓ, r } since every vertex is adjacent to l or r because G is universal-edged and { ℓ, r } thus universal. Wenow consider the case that NO is returned with the fifth return statement. This happens only if g = Subcol ( G [ M ] , k − 2) = NO. Thus, G [ M ] is not ( k − k -coloring of G yields a ( k − G [ M ], thus proving by contradiction that G is not k -colorable. Assume that there is a k -coloring of G . Due to the edge { ℓ, r } , the two vertices ℓ and r have two different colors out of the k available ones. Since all vertices of M are adjacent toboth ℓ and r , the subgraph G [ M ] is indeed colored by f with the k − f ( x ) is a k -coloring of G . By the induction hypothesis, we know that g is a( k − G [ M ] using the colors 1 , . . . , k − 2. The remaining vertices L ∪ { r } and R ∪ { ℓ } arecolored with k − k , respectively. Thus, it suffices to show that G [ L ∪ { r } ] and G [ R ∪ { ℓ } ] areindependent sets. Consider first G [ L ∪ { r } ]. On the one hand, none of the vertices in L are adjacentto r by the definition of L . On the other hand, if there were x, y ∈ L with { x, y } ∈ E , this wouldcontradict the universality of { x, y } for r . Analogously, we see that G [ R ∪ { ℓ } ] is an independent set,concluding the proof. F Analogue of Theorem 12 We prove the analogue of Theorem 12 for adding edges instead of deleting vertices. Theorem 19. There is a polynomial-time algorithm that computes an optimal vertex cover for a graphfrom two optimal vertex covers for some one-edge-added supergraphs.Proof. Observe first what can happen when an edge e is added to a graph G with an optimal vertexcover of size k . If one of its endpoints v is part of any optimal vertex cover of G , then the optimalvertex cover of G containing v is also an optimal vertex cover for G ∪ e . Given any graph G , thealgorithm picks any two non-universal vertices v and v that are adjacent. Since there is an edgebetween them, any given optimal vertex cover contains at least one of them. If edges are added tothis vertex, the vertex cover of size k thus remains optimal. Because v and v are non-universal, thealgorithm can add to G an edge e that is incident to v and an edge e incident to v . Now thealgorithm queries the oracle for two optimal vertex covers, one for G ∪ e and one for G ∪ e . At leastone of them has size k (as opposed to k + 1) and is thus optimal for G as well. If G has the propertythat, for every pair of adjacent vertices v and v , one of them is universal, then the set of all universalvertices constitutes an optimal vertex cover. G Full Proof of Theorem 13 In this appendix, we provide the formal proof of Theorem 13, after restating it for convenience.22 heorem 13.